Quaternion type
From Rosetta Code
You are encouraged to solve this task according to the task description, using any language you may know.
Quaternions are an extension of the idea of complex numbers.
A complex number has a real and complex part written sometimes as a + bi, where a and b stand for real numbers and i stands for the square root of minus 1. An example of a complex number might be -3 + 2i, where the real part, a is -3.0 and the complex part, b is +2.0.
A quaternion has one real part and three imaginary parts, i, j, and k. A quaternion might be written as a + bi + cj + dk. In this numbering system, ii = jj = kk = ijk = -1. The order of multiplication is important, as, in general, for two quaternions q1 and q2; q1q2 != q2q1. An example of a quaternion might be 1 +2i +3j +4k
There is a list form of notation where just the numbers are shown and the imaginary multipliers i, j, and k are assumed by position. So the example above would be written as (1, 2, 3, 4)
Task Description
Given the three quaternions and their components:
q = (1, 2, 3, 4) = (a, b, c, d ) q1 = (2, 3, 4, 5) = (a1, b1, c1, d1) q2 = (3, 4, 5, 6) = (a2, b2, c2, d2)
And a wholly real number r = 7.
Your task is to create functions or classes to perform simple maths with quaternions including computing:
- The norm of a quaternion:
- The negative of a quaternion:
=(-a, -b, -c, -d) - The conjugate of a quaternion:
=( a, -b, -c, -d) - Addition of a real number r and a quaternion q:
r + q = q + r = (a+r, b, c, d) - Addition of two quaternions:
q1 + q2 = (a1+a2, b1+b2, c1+c2, d1+d2) - Multiplication of a real number and a quaternion:
qr = rq = (ar, br, cr, dr) - Multiplication of two quaternions q1 and q2 is given by:
( a1a2 − b1b2 − c1c2 − d1d2,a1b2 + b1a2 + c1d2 − d1c2,a1c2 − b1d2 + c1a2 + d1b2,a1d2 + b1c2 − c1b2 + d1a2 ) - Show that, for the two quaternions q1 and q2:
q1q2 != q2q1
If your language has built-in support for quaternions then use it.
C.f.
- Vector products
- On Quaternions; or on a new System of Imaginaries in Algebra. By Sir William Rowan Hamilton LL.D, P.R.I.A., F.R.A.S., Hon. M. R. Soc. Ed. and Dub., Hon. or Corr. M. of the Royal or Imperial Academies of St. Petersburgh, Berlin, Turin and Paris, Member of the American Academy of Arts and Sciences, and of other Scientific Societies at Home and Abroad, Andrews' Prof. of Astronomy in the University of Dublin, and Royal Astronomer of Ireland.
[edit] Ada
The package specification (works with any floating-point type):
generic
type Real is digits <>;
package Quaternions is
type Quaternion is record
A, B, C, D : Real;
end record;
function "abs" (Left : Quaternion) return Real;
function Conj (Left : Quaternion) return Quaternion;
function "-" (Left : Quaternion) return Quaternion;
function "+" (Left, Right : Quaternion) return Quaternion;
function "-" (Left, Right : Quaternion) return Quaternion;
function "*" (Left : Quaternion; Right : Real) return Quaternion;
function "*" (Left : Real; Right : Quaternion) return Quaternion;
function "*" (Left, Right : Quaternion) return Quaternion;
function Image (Left : Quaternion) return String;
end Quaternions;
The package implementation:
with Ada.Numerics.Generic_Elementary_Functions;
package body Quaternions is
package Elementary_Functions is
new Ada.Numerics.Generic_Elementary_Functions (Real);
use Elementary_Functions;
function "abs" (Left : Quaternion) return Real is
begin
return Sqrt (Left.A**2 + Left.B**2 + Left.C**2 + Left.D**2);
end "abs";
function Conj (Left : Quaternion) return Quaternion is
begin
return (A => Left.A, B => -Left.B, C => -Left.C, D => -Left.D);
end Conj;
function "-" (Left : Quaternion) return Quaternion is
begin
return (A => -Left.A, B => -Left.B, C => -Left.C, D => -Left.D);
end "-";
function "+" (Left, Right : Quaternion) return Quaternion is
begin
return
( A => Left.A + Right.A, B => Left.B + Right.B,
C => Left.C + Right.C, D => Left.D + Right.D
);
end "+";
function "-" (Left, Right : Quaternion) return Quaternion is
begin
return
( A => Left.A - Right.A, B => Left.B - Right.B,
C => Left.C - Right.C, D => Left.D - Right.D
);
end "-";
function "*" (Left : Quaternion; Right : Real) return Quaternion is
begin
return
( A => Left.A * Right, B => Left.B * Right,
C => Left.C * Right, D => Left.D * Right
);
end "*";
function "*" (Left : Real; Right : Quaternion) return Quaternion is
begin
return Right * Left;
end "*";
function "*" (Left, Right : Quaternion) return Quaternion is
begin
return
( A => Left.A * Right.A - Left.B * Right.B - Left.C * Right.C - Left.D * Right.D,
B => Left.A * Right.B + Left.B * Right.A + Left.C * Right.D - Left.D * Right.C,
C => Left.A * Right.C - Left.B * Right.D + Left.C * Right.A + Left.D * Right.B,
D => Left.A * Right.D + Left.B * Right.C - Left.C * Right.B + Left.D * Right.A
);
end "*";
function Image (Left : Quaternion) return String is
begin
return Real'Image (Left.A) & " +" &
Real'Image (Left.B) & "i +" &
Real'Image (Left.C) & "j +" &
Real'Image (Left.D) & "k";
end Image;
end Quaternions;
Test program:
with Ada.Text_IO; use Ada.Text_IO;
with Quaternions;
procedure Test_Quaternion is
package Float_Quaternion is new Quaternions (Float);
use Float_Quaternion;
q : Quaternion := (1.0, 2.0, 3.0, 4.0);
q1 : Quaternion := (2.0, 3.0, 4.0, 5.0);
q2 : Quaternion := (3.0, 4.0, 5.0, 6.0);
r : Float := 7.0;
begin
Put_Line ("q = " & Image (q));
Put_Line ("q1 = " & Image (q1));
Put_Line ("q2 = " & Image (q2));
Put_Line ("r =" & Float'Image (r));
Put_Line ("abs q =" & Float'Image (abs q));
Put_Line ("abs q1 =" & Float' Image (abs q1));
Put_Line ("abs q2 =" & Float' Image (abs q2));
Put_Line ("-q = " & Image (-q));
Put_Line ("conj q = " & Image (Conj (q)));
Put_Line ("q1 + q2 = " & Image (q1 + q2));
Put_Line ("q2 + q1 = " & Image (q2 + q1));
Put_Line ("q * r = " & Image (q * r));
Put_Line ("r * q = " & Image (r * q));
Put_Line ("q1 * q2 = " & Image (q1 * q2));
Put_Line ("q2 * q1 = " & Image (q2 * q1));
end Test_Quaternion;
Sample output:
q = 1.00000E+00 + 2.00000E+00i + 3.00000E+00j + 4.00000E+00k q1 = 2.00000E+00 + 3.00000E+00i + 4.00000E+00j + 5.00000E+00k q2 = 3.00000E+00 + 4.00000E+00i + 5.00000E+00j + 6.00000E+00k r = 7.00000E+00 abs q = 5.47723E+00 abs q1 = 7.34847E+00 abs q2 = 9.27362E+00 -q = -1.00000E+00 +-2.00000E+00i +-3.00000E+00j +-4.00000E+00k conj q = 1.00000E+00 +-2.00000E+00i +-3.00000E+00j +-4.00000E+00k q1 + q2 = 5.00000E+00 + 7.00000E+00i + 9.00000E+00j + 1.10000E+01k q2 + q1 = 5.00000E+00 + 7.00000E+00i + 9.00000E+00j + 1.10000E+01k q * r = 7.00000E+00 + 1.40000E+01i + 2.10000E+01j + 2.80000E+01k r * q = 7.00000E+00 + 1.40000E+01i + 2.10000E+01j + 2.80000E+01k q1 * q2 = -5.60000E+01 + 1.60000E+01i + 2.40000E+01j + 2.60000E+01k q2 * q1 = -5.60000E+01 + 1.80000E+01i + 2.00000E+01j + 2.80000E+01k
[edit] ALGOL 68
- note: This specimen retains the original python coding style.MODE QUAT = STRUCT(REAL re, i, j, k);
MODE QUATERNION = QUAT;
MODE SUBQUAT = UNION(QUAT, #COMPL, # REAL#, INT, [4]REAL, [4]INT # );
MODE CLASSQUAT = STRUCT(
PROC (REF QUAT #new#, REAL #re#, REAL #i#, REAL #j#, REAL #k#)REF QUAT new,
PROC (REF QUAT #self#)QUAT conjugate,
PROC (REF QUAT #self#)REAL norm sq,
PROC (REF QUAT #self#)REAL norm,
PROC (REF QUAT #self#)QUAT reciprocal,
PROC (REF QUAT #self#)STRING repr,
PROC (REF QUAT #self#)QUAT neg,
PROC (REF QUAT #self#, SUBQUAT #other#)QUAT add,
PROC (REF QUAT #self#, SUBQUAT #other#)QUAT radd,
PROC (REF QUAT #self#, SUBQUAT #other#)QUAT sub,
PROC (REF QUAT #self#, SUBQUAT #other#)QUAT mul,
PROC (REF QUAT #self#, SUBQUAT #other#)QUAT rmul,
PROC (REF QUAT #self#, SUBQUAT #other#)QUAT div,
PROC (REF QUAT #self#, SUBQUAT #other#)QUAT rdiv,
PROC (REF QUAT #self#)QUAT exp
);
CLASSQUAT class quat = (
# PROC new =#(REF QUAT new, REAL re, i, j, k)REF QUAT: (
# 'Defaults all parts of quaternion to zero' #
IF new ISNT REF QUAT(NIL) THEN new ELSE HEAP QUAT FI := (re, i, j, k)
),
# PROC conjugate =#(REF QUAT self)QUAT:
(re OF self, -i OF self, -j OF self, -k OF self),
# PROC norm sq =#(REF QUAT self)REAL:
re OF self**2 + i OF self**2 + j OF self**2 + k OF self**2,
# PROC norm =#(REF QUAT self)REAL:
sqrt((norm sq OF class quat)(self)),
# PROC reciprocal =#(REF QUAT self)QUAT:(
REAL n2 = (norm sq OF class quat)(self);
QUAT conj = (conjugate OF class quat)(self);
(re OF conj/n2, i OF conj/n2, j OF conj/n2, k OF conj/n2)
),
# PROC repr =#(REF QUAT self)STRING: (
# 'Shorter form of Quaternion as string' #
FILE f; STRING s; associate(f, s);
putf(f, (squat fmt, re OF self>=0, re OF self,
i OF self>=0, i OF self, j OF self>=0, j OF self, k OF self>=0, k OF self));
close(f);
s
),
# PROC neg =#(REF QUAT self)QUAT:
(-re OF self, -i OF self, -j OF self, -k OF self),
# PROC add =#(REF QUAT self, SUBQUAT other)QUAT:
CASE other IN
(QUAT other): (re OF self + re OF other, i OF self + i OF other, j OF self + j OF other, k OF self + k OF other),
(REAL other): (re OF self + other, i OF self, j OF self, k OF self)
ESAC,
# PROC radd =#(REF QUAT self, SUBQUAT other)QUAT:
(add OF class quat)(self, other),
# PROC sub =#(REF QUAT self, SUBQUAT other)QUAT:
CASE other IN
(QUAT other): (re OF self - re OF other, i OF self - i OF other, j OF self - j OF other, k OF self - k OF other),
(REAL other): (re OF self - other, i OF self, j OF self, k OF self)
ESAC,
# PROC mul =#(REF QUAT self, SUBQUAT other)QUAT:
CASE other IN
(QUAT other):(
re OF self*re OF other - i OF self*i OF other - j OF self*j OF other - k OF self*k OF other,
re OF self*i OF other + i OF self*re OF other + j OF self*k OF other - k OF self*j OF other,
re OF self*j OF other - i OF self*k OF other + j OF self*re OF other + k OF self*i OF other,
re OF self*k OF other + i OF self*j OF other - j OF self*i OF other + k OF self*re OF other
),
(REAL other): ( re OF self * other, i OF self * other, j OF self * other, k OF self * other)
ESAC,
# PROC rmul =#(REF QUAT self, SUBQUAT other)QUAT:
CASE other IN
(QUAT other): (mul OF class quat)(LOC QUAT := other, self),
(REAL other): (mul OF class quat)(self, other)
ESAC,
# PROC div =#(REF QUAT self, SUBQUAT other)QUAT:
CASE other IN
(QUAT other): (mul OF class quat)(self, (reciprocal OF class quat)(LOC QUAT := other)),
(REAL other): (mul OF class quat)(self, 1/other)
ESAC,
# PROC rdiv =#(REF QUAT self, SUBQUAT other)QUAT:
CASE other IN
(QUAT other): (div OF class quat)(LOC QUAT := other, self),
(REAL other): (div OF class quat)(LOC QUAT := (other, 0, 0, 0), self)
ESAC,
# PROC exp =#(REF QUAT self)QUAT: (
QUAT fac := self;
QUAT sum := 1.0 + fac;
FOR i FROM 2 WHILE ABS(fac + small real) /= small real DO
VOID(sum +:= (fac *:= self / REAL(i)))
OD;
sum
)
);
FORMAT real fmt = $g(-0, 4)$;
FORMAT signed fmt = $b("+", "")f(real fmt)$;
FORMAT quat fmt = $f(real fmt)"+"f(real fmt)"i+"f(real fmt)"j+"f(real fmt)"k"$;
FORMAT squat fmt = $f(signed fmt)f(signed fmt)"i"f(signed fmt)"j"f(signed fmt)"k"$;
PRIO INIT = 1;
OP INIT = (REF QUAT new)REF QUAT: new := (0, 0, 0, 0);
OP INIT = (REF QUAT new, []REAL rijk)REF QUAT:
(new OF class quat)(LOC QUAT := new, rijk[1], rijk[2], rijk[3], rijk[4]);
OP + = (QUAT q)QUAT: q,
- = (QUAT q)QUAT: (neg OF class quat)(LOC QUAT := q),
CONJ = (QUAT q)QUAT: (conjugate OF class quat)(LOC QUAT := q),
ABS = (QUAT q)REAL: (norm OF class quat)(LOC QUAT := q),
REPR = (QUAT q)STRING: (repr OF class quat)(LOC QUAT := q);
# missing: Diadic: I, J, K END #
OP +:= = (REF QUAT a, QUAT b)QUAT: a:=( add OF class quat)(a, b),
+:= = (REF QUAT a, REAL b)QUAT: a:=( add OF class quat)(a, b),
+=: = (QUAT a, REF QUAT b)QUAT: b:=(radd OF class quat)(b, a),
+=: = (REAL a, REF QUAT b)QUAT: b:=(radd OF class quat)(b, a);
# missing: Worthy PLUSAB, PLUSTO for SHORT/LONG INT REAL & COMPL #
OP -:= = (REF QUAT a, QUAT b)QUAT: a:=( sub OF class quat)(a, b),
-:= = (REF QUAT a, REAL b)QUAT: a:=( sub OF class quat)(a, b);
# missing: Worthy MINUSAB for SHORT/LONG INT REAL & COMPL #
PRIO *=: = 1, /=: = 1;
OP *:= = (REF QUAT a, QUAT b)QUAT: a:=( mul OF class quat)(a, b),
*:= = (REF QUAT a, REAL b)QUAT: a:=( mul OF class quat)(a, b),
*=: = (QUAT a, REF QUAT b)QUAT: b:=(rmul OF class quat)(b, a),
*=: = (REAL a, REF QUAT b)QUAT: b:=(rmul OF class quat)(b, a);
# missing: Worthy TIMESAB, TIMESTO for SHORT/LONG INT REAL & COMPL #
OP /:= = (REF QUAT a, QUAT b)QUAT: a:=( div OF class quat)(a, b),
/:= = (REF QUAT a, REAL b)QUAT: a:=( div OF class quat)(a, b),
/=: = (QUAT a, REF QUAT b)QUAT: b:=(rdiv OF class quat)(b, a),
/=: = (REAL a, REF QUAT b)QUAT: b:=(rdiv OF class quat)(b, a);
# missing: Worthy OVERAB, OVERTO for SHORT/LONG INT REAL & COMPL #
OP + = (QUAT a, b)QUAT: ( add OF class quat)(LOC QUAT := a, b),
+ = (QUAT a, REAL b)QUAT: ( add OF class quat)(LOC QUAT := a, b),
+ = (REAL a, QUAT b)QUAT: (radd OF class quat)(LOC QUAT := b, a);
OP - = (QUAT a, b)QUAT: ( sub OF class quat)(LOC QUAT := a, b),
- = (QUAT a, REAL b)QUAT: ( sub OF class quat)(LOC QUAT := a, b),
- = (REAL a, QUAT b)QUAT:-( sub OF class quat)(LOC QUAT := b, a);
OP * = (QUAT a, b)QUAT: ( mul OF class quat)(LOC QUAT := a, b),
* = (QUAT a, REAL b)QUAT: ( mul OF class quat)(LOC QUAT := a, b),
* = (REAL a, QUAT b)QUAT: (rmul OF class quat)(LOC QUAT := b, a);
OP / = (QUAT a, b)QUAT: ( div OF class quat)(LOC QUAT := a, b),
/ = (QUAT a, REAL b)QUAT: ( div OF class quat)(LOC QUAT := a, b),
/ = (REAL a, QUAT b)QUAT: ( div OF class quat)(LOC QUAT := b, 1/a);
PROC quat exp = (QUAT q)QUAT: (exp OF class quat)(LOC QUAT := q);
# missing: quat arc{sin, cos, tan}h, log, exp, ln etc END #
test:(
REAL r = 7;
QUAT q = (1, 2, 3, 4),
q1 = (2, 3, 4, 5),
q2 = (3, 4, 5, 6);
printf((
$"r = " f(real fmt)l$, r,
$"q = " f(quat fmt)l$, q,
$"q1 = " f(quat fmt)l$, q1,
$"q2 = " f(quat fmt)l$, q2,
$"ABS q = " f(real fmt)", "$, ABS q,
$"ABS q1 = " f(real fmt)", "$, ABS q1,
$"ABS q2 = " f(real fmt)l$, ABS q2,
$"-q = " f(quat fmt)l$, -q,
$"CONJ q = " f(quat fmt)l$, CONJ q,
$"r + q = " f(quat fmt)l$, r + q,
$"q + r = " f(quat fmt)l$, q + r,
$"q1 + q2 = "f(quat fmt)l$, q1 + q2,
$"q2 + q1 = "f(quat fmt)l$, q2 + q1,
$"q * r = " f(quat fmt)l$, q * r,
$"r * q = " f(quat fmt)l$, r * q,
$"q1 * q2 = "f(quat fmt)l$, q1 * q2,
$"q2 * q1 = "f(quat fmt)l$, q2 * q1
));
CO
$"ASSERT q1 * q2 != q2 * q1 = "f(quat fmt)l$, ASSERT q1 * q2 != q2 * q1, $l$);
END CO
QUAT i=(0, 1, 0, 0),
j=(0, 0, 1, 0),
k=(0, 0, 0, 1);
printf((
$"i*i = " f(quat fmt)l$, i*i,
$"j*j = " f(quat fmt)l$, j*j,
$"k*k = " f(quat fmt)l$, k*k,
$"i*j*k = " f(quat fmt)l$, i*j*k,
$"q1 / q2 = " f(quat fmt)l$, q1 / q2,
$"q1 / q2 * q2 = "f(quat fmt)l$, q1 / q2 * q2,
$"q2 * q1 / q2 = "f(quat fmt)l$, q2 * q1 / q2,
$"1/q1 * q1 = " f(quat fmt)l$, 1.0/q1 * q1,
$"q1 / q1 = " f(quat fmt)l$, q1 / q1,
$"quat exp(pi * i) = " f(quat fmt)l$, quat exp(pi * i),
$"quat exp(pi * j) = " f(quat fmt)l$, quat exp(pi * j),
$"quat exp(pi * i) = " f(quat fmt)l$, quat exp(pi * k)
));
print((REPR(-q1*q2), ", ", REPR(-q2*q1), new line))
)
Output:
r = 7.0000 q = 1.0000+2.0000i+3.0000j+4.0000k q1 = 2.0000+3.0000i+4.0000j+5.0000k q2 = 3.0000+4.0000i+5.0000j+6.0000k ABS q = 5.4772, ABS q1 = 7.3485, ABS q2 = 9.2736 -q = -1.0000+-2.0000i+-3.0000j+-4.0000k CONJ q = 1.0000+-2.0000i+-3.0000j+-4.0000k r + q = 8.0000+2.0000i+3.0000j+4.0000k q + r = 8.0000+2.0000i+3.0000j+4.0000k q1 + q2 = 5.0000+7.0000i+9.0000j+11.0000k q2 + q1 = 5.0000+7.0000i+9.0000j+11.0000k q * r = 7.0000+14.0000i+21.0000j+28.0000k r * q = 7.0000+14.0000i+21.0000j+28.0000k q1 * q2 = -56.0000+16.0000i+24.0000j+26.0000k q2 * q1 = -56.0000+18.0000i+20.0000j+28.0000k i*i = -1.0000+.0000i+.0000j+.0000k j*j = -1.0000+.0000i+.0000j+.0000k k*k = -1.0000+.0000i+.0000j+.0000k i*j*k = -1.0000+.0000i+.0000j+.0000k q1 / q2 = .7907+.0233i+-.0000j+.0465k q1 / q2 * q2 = 2.0000+3.0000i+4.0000j+5.0000k q2 * q1 / q2 = 2.0000+3.4651i+3.9070j+4.7674k 1/q1 * q1 = -46.0000+12.0000i+16.0000j+20.0000k q1 / q1 = 1.0000+.0000i+.0000j+.0000k quat exp(pi * i) = -1.0000+.0000i+.0000j+.0000k quat exp(pi * j) = -1.0000+.0000i+.0000j+.0000k quat exp(pi * i) = -1.0000+.0000i+.0000j+.0000k +56.0000-16.0000i-24.0000j-26.0000k, +56.0000-18.0000i-20.0000j-28.0000k
[edit] BBC BASIC
Although BBC BASIC doesn't have native support for quaternions its array arithmetic provides all of the required operations either directly or very straightforwardly.
DIM q(3), q1(3), q2(3), t(3)
q() = 1, 2, 3, 4
q1() = 2, 3, 4, 5
q2() = 3, 4, 5, 6
r = 7
PRINT "q = " FNq_show(q())
PRINT "q1 = " FNq_show(q1())
PRINT "q2 = " FNq_show(q2())
PRINT "r = "; r
PRINT "norm(q) = "; FNq_norm(q())
t() = q() : PROCq_neg(t()) : PRINT "neg(q) = " FNq_show(t())
t() = q() : PROCq_conj(t()) : PRINT "conjugate(q) = " FNq_show(t())
t() = q() : PROCq_addreal(t(),r) : PRINT "q + r = " FNq_show(t())
t() = q1() : PROCq_add(t(),q2()) : PRINT "q1 + q2 = " FNq_show(t())
t() = q2() : PROCq_add(t(),q1()) : PRINT "q2 + q1 = " FNq_show(t())
t() = q() : PROCq_mulreal(t(),r) : PRINT "qr = " FNq_show(t())
t() = q1() : PROCq_mul(t(),q2()) : PRINT "q1q2 = " FNq_show(t())
t() = q2() : PROCq_mul(t(),q1()) : PRINT "q2q1 = " FNq_show(t())
END
DEF FNq_norm(q()) = MOD(q())
DEF PROCq_neg(q()) : q() *= -1 : ENDPROC
DEF PROCq_conj(q()) : q() *= -1 : q(0) *= -1 : ENDPROC
DEF PROCq_addreal(q(), r) : q(0) += r : ENDPROC
DEF PROCq_add(q(), r()) : q() += r() : ENDPROC
DEF PROCq_mulreal(q(), r) : q() *= r : ENDPROC
DEF PROCq_mul(q(), r()) : LOCAL s() : DIM s(3,3)
s() = r(0), -r(1), -r(2), -r(3), r(1), r(0), r(3), -r(2), \
\ r(2), -r(3), r(0), r(1), r(3), r(2), -r(1), r(0)
q() = s() . q()
ENDPROC
DEF FNq_show(q()) : LOCAL i%, a$ : a$ = "("
FOR i% = 0 TO 3 : a$ += STR$(q(i%)) + ", " : NEXT
= LEFT$(LEFT$(a$)) + ")"
Output:
q = (1, 2, 3, 4) q1 = (2, 3, 4, 5) q2 = (3, 4, 5, 6) r = 7 norm(q) = 5.47722558 neg(q) = (-1, -2, -3, -4) conjugate(q) = (1, -2, -3, -4) q + r = (8, 2, 3, 4) q1 + q2 = (5, 7, 9, 11) q2 + q1 = (5, 7, 9, 11) qr = (7, 14, 21, 28) q1q2 = (-56, 16, 24, 26) q2q1 = (-56, 18, 20, 28)
[edit] C
#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <math.h>
typedef struct quaternion
{
double q[4];
} quaternion_t;
quaternion_t *quaternion_new(void)
{
return malloc(sizeof(quaternion_t));
}
quaternion_t *quaternion_new_set(double q1,
double q2,
double q3,
double q4)
{
quaternion_t *q = malloc(sizeof(quaternion_t));
if (q != NULL) {
q->q[0] = q1; q->q[1] = q2; q->q[2] = q3; q->q[3] = q4;
}
return q;
}
void quaternion_copy(quaternion_t *r, quaternion_t *q)
{
size_t i;
if (r == NULL || q == NULL) return;
for(i = 0; i < 4; i++) r->q[i] = q->q[i];
}
double quaternion_norm(quaternion_t *q)
{
size_t i;
double r = 0.0;
if (q == NULL) {
fprintf(stderr, "NULL quaternion in norm\n");
return 0.0;
}
for(i = 0; i < 4; i++) r += q->q[i] * q->q[i];
return sqrt(r);
}
void quaternion_neg(quaternion_t *r, quaternion_t *q)
{
size_t i;
if (q == NULL || r == NULL) return;
for(i = 0; i < 4; i++) r->q[i] = -q->q[i];
}
void quaternion_conj(quaternion_t *r, quaternion_t *q)
{
size_t i;
if (q == NULL || r == NULL) return;
r->q[0] = q->q[0];
for(i = 1; i < 4; i++) r->q[i] = -q->q[i];
}
void quaternion_add_d(quaternion_t *r, quaternion_t *q, double d)
{
if (q == NULL || r == NULL) return;
quaternion_copy(r, q);
r->q[0] += d;
}
void quaternion_add(quaternion_t *r, quaternion_t *a, quaternion_t *b)
{
size_t i;
if (r == NULL || a == NULL || b == NULL) return;
for(i = 0; i < 4; i++) r->q[i] = a->q[i] + b->q[i];
}
void quaternion_mul_d(quaternion_t *r, quaternion_t *q, double d)
{
size_t i;
if (r == NULL || q == NULL) return;
for(i = 0; i < 4; i++) r->q[i] = q->q[i] * d;
}
bool quaternion_equal(quaternion_t *a, quaternion_t *b)
{
size_t i;
for(i = 0; i < 4; i++) if (a->q[i] != b->q[i]) return false;
return true;
}
#define A(N) (a->q[(N)])
#define B(N) (b->q[(N)])
#define R(N) (r->q[(N)])
void quaternion_mul(quaternion_t *r, quaternion_t *a, quaternion_t *b)
{
size_t i;
double ri = 0.0;
if (r == NULL || a == NULL || b == NULL) return;
R(0) = A(0)*B(0) - A(1)*B(1) - A(2)*B(2) - A(3)*B(3);
R(1) = A(0)*B(1) + A(1)*B(0) + A(2)*B(3) - A(3)*B(2);
R(2) = A(0)*B(2) - A(1)*B(3) + A(2)*B(0) + A(3)*B(1);
R(3) = A(0)*B(3) + A(1)*B(2) - A(2)*B(1) + A(3)*B(0);
}
#undef A
#undef B
#undef R
void quaternion_print(quaternion_t *q)
{
if (q == NULL) return;
printf("(%lf, %lf, %lf, %lf)\n",
q->q[0], q->q[1], q->q[2], q->q[3]);
}
int main()
{
size_t i;
double d = 7.0;
quaternion_t *q[3];
quaternion_t *r = quaternion_new();
quaternion_t *qd = quaternion_new_set(7.0, 0.0, 0.0, 0.0);
q[0] = quaternion_new_set(1.0, 2.0, 3.0, 4.0);
q[1] = quaternion_new_set(2.0, 3.0, 4.0, 5.0);
q[2] = quaternion_new_set(3.0, 4.0, 5.0, 6.0);
printf("r = %lf\n", d);
for(i = 0; i < 3; i++) {
printf("q[%u] = ", i);
quaternion_print(q[i]);
printf("abs q[%u] = %lf\n", i, quaternion_norm(q[i]));
}
printf("-q[0] = ");
quaternion_neg(r, q[0]);
quaternion_print(r);
printf("conj q[0] = ");
quaternion_conj(r, q[0]);
quaternion_print(r);
printf("q[1] + q[2] = ");
quaternion_add(r, q[1], q[2]);
quaternion_print(r);
printf("q[2] + q[1] = ");
quaternion_add(r, q[2], q[1]);
quaternion_print(r);
printf("q[0] * r = ");
quaternion_mul_d(r, q[0], d);
quaternion_print(r);
printf("q[0] * (r, 0, 0, 0) = ");
quaternion_mul(r, q[0], qd);
quaternion_print(r);
printf("q[1] * q[2] = ");
quaternion_mul(r, q[1], q[2]);
quaternion_print(r);
printf("q[2] * q[1] = ");
quaternion_mul(r, q[2], q[1]);
quaternion_print(r);
free(q[0]); free(q[1]); free(q[2]); free(r);
return EXIT_SUCCESS;
}
[edit] C++
This example uses templates to provide the underlying data-type, and includes several extra functions and constructors that often come up when using quaternions.
#include <iostream>
using namespace std;
template<class T = double>
class Quaternion
{
public:
T w, x, y, z;
// Numerical constructor
Quaternion(const T &w, const T &x, const T &y, const T &z): w(w), x(x), y(y), z(z) {};
Quaternion(const T &x, const T &y, const T &z): w(T()), x(x), y(y), z(z) {}; // For 3-rotations
Quaternion(const T &r): w(r), x(T()), y(T()), z(T()) {};
Quaternion(): w(T()), x(T()), y(T()), z(T()) {};
// Copy constructor and assignment
Quaternion(const Quaternion &q): w(q.w), x(q.x), y(q.y), z(q.z) {};
Quaternion& operator=(const Quaternion &q) { w=q.w; x=q.x; y=q.y; z=q.z; return *this; }
// Unary operators
Quaternion operator-() const { return Quaternion(-w, -x, -y, -z); }
Quaternion operator~() const { return Quaternion(w, -x, -y, -z); } // Conjugate
// Norm-squared. SQRT would have to be made generic to be used here
T normSquared() const { return w*w + x*x + y*y + z*z; }
// In-place operators
Quaternion& operator+=(const T &r)
{ w += r; return *this; }
Quaternion& operator+=(const Quaternion &q)
{ w += q.w; x += q.x; y += q.y; z += q.z; return *this; }
Quaternion& operator-=(const T &r)
{ w -= r; return *this; }
Quaternion& operator-=(const Quaternion &q)
{ w -= q.w; x -= q.x; y -= q.y; z -= q.z; return *this; }
Quaternion& operator*=(const T &r)
{ w *= r; x *= r; y *= r; z *= r; return *this; }
Quaternion& operator*=(const Quaternion &q)
{
T oldW(w), oldX(x), oldY(y), oldZ(z);
w = oldW*q.w - oldX*q.x - oldY*q.y - oldZ*q.z;
x = oldW*q.x + oldX*q.w + oldY*q.z - oldZ*q.y;
y = oldW*q.y + oldY*q.w + oldZ*q.x - oldX*q.z;
z = oldW*q.z + oldZ*q.w + oldX*q.y - oldY*q.x;
return *this;
}
Quaternion& operator/=(const T &r)
{ w /= r; x /= r; y /= r; z /= r; return *this; }
Quaternion& operator/=(const Quaternion &q)
{
T oldW(w), oldX(x), oldY(y), oldZ(z), n(q.normSquared());
w = (oldW*q.w + oldX*q.x + oldY*q.y + oldZ*q.z) / n;
x = (oldX*q.w - oldW*q.x + oldY*q.z - oldZ*q.y) / n;
y = (oldY*q.w - oldW*q.y + oldZ*q.x - oldX*q.z) / n;
z = (oldZ*q.w - oldW*q.z + oldX*q.y - oldY*q.x) / n;
return *this;
}
// Binary operators based on in-place operators
Quaternion operator+(const T &r) const { return Quaternion(*this) += r; }
Quaternion operator+(const Quaternion &q) const { return Quaternion(*this) += q; }
Quaternion operator-(const T &r) const { return Quaternion(*this) -= r; }
Quaternion operator-(const Quaternion &q) const { return Quaternion(*this) -= q; }
Quaternion operator*(const T &r) const { return Quaternion(*this) *= r; }
Quaternion operator*(const Quaternion &q) const { return Quaternion(*this) *= q; }
Quaternion operator/(const T &r) const { return Quaternion(*this) /= r; }
Quaternion operator/(const Quaternion &q) const { return Quaternion(*this) /= q; }
// Comparison operators, as much as they make sense
bool operator==(const Quaternion &q) const
{ return (w == q.w) && (x == q.x) && (y == q.y) && (z == q.z); }
bool operator!=(const Quaternion &q) const { return !operator==(q); }
// The operators above allow quaternion op real. These allow real op quaternion.
// Uses the above where appropriate.
template<class T> friend Quaternion<T> operator+(const T &r, const Quaternion<T> &q);
template<class T> friend Quaternion<T> operator-(const T &r, const Quaternion<T> &q);
template<class T> friend Quaternion<T> operator*(const T &r, const Quaternion<T> &q);
template<class T> friend Quaternion<T> operator/(const T &r, const Quaternion<T> &q);
// Allows cout << q
template<class T> friend ostream& operator<<(ostream &io, const Quaternion<T> &q);
};
// Friend functions need to be outside the actual class definition
template<class T>
Quaternion<T> operator+(const T &r, const Quaternion<T> &q)
{ return q+r; }
template<class T>
Quaternion<T> operator-(const T &r, const Quaternion<T> &q)
{ return Quaternion<T>(r-q.w, q.x, q.y, q.z); }
template<class T>
Quaternion<T> operator*(const T &r, const Quaternion<T> &q)
{ return q*r; }
template<class T>
Quaternion<T> operator/(const T &r, const Quaternion<T> &q)
{
T n(q.normSquared());
return Quaternion(r*q.w/n, -r*q.x/n, -r*q.y/n, -r*q.z/n);
}
template<class T>
ostream& operator<<(ostream &io, const Quaternion<T> &q)
{
io << q.w;
(q.x < T()) ? (io << " - " << (-q.x) << "i") : (io << " + " << q.x << "i");
(q.y < T()) ? (io << " - " << (-q.y) << "j") : (io << " + " << q.y << "j");
(q.z < T()) ? (io << " - " << (-q.z) << "k") : (io << " + " << q.z << "k");
return io;
}
Test program:
int main()
{
Quaternion<> q0(1, 2, 3, 4);
Quaternion<> q1(2, 3, 4, 5);
Quaternion<> q2(3, 4, 5, 6);
double r = 7;
cout << "q0: " << q0 << endl;
cout << "q1: " << q1 << endl;
cout << "q2: " << q2 << endl;
cout << "r: " << r << endl;
cout << endl;
cout << "-q0: " << -q0 << endl;
cout << "~q0: " << ~q0 << endl;
cout << endl;
cout << "r * q0: " << r*q0 << endl;
cout << "r + q0: " << r+q0 << endl;
cout << "q0 / r: " << q0/r << endl;
cout << "q0 - r: " << q0-r << endl;
cout << endl;
cout << "q0 + q1: " << q0+q1 << endl;
cout << "q0 - q1: " << q0-q1 << endl;
cout << "q0 * q1: " << q0*q1 << endl;
cout << "q0 / q1: " << q0/q1 << endl;
cout << endl;
cout << "q0 * ~q0: " << q0*~q0 << endl;
cout << "q0 + q1*q2: " << q0+q1*q2 << endl;
cout << "(q0 + q1)*q2: " << (q0+q1)*q2 << endl;
cout << "q0*q1*q2: " << q0*q1*q2 << endl;
cout << "(q0*q1)*q2: " << (q0*q1)*q2 << endl;
cout << "q0*(q1*q2): " << q0*(q1*q2) << endl;
cout << endl;
cout << "||q0||: " << sqrt(q0.normSquared()) << endl;
cout << endl;
cout << "q0*q1 - q1*q0: " << (q0*q1 - q1*q0) << endl;
// Other base types
Quaternion<int> q5(2), q6(3);
cout << endl << q5*q6 << endl;
}
Output:
q0: 1 + 2i + 3j + 4k q1: 2 + 3i + 4j + 5k q2: 3 + 4i + 5j + 6k r: 7 -q0: -1 - 2i - 3j - 4k ~q0: 1 - 2i - 3j - 4k r * q0: 7 + 14i + 21j + 28k r + q0: 8 + 2i + 3j + 4k q0 / r: 0.142857 + 0.285714i + 0.428571j + 0.571429k q0 - r: -6 + 2i + 3j + 4k q0 + q1: 3 + 5i + 7j + 9k q0 - q1: -1 - 1i - 1j - 1k q0 * q1: -36 + 6i + 12j + 12k q0 / q1: 0.740741 + 0i + 0.0740741j + 0.037037k q0 * ~q0: 30 + 0i + 0j + 0k q0 + q1*q2: -55 + 18i + 27j + 30k (q0 + q1)*q2: -100 + 24i + 42j + 42k q0*q1*q2: -264 - 114i - 132j - 198k (q0*q1)*q2: -264 - 114i - 132j - 198k q0*(q1*q2): -264 - 114i - 132j - 198k ||q0||: 5.47723 q0*q1 - q1*q0: 0 - 2i + 4j - 2k 6 + 0i + 0j + 0k
[edit] C#
using System;
struct Quaternion : IEquatable<Quaternion>
{
public readonly double A, B, C, D;
public Quaternion(double a, double b, double c, double d)
{
this.A = a;
this.B = b;
this.C = c;
this.D = d;
}
public double Norm()
{
return Math.Sqrt(A * A + B * B + C * C + D * D);
}
public static Quaternion operator -(Quaternion q)
{
return new Quaternion(-q.A, -q.B, -q.C, -q.D);
}
public Quaternion Conjugate()
{
return new Quaternion(A, -B, -C, -D);
}
// implicit conversion takes care of real*quaternion and real+quaternion
public static implicit operator Quaternion(double d)
{
return new Quaternion(d, 0, 0, 0);
}
public static Quaternion operator +(Quaternion q1, Quaternion q2)
{
return new Quaternion(q1.A + q2.A, q1.B + q2.B, q1.C + q2.C, q1.D + q2.D);
}
public static Quaternion operator *(Quaternion q1, Quaternion q2)
{
return new Quaternion(
q1.A * q2.A - q1.B * q2.B - q1.C * q2.C - q1.D * q2.D,
q1.A * q2.B + q1.B * q2.A + q1.C * q2.D - q1.D * q2.C,
q1.A * q2.C - q1.B * q2.D + q1.C * q2.A + q1.D * q2.B,
q1.A * q2.D + q1.B * q2.C - q1.C * q2.B + q1.D * q2.A);
}
public static bool operator ==(Quaternion q1, Quaternion q2)
{
return q1.A == q2.A && q1.B == q2.B && q1.C == q2.C && q1.D == q2.D;
}
public static bool operator !=(Quaternion q1, Quaternion q2)
{
return !(q1 == q2);
}
#region Object Members
public override bool Equals(object obj)
{
if (obj is Quaternion)
return Equals((Quaternion)obj);
return false;
}
public override int GetHashCode()
{
return A.GetHashCode() ^ B.GetHashCode() ^ C.GetHashCode() ^ D.GetHashCode();
}
public override string ToString()
{
return string.Format("Q({0}, {1}, {2}, {3})", A, B, C, D);
}
#endregion
#region IEquatable<Quaternion> Members
public bool Equals(Quaternion other)
{
return other == this;
}
#endregion
}
Demonstration:
using System;
static class Program
{
static void Main(string[] args)
{
Quaternion q = new Quaternion(1, 2, 3, 4);
Quaternion q1 = new Quaternion(2, 3, 4, 5);
Quaternion q2 = new Quaternion(3, 4, 5, 6);
double r = 7;
Console.WriteLine("q = {0}", q);
Console.WriteLine("q1 = {0}", q1);
Console.WriteLine("q2 = {0}", q2);
Console.WriteLine("r = {0}", r);
Console.WriteLine("q.Norm() = {0}", q.Norm());
Console.WriteLine("q1.Norm() = {0}", q1.Norm());
Console.WriteLine("q2.Norm() = {0}", q2.Norm());
Console.WriteLine("-q = {0}", -q);
Console.WriteLine("q.Conjugate() = {0}", q.Conjugate());
Console.WriteLine("q + r = {0}", q + r);
Console.WriteLine("q1 + q2 = {0}", q1 + q2);
Console.WriteLine("q2 + q1 = {0}", q2 + q1);
Console.WriteLine("q * r = {0}", q * r);
Console.WriteLine("q1 * q2 = {0}", q1 * q2);
Console.WriteLine("q2 * q1 = {0}", q2 * q1);
Console.WriteLine("q1*q2 {0} q2*q1", (q1 * q2) == (q2 * q1) ? "==" : "!=");
}
}
Output:
q = Q(1, 2, 3, 4) q1 = Q(2, 3, 4, 5) q2 = Q(3, 4, 5, 6) r = 7 q.Norm() = 5.47722557505166 q1.Norm() = 7.34846922834953 q2.Norm() = 9.2736184954957 -q = Q(-1, -2, -3, -4) q.Conjugate() = Q(1, -2, -3, -4) q + r = Q(8, 2, 3, 4) q1 + q2 = Q(5, 7, 9, 11) q2 + q1 = Q(5, 7, 9, 11) q * r = Q(7, 14, 21, 28) q1 * q2 = Q(-56, 16, 24, 26) q2 * q1 = Q(-56, 18, 20, 28) q1*q2 != q2*q1
[edit] D
import std.math, std.numeric, std.traits, std.conv, std.complex;
struct Quat(T) if (isFloatingPoint!T) {
alias Complex!T CT;
union {
struct { T re, i, j, k; } // Default init to NaN
struct { CT x, y; }
struct { T[4] vector; }
}
string toString() const /*pure nothrow*/ {
return text(vector);
}
@property T norm2() const pure nothrow { /// Norm squared
return re ^^ 2 + i ^^ 2 + j ^^ 2 + k ^^ 2;
}
@property T abs() const pure nothrow { /// Norm
return sqrt(norm2);
}
@property T arg() const pure nothrow { /// Theta
return acos(re / abs); // this may be incorrect...
}
@property Quat!T conj() const pure nothrow { /// Conjugate
return Quat!T(re, -i, -j, -k);
}
@property Quat!T recip() const pure nothrow { /// Reciprocal
return Quat!T(re / norm2, -i / norm2, -j / norm2, -k / norm2);
}
@property Quat!T pureim() const pure nothrow { /// Pure imagery
return Quat!T(0, i, j, k);
}
@property Quat!T versor() const pure nothrow { /// Unit versor
return this / abs;
}
/// Unit versor of imagery part
@property Quat!T iversor() const pure nothrow {
return pureim / pureim.abs;
}
/// Assignment
Quat!T opAssign(U : T)(Quat!U z) pure nothrow {
x = z.x; y = z.y;
return this;
}
Quat!T opAssign(U : T)(Complex!U c) pure nothrow {
x = c; y = 0;
return this;
}
Quat!T opAssign(U : T)(U r) pure nothrow if (isNumeric!U) {
re = r; i = 0; y = 0;
return this;
}
/// Test for equal, not ordered so no opCmp
bool opEquals(U : T)(Quat!U z) const pure nothrow {
return re == z.re && i == z.i && j == z.j && k == z.k;
}
bool opEquals(U : T)(Complex!U c) const pure nothrow {
return re == c.re && i == c.im && j == 0 && k == 0;
}
bool opEquals(U : T)(U r) const pure nothrow if (isNumeric!U) {
return re == r && i == 0 && j == 0 && k == 0;
}
/// Unary op
Quat!T opUnary(string op)() const pure nothrow if (op == "+") {
return this;
}
Quat!T opUnary(string op)() const pure nothrow if (op == "-") {
return Quat!T(-re, -i, -j, -k);
}
/// Binary op, Quaternion on left of op
Quat!(CommonType!(T,U)) opBinary(string op, U)(Quat!U z)
const pure nothrow {
alias typeof(return) C;
static if (op == "+" ) {
return C(re + z.re, i + z.i, j + z.j, k + z.k);
} else static if (op == "-") {
return C(re - z.re, i - z.i, j - z.j, k - z.k);
} else static if (op == "*") {
return C(re * z.re - i * z.i - j * z.j - k * z.k,
re * z.i + i * z.re + j * z.k - k * z.j,
re * z.j - i * z.k + j * z.re + k * z.i,
re * z.k + i * z.j - j * z.i + k * z.re);
} else static if (op == "/") {
return this * z.recip;
}
}
/// Extend complex to quaternion
Quat!(CommonType!(T,U)) opBinary(string op, U)(Complex!U c)
const pure nothrow {
return opBinary!op(typeof(return)(c.re, c.im, 0, 0));
}
/// For scalar
Quat!(CommonType!(T,U)) opBinary(string op, U)(U r)
const pure nothrow if (isNumeric!U) {
alias typeof(return) C;
static if (op == "+" ) {
return C(re + r, i, j, k);
} else static if (op == "-") {
return C(re - r, i, j, k);
} else static if (op == "*") {
return C(re * r, i * r, j * r, k * r);
} else static if (op == "/") {
return C(re / r, i / r, j / r, k / r);
} else static if (op == "^^") {
return pow(r);
}
}
/// Power function
Quat!(CommonType!(T,U)) pow(U)(U r)
const pure nothrow if (isNumeric!U) {
return (abs^^r) * exp(r * iversor * arg);
}
/// Handle binary op if Quaternion on right of op and left is
/// not quaternion.
Quat!(CommonType!(T,U)) opBinaryRight(string op, U)(Complex!U c)
const pure nothrow {
alias typeof(return) C;
auto w = C(c.re, c.im, 0, 0);
return w.opBinary!(op)(this);
}
Quat!(CommonType!(T,U)) opBinaryRight(string op, U)(U r)
const pure nothrow if (isNumeric!U) {
alias typeof(return) C;
static if (op == "+" || op == "*") {
return opBinary!op(r);
} else static if (op == "-") {
return C(r - re , -i, -j, -k);
} else static if (op == "/") {
auto w = C(re, i, j, k);
return w.recip * r;
}
}
}
HT exp(HT)(HT z) pure nothrow if (is(HT T == Quat!T)) {
immutable inorm = z.pureim.abs;
return std.math.exp(z.re) * (cos(inorm) + z.iversor * sin(inorm));
}
HT log(HT)(HT z) pure nothrow if (is(HT T == Quat!T)) {
return std.math.log(z.abs) + z.iversor * acos(z.re / z.abs);
}
void main() { // Demo code
import std.stdio;
alias Quat!real QR;
immutable real r = 7;
immutable QR q = QR(2, 3, 4, 5),
q1 = QR(2, 3, 4, 5),
q2 = QR(3, 4, 5, 6);
writeln("1. q - norm: ", q.abs);
writeln("2. q - negative: ", -q);
writeln("3. q - conjugate: ", q.conj);
writeln("4. r + q: ", r + q);
writeln(" q + r: ", q + r);
writeln("5. q1 + q2: ", q1 + q2);
writeln("6. r * q: ", r * q);
writeln(" q * r: ", q * r);
writeln("7. q1 * q2: ", q1 * q2);
writeln(" q2 * q1: ", q2 * q1);
writeln("8. q1 * q2 != q2 * Q1 ? ", q1 * q2 != q2 * q1);
immutable QR i = QR(0, 1, 0, 0),
j = QR(0, 0, 1, 0),
k = QR(0, 0, 0, 1);
writeln("9.1 i * i: ", i * i);
writeln(" J * j: ", j * j);
writeln(" k * k: ", k * k);
writeln(" i * j * k: ", i * j * k);
writeln("9.2 q1 / q2: ", q1 / q2);
writeln("9.3 q1 / q2 * q2: ", q1 / q2 * q2);
writeln(" q2 * q1 / q2: ", q2 * q1 / q2);
writeln("9.4 exp(pi * i): ", exp(PI * i));
writeln(" exp(pi * j): ", exp(PI * j));
writeln(" exp(pi * k): ", exp(PI * k));
writeln(" exp(q): ", exp(q));
writeln(" log(q): ", log(q));
writeln(" exp(log(q)): ", exp(log(q)));
writeln(" log(exp(q)): ", log(exp(q)));
immutable s = log(exp(q));
writeln("9.5 let s = log(exp(q)): ", s);
writeln(" exp(s): ", exp(s));
writeln(" log(s): ", log(s));
writeln(" exp(log(s)): ", exp(log(s)));
writeln(" log(exp(s)): ", log(exp(s)));
}
- Output:
1. q - norm: 7.34847
2. q - negative: [-2, -3, -4, -5]
3. q - conjugate: [2, -3, -4, -5]
4. r + q: [9, 3, 4, 5]
q + r: [9, 3, 4, 5]
5. q1 + q2: [5, 7, 9, 11]
6. r * q: [14, 21, 28, 35]
q * r: [14, 21, 28, 35]
7. q1 * q2: [-56, 16, 24, 26]
q2 * q1: [-56, 18, 20, 28]
8. q1 * q2 != q2 * Q1 ? true
9.1 i * i: [-1, 0, 0, 0]
J * j: [-1, 0, 0, 0]
k * k: [-1, 0, 0, 0]
i * j * k: [-1, 0, 0, 0]
9.2 q1 / q2: [0.790698, 0.0232558, -1.35525e-20, 0.0465116]
9.3 q1 / q2 * q2: [2, 3, 4, 5]
q2 * q1 / q2: [2, 3.46512, 3.90698, 4.76744]
9.4 exp(pi * i): [-1, -5.42101e-20, -0, -0]
exp(pi * j): [-1, -0, -5.42101e-20, -0]
exp(pi * k): [-1, -0, -0, -5.42101e-20]
exp(q): [5.21186, 2.22222, 2.96296, 3.7037]
log(q): [1.99449, 0.549487, 0.732649, 0.915812]
exp(log(q)): [2, 3, 4, 5]
log(exp(q)): [2, 0.33427, 0.445694, 0.557117]
9.5 let s = log(exp(q)): [2, 0.33427, 0.445694, 0.557117]
exp(s): [5.21186, 2.22222, 2.96296, 3.7037]
log(s): [0.765279, 0.159215, 0.212286, 0.265358]
exp(log(s)): [2, 0.33427, 0.445694, 0.557117]
log(exp(s)): [2, 0.33427, 0.445694, 0.557117]
[edit] Delphi
unit Quaternions;
interface
type
TQuaternion = record
A, B, C, D: double;
function Init (aA, aB, aC, aD : double): TQuaternion;
function Norm : double;
function Conjugate : TQuaternion;
function ToString : string;
class operator Negative (Left : TQuaternion): TQuaternion;
class operator Positive (Left : TQuaternion): TQuaternion;
class operator Add (Left, Right : TQuaternion): TQuaternion;
class operator Add (Left : TQuaternion; Right : double): TQuaternion; overload;
class operator Add (Left : double; Right : TQuaternion): TQuaternion; overload;
class operator Subtract (Left, Right : TQuaternion): TQuaternion;
class operator Multiply (Left, Right : TQuaternion): TQuaternion;
class operator Multiply (Left : TQuaternion; Right : double): TQuaternion; overload;
class operator Multiply (Left : double; Right : TQuaternion): TQuaternion; overload;
end;
implementation
uses
SysUtils;
{ TQuaternion }
function TQuaternion.Init(aA, aB, aC, aD: double): TQuaternion;
begin
A := aA;
B := aB;
C := aC;
D := aD;
result := Self;
end;
function TQuaternion.Norm: double;
begin
result := sqrt(sqr(A) + sqr(B) + sqr(C) + sqr(D));
end;
function TQuaternion.Conjugate: TQuaternion;
begin
result.B := -B;
result.C := -C;
result.D := -D;
end;
class operator TQuaternion.Negative(Left: TQuaternion): TQuaternion;
begin
result.A := -Left.A;
result.B := -Left.B;
result.C := -Left.C;
result.D := -Left.D;
end;
class operator TQuaternion.Positive(Left: TQuaternion): TQuaternion;
begin
result := Left;
end;
class operator TQuaternion.Add(Left, Right: TQuaternion): TQuaternion;
begin
result.A := Left.A + Right.A;
result.B := Left.B + Right.B;
result.C := Left.C + Right.C;
result.D := Left.D + Right.D;
end;
class operator TQuaternion.Add(Left: TQuaternion; Right: double): TQuaternion;
begin
result.A := Left.A + Right;
result.B := Left.B;
result.C := Left.C;
result.D := Left.D;
end;
class operator TQuaternion.Add(Left: double; Right: TQuaternion): TQuaternion;
begin
result.A := Left + Right.A;
result.B := Right.B;
result.C := Right.C;
result.D := Right.D;
end;
class operator TQuaternion.Subtract(Left, Right: TQuaternion): TQuaternion;
begin
result.A := Left.A - Right.A;
result.B := Left.B - Right.B;
result.C := Left.C - Right.C;
result.D := Left.D - Right.D;
end;
class operator TQuaternion.Multiply(Left, Right: TQuaternion): TQuaternion;
begin
result.A := Left.A * Right.A - Left.B * Right.B - Left.C * Right.C - Left.D * Right.D;
result.B := Left.A * Right.B + Left.B * Right.A + Left.C * Right.D - Left.D * Right.C;
result.C := Left.A * Right.C - Left.B * Right.D + Left.C * Right.A + Left.D * Right.B;
result.D := Left.A * Right.D + Left.B * Right.C - Left.C * Right.B + Left.D * Right.A;
end;
class operator TQuaternion.Multiply(Left: double; Right: TQuaternion): TQuaternion;
begin
result.A := Left * Right.A;
result.B := Left * Right.B;
result.C := Left * Right.C;
result.D := Left * Right.D;
end;
class operator TQuaternion.Multiply(Left: TQuaternion; Right: double): TQuaternion;
begin
result.A := Left.A * Right;
result.B := Left.B * Right;
result.C := Left.C * Right;
result.D := Left.D * Right;
end;
function TQuaternion.ToString: string;
begin
result := Format('%f + %fi + %fj + %fk', [A, B, C, D]);
end;
end.
Test program
program QuaternionTest;
{$APPTYPE CONSOLE}
uses
Quaternions in 'Quaternions.pas';
var
r : double;
q, q1, q2 : TQuaternion;
begin
r := 7;
q := q .Init(1, 2, 3, 4);
q1 := q1.Init(2, 3, 4, 5);
q2 := q2.Init(3, 4, 5, 6);
writeln('q = ', q.ToString);
writeln('q1 = ', q1.ToString);
writeln('q2 = ', q2.ToString);
writeln('r = ', r);
writeln('Norm(q ) = ', q.Norm);
writeln('Norm(q1) = ', q1.Norm);
writeln('Norm(q2) = ', q2.Norm);
writeln('-q = ', (-q).ToString);
writeln('Conjugate q = ', q.Conjugate.ToString);
writeln('q1 + q2 = ', (q1 + q2).ToString);
writeln('q2 + q1 = ', (q2 + q1).ToString);
writeln('q * r = ', (q * r).ToString);
writeln('r * q = ', (r * q).ToString);
writeln('q1 * q2 = ', (q1 * q2).ToString);
writeln('q2 * q1 = ', (q2 * q1).ToString);
end.
Output:
q = 1.00 + 2.00i + 3.00j + 4.00k q1 = 2.00 + 3.00i + 4.00j + 5.00k q2 = 3.00 + 4.00i + 5.00j + 6.00k r = 7.00000000000000E+0000 Norm(q ) = 5.47722557505166E+0000 Norm(q1) = 7.34846922834953E+0000 Norm(q2) = 9.27361849549570E+0000 -q = -1.00 + -2.00i + -3.00j + -4.00k Conjugate q = -1.00 + -2.00i + -3.00j + -4.00k q1 + q2 = 5.00 + 7.00i + 9.00j + 11.00k q2 + q1 = 5.00 + 7.00i + 9.00j + 11.00k q * r = 7.00 + 14.00i + 21.00j + 28.00k r * q = 7.00 + 14.00i + 21.00j + 28.00k q1 * q2 = -56.00 + 16.00i + 24.00j + 26.00k q2 * q1 = -56.00 + 18.00i + 20.00j + 28.00k
--DavidIzadaR 20:33, 7 August 2011 (UTC)
[edit] E
interface Quaternion guards QS {}
def makeQuaternion(a, b, c, d) {
return def quaternion implements QS {
to __printOn(out) {
out.print("(", a, " + ", b, "i + ")
out.print(c, "j + ", d, "k)")
}
# Task requirement 1
to norm() {
return (a**2 + b**2 + c**2 + d**2).sqrt()
}
# Task requirement 2
to negate() {
return makeQuaternion(-a, -b, -c, -d)
}
# Task requirement 3
to conjugate() {
return makeQuaternion(a, -b, -c, -d)
}
# Task requirement 4, 5
# This implements q + r; r + q is deliberately prohibited by E
to add(other :any[Quaternion, int, float64]) {
switch (other) {
match q :Quaternion {
return makeQuaternion(
a+q.a(), b+q.b(), c+q.c(), d+q.d())
}
match real {
return makeQuaternion(a+real, b, c, d)
}
}
}
# Task requirement 6, 7
# This implements q * r; r * q is deliberately prohibited by E
to multiply(other :any[Quaternion, int, float64]) {
switch (other) {
match q :Quaternion {
return makeQuaternion(
a*q.a() - b*q.b() - c*q.c() - d*q.d(),
a*q.b() + b*q.a() + c*q.d() - d*q.c(),
a*q.c() - b*q.d() + c*q.a() + d*q.b(),
a*q.d() + b*q.c() - c*q.b() + d*q.a())
}
match real {
return makeQuaternion(real*a, real*b, real*c, real*d)
}
}
}
to a() { return a }
to b() { return b }
to c() { return c }
to d() { return d }
}
}
? def q1 := makeQuaternion(2,3,4,5)
# value: (2 + 3i + 4j + 5k)
? def q2 := makeQuaternion(3,4,5,6)
# value: (3 + 4i + 5j + 6k)
? q1+q2
# value: (5 + 7i + 9j + 11k)
? q1*q2
# value: (-56 + 16i + 24j + 26k)
? q2*q1
# value: (-56 + 18i + 20j + 28k)
? q1+(-2)
# value: (0 + 3i + 4j + 5k)
[edit] Euphoria
function norm(sequence q)
return sqrt(power(q[1],2)+power(q[2],2)+power(q[3],2)+power(q[4],2))
end function
function conj(sequence q)
q[2..4] = -q[2..4]
return q
end function
function add(object q1, object q2)
if atom(q1) != atom(q2) then
if atom(q1) then
q1 = {q1,0,0,0}
else
q2 = {q2,0,0,0}
end if
end if
return q1+q2
end function
function mul(object q1, object q2)
if sequence(q1) and sequence(q2) then
return { q1[1]*q2[1] - q1[2]*q2[2] - q1[3]*q2[3] - q1[4]*q2[4],
q1[1]*q2[2] + q1[2]*q2[1] + q1[3]*q2[4] - q1[4]*q2[3],
q1[1]*q2[3] - q1[2]*q2[4] + q1[3]*q2[1] + q1[4]*q2[2],
q1[1]*q2[4] + q1[2]*q2[3] - q1[3]*q2[2] + q1[4]*q2[1] }
else
return q1*q2
end if
end function
function quats(sequence q)
return sprintf("%g + %gi + %gj + %gk",q)
end function
constant
q = {1, 2, 3, 4},
q1 = {2, 3, 4, 5},
q2 = {5, 6, 7, 8},
r = 7
printf(1, "norm(q) = %g\n", norm(q))
printf(1, "-q = %s\n", {quats(-q)})
printf(1, "conj(q) = %s\n", {quats(conj(q))})
printf(1, "q + r = %s\n", {quats(add(q,r))})
printf(1, "q1 + q2 = %s\n", {quats(add(q1,q2))})
printf(1, "q1 * q2 = %s\n", {quats(mul(q1,q2))})
printf(1, "q2 * q1 = %s\n", {quats(mul(q2,q1))})
Output:
norm(q) = 5.47723 -q = -1 + -2i + -3j + -4k conj(q) = 1 + -2i + -3j + -4k q + r = 8 + 2i + 3j + 4k q1 + q2 = 7 + 9i + 11j + 13k q1 * q2 = -76 + 24i + 40j + 38k q2 * q1 = -76 + 30i + 28j + 44k
[edit] F#
Mainly a On the minus side we have no way to define a conversion to Quaternion from any suitable (numeric) type. On the plus side we can avoid the stuff to make the equality structual (from the referential equality default) by just declaring it as an attribute to the type and let the compiler handle the details.open System
[<Struct; StructuralEquality; NoComparison>]
type Quaternion(r : float, i : float, j : float, k : float) =
member this.A = r
member this.B = i
member this.C = j
member this.D = k
new (f : float) = Quaternion(f, 0., 0., 0.)
static member (~-) (q : Quaternion) = Quaternion(-q.A, -q.B, -q.C, -q.D)
static member (+) (q1 : Quaternion, q2 : Quaternion) =
Quaternion(q1.A + q2.A, q1.B + q2.B, q1.C + q2.C, q1.D + q2.D)
static member (+) (q : Quaternion, r : float) = q + Quaternion(r)
static member (+) (r : float, q: Quaternion) = Quaternion(r) + q
static member (*) (q1 : Quaternion, q2 : Quaternion) =
Quaternion(
q1.A * q2.A - q1.B * q2.B - q1.C * q2.C - q1.D * q2.D,
q1.A * q2.B + q1.B * q2.A + q1.C * q2.D - q1.D * q2.C,
q1.A * q2.C - q1.B * q2.D + q1.C * q2.A + q1.D * q2.B,
q1.A * q2.D + q1.B * q2.C - q1.C * q2.B + q1.D * q2.A)
static member (*) (q : Quaternion, r : float) = q * Quaternion(r)
static member (*) (r : float, q: Quaternion) = Quaternion(r) * q
member this.Norm = Math.Sqrt(r * r + i * i + j * j + k * k)
member this.Conjugate = Quaternion(r, -i, -j, -k)
override this.ToString() = sprintf "Q(%f, %f, %f, %f)" r i j k
[<EntryPoint>]
let main argv =
let q = Quaternion(1., 2., 3., 4.)
let q1 = Quaternion(2., 3., 4., 5.)
let q2 = Quaternion(3., 4., 5., 6.)
let r = 7.
printfn "q = %A" q
printfn "q1 = %A" q1
printfn "q2 = %A" q2
printfn "r = %A" r
printfn "q.Norm = %A" q.Norm
printfn "q1.Norm = %A" q1.Norm
printfn "q2.Norm = %A" q2.Norm
printfn "-q = %A" -q
printfn "q.Conjugate = %A" q.Conjugate
printfn "q + r = %A" (q + (Quaternion r))
printfn "q1 + q2 = %A" (q1 + q2)
printfn "q2 + q1 = %A" (q2 + q1)
printfn "q * r = %A" (q * r)
printfn "q1 * q2 = %A" (q1 * q2)
printfn "q2 * q1 = %A" (q2 * q1)
printfn "q1*q2 %s q2*q1" (if (q1 * q2) = (q2 * q1) then "=" else "<>")
printfn "q %s Q(1.,2.,3.,4.)" (if q = Quaternion(1., 2., 3., 4.) then "=" else "<>")
0
Output
q = Q(1.000000, 2.000000, 3.000000, 4.000000) q1 = Q(2.000000, 3.000000, 4.000000, 5.000000) q2 = Q(3.000000, 4.000000, 5.000000, 6.000000) r = 7.0 q.Norm = 5.477225575 q1.Norm = 7.348469228 q2.Norm = 9.273618495 -q = Q(-1.000000, -2.000000, -3.000000, -4.000000) q.Conjugate = Q(1.000000, -2.000000, -3.000000, -4.000000) q + r = Q(8.000000, 2.000000, 3.000000, 4.000000) q1 + q2 = Q(5.000000, 7.000000, 9.000000, 11.000000) q2 + q1 = Q(5.000000, 7.000000, 9.000000, 11.000000) q * r = Q(7.000000, 14.000000, 21.000000, 28.000000) q1 * q2 = Q(-56.000000, 16.000000, 24.000000, 26.000000) q2 * q1 = Q(-56.000000, 18.000000, 20.000000, 28.000000) q1*q2 <> q2*q1 q = Q(1.,2.,3.,4.)
[edit] Forth
: quaternions 4 * floats ;
: qvariable create 1 quaternions allot ;
: q! ( a b c d q -- )
dup 3 floats + f! dup 2 floats + f! dup float+ f! f! ;
: qcopy ( src dest -- ) 1 quaternions move ;
: qnorm ( q -- f )
0e 4 0 do dup f@ fdup f* f+ float+ loop drop fsqrt ;
: qf* ( q f -- )
4 0 do dup f@ fover f* dup f! float+ loop fdrop drop ;
: qnegate ( q -- ) -1e qf* ;
: qconj ( q -- )
float+ 3 0 do dup f@ fnegate dup f! float+ loop drop ;
: qf+ ( q f -- ) dup f@ f+ f! ;
: q+ ( q1 q2 -- )
4 0 do over f@ dup f@ f+ dup f! float+ swap float+ swap loop 2drop ;
\ access
: q.a f@ ;
: q.b float+ f@ ;
: q.c 2 floats + f@ ;
: q.d 3 floats + f@ ;
: q* ( dest q1 q2 -- )
over q.a dup q.d f* over q.b dup q.c f* f+ over q.c dup q.b f* f- over q.d dup q.a f* f+
over q.a dup q.c f* over q.b dup q.d f* f- over q.c dup q.a f* f+ over q.d dup q.b f* f+
over q.a dup q.b f* over q.b dup q.a f* f+ over q.c dup q.d f* f+ over q.d dup q.c f* f-
over q.a dup q.a f* over q.b dup q.b f* f- over q.c dup q.c f* f- over q.d dup q.d f* f-
2drop 4 0 do dup f! float+ loop drop ;
: q= ( q1 q2 -- ? )
4 0 do
over f@ dup f@ f<> if 2drop false unloop exit then
float+ swap float+
loop
2drop true ;
\ testing
: q. ( q -- )
[char] ( emit space
4 0 do dup f@ f. float+ loop drop
[char] ) emit space ;
qvariable q 1e 2e 3e 4e q q!
qvariable q1 2e 3e 4e 5e q1 q!
create q2 3e f, 4e f, 5e f, 6e f, \ by hand
qvariable tmp
qvariable m1
qvariable m2
q qnorm f. \ 5.47722557505166
q tmp qcopy tmp qnegate tmp q. \ ( -1. -2. -3. -4. )
q tmp qcopy tmp qconj tmp q. \ ( 1. -2. -3. -4. )
q m1 qcopy m1 7e qf+ m1 q. \ ( 8. 2. 3. 4. )
q m2 qcopy 7e m2 qf+ m2 q. \ ( 8. 2. 3. 4. )
m1 m2 q= . \ -1 (true)
q2 tmp qcopy q1 tmp q+ tmp q. \ ( 5. 7. 9. 11. )
q m1 qcopy m1 7e qf* m1 q. \ ( 7. 14. 21. 28. )
q m2 qcopy 7e m2 qf* m2 q. \ ( 7. 14. 21. 28. )
m1 m2 q= . \ -1 (true)
m1 q1 q2 q* m1 q. \ ( -56. 16. 24. 26. )
m2 q2 q1 q* m2 q. \ ( -56. 18. 20. 28. )
m1 m2 q= . \ 0 (false)
[edit] Fortran
module Q_mod
implicit none
type quaternion
real :: a, b, c, d
end type
public :: norm, neg, conj
public :: operator (+)
public :: operator (*)
private :: q_plus_q, q_plus_r, r_plus_q, &
q_mult_q, q_mult_r, r_mult_q, &
norm_q, neg_q, conj_q
interface norm
module procedure norm_q
end interface
interface neg
module procedure neg_q
end interface
interface conj
module procedure conj_q
end interface
interface operator (+)
module procedure q_plus_q, q_plus_r, r_plus_q
end interface
interface operator (*)
module procedure q_mult_q, q_mult_r, r_mult_q
end interface
contains
function norm_q(x) result(res)
real :: res
type (quaternion), intent (in) :: x
res = sqrt(x%a*x%a + x%b*x%b + x%c*x%c + x%d*x%d)
end function norm_q
function neg_q(x) result(res)
type (quaternion) :: res
type (quaternion), intent (in) :: x
res%a = -x%a
res%b = -x%b
res%c = -x%c
res%d = -x%d
end function neg_q
function conj_q(x) result(res)
type (quaternion) :: res
type (quaternion), intent (in) :: x
res%a = x%a
res%b = -x%b
res%c = -x%c
res%d = -x%d
end function conj_q
function q_plus_q(x, y) result (res)
type (quaternion) :: res
type (quaternion), intent (in) :: x, y
res%a = x%a + y%a
res%b = x%b + y%b
res%c = x%c + y%c
res%d = x%d + y%d
end function q_plus_q
function q_plus_r(x, r) result (res)
type (quaternion) :: res
type (quaternion), intent (in) :: x
real, intent(in) :: r
res = x
res%a = x%a + r
end function q_plus_r
function r_plus_q(r, x) result (res)
type (quaternion) :: res
type (quaternion), intent (in) :: x
real, intent(in) :: r
res = x
res%a = x%a + r
end function r_plus_q
function q_mult_q(x, y) result (res)
type (quaternion) :: res
type (quaternion), intent (in) :: x, y
res%a = x%a*y%a - x%b*y%b - x%c*y%c - x%d*y%d
res%b = x%a*y%b + x%b*y%a + x%c*y%d - x%d*y%c
res%c = x%a*y%c - x%b*y%d + x%c*y%a + x%d*y%b
res%d = x%a*y%d + x%b*y%c - x%c*y%b + x%d*y%a
end function q_mult_q
function q_mult_r(x, r) result (res)
type (quaternion) :: res
type (quaternion), intent (in) :: x
real, intent(in) :: r
res%a = x%a*r
res%b = x%b*r
res%c = x%c*r
res%d = x%d*r
end function q_mult_r
function r_mult_q(r, x) result (res)
type (quaternion) :: res
type (quaternion), intent (in) :: x
real, intent(in) :: r
res%a = x%a*r
res%b = x%b*r
res%c = x%c*r
res%d = x%d*r
end function r_mult_q
end module Q_mod
program Quaternions
use Q_mod
implicit none
real :: r = 7.0
type(quaternion) :: q, q1, q2
q = quaternion(1, 2, 3, 4)
q1 = quaternion(2, 3, 4, 5)
q2 = quaternion(3, 4, 5, 6)
write(*, "(a, 4f8.3)") " q = ", q
write(*, "(a, 4f8.3)") " q1 = ", q1
write(*, "(a, 4f8.3)") " q2 = ", q2
write(*, "(a, f8.3)") " r = ", r
write(*, "(a, f8.3)") " Norm of q = ", norm(q)
write(*, "(a, 4f8.3)") " Negative of q = ", neg(q)
write(*, "(a, 4f8.3)") "Conjugate of q = ", conj(q)
write(*, "(a, 4f8.3)") " q + r = ", q + r
write(*, "(a, 4f8.3)") " r + q = ", r + q
write(*, "(a, 4f8.3)") " q1 + q2 = ", q1 + q2
write(*, "(a, 4f8.3)") " q * r = ", q * r
write(*, "(a, 4f8.3)") " r * q = ", r * q
write(*, "(a, 4f8.3)") " q1 * q2 = ", q1 * q2
write(*, "(a, 4f8.3)") " q2 * q1 = ", q2 * q1
end program
Output
q = 1.000 2.000 3.000 4.000
q1 = 2.000 3.000 4.000 5.000
q2 = 3.000 4.000 5.000 6.000
r = 7.000
Norm of q = 5.477
Negative of q = -1.000 -2.000 -3.000 -4.000
Conjugate of q = 1.000 -2.000 -3.000 -4.000
q + r = 8.000 2.000 3.000 4.000
r + q = 8.000 2.000 3.000 4.000
q1 + q2 = 5.000 7.000 9.000 11.000
q * r = 7.000 14.000 21.000 28.000
r * q = 7.000 14.000 21.000 28.000
q1 * q2 = -56.000 16.000 24.000 26.000
q2 * q1 = -56.000 18.000 20.000 28.000
[edit] GAP
# GAP has built-in support for quaternions
A := QuaternionAlgebra(Rationals);
# <algebra-with-one of dimension 4 over Rationals>
b := BasisVectors(Basis(A));
# [ e, i, j, k ]
q := [1, 2, 3, 4]*b;
# e+(2)*i+(3)*j+(4)*k
# Computing norm may be difficult, since the result would be in a quadratic field (Sqrt exists in GAP, is quite unusual, see ?E in GAP, and the following example
Sqrt(5/3);
# 1/3*E(60)^7+1/3*E(60)^11-1/3*E(60)^19-1/3*E(60)^23-1/3*E(60)^31+1/3*E(60)^43-1/3*E(60)^47+1/3*E(60)^59
q1 := [2, 3, 4, 5]*b;
# (2)*e+(3)*i+(4)*j+(5)*k
q2 := [3, 4, 5, 6]*b;
# (3)*e+(4)*i+(5)*j+(6)*k
q1*q2 - q2*q1;
# (-2)*i+(4)*j+(-2)*k
# Can't add directly to a rational, one must make a quaternion of it
r:=5/3*b[1];
# (5/3)*e
r + q;
# (8/3)*e+(2)*i+(3)*j+(4)*k
# For multiplication, no problem (we are in an algebra over rationals !)
r * q;
# (5/3)*e+(10/3)*i+(5)*j+(20/3)*k
5/3 * q;
# (5/3)*e+(10/3)*i+(5)*j+(20/3)*k
# Negative
-q;
(-1)*e+(-2)*i+(-3)*j+(-4)*k
[edit] Go
Conventions for method receiver, parameter, and return values modeled after Go's big number package. It provides flexibility without requiring unnecessary object creation. The test program creates only four quaternion objects, the three inputs and one more for an output. The three inputs are reused repeatedly without being modified. The output is also reused repeatedly, being overwritten for each operation.
package main
import (
"fmt"
"math"
)
type qtn struct {
r, i, j, k float64
}
var (
q = &qtn{1, 2, 3, 4}
q1 = &qtn{2, 3, 4, 5}
q2 = &qtn{3, 4, 5, 6}
r float64 = 7
)
func main() {
fmt.Println("Inputs")
fmt.Println("q:", q)
fmt.Println("q1:", q1)
fmt.Println("q2:", q2)
fmt.Println("r:", r)
var qr qtn
fmt.Println("\nFunctions")
fmt.Println("q.norm():", q.norm())
fmt.Println("neg(q):", qr.neg(q))
fmt.Println("conj(q):", qr.conj(q))
fmt.Println("addF(q, r):", qr.addF(q, r))
fmt.Println("addQ(q1, q2):", qr.addQ(q1, q2))
fmt.Println("mulF(q, r):", qr.mulF(q, r))
fmt.Println("mulQ(q1, q2):", qr.mulQ(q1, q2))
fmt.Println("mulQ(q2, q1):", qr.mulQ(q2, q1))
}
func (q *qtn) String() string {
return fmt.Sprintf("(%g, %g, %g, %g)", q.r, q.i, q.j, q.k)
}
func (q *qtn) norm() float64 {
return math.Sqrt(q.r*q.r + q.i*q.i + q.j*q.j + q.k*q.k)
}
func (z *qtn) neg(q *qtn) *qtn {
z.r, z.i, z.j, z.k = -q.r, -q.i, -q.j, -q.k
return z
}
func (z *qtn) conj(q *qtn) *qtn {
z.r, z.i, z.j, z.k = q.r, -q.i, -q.j, -q.k
return z
}
func (z *qtn) addF(q *qtn, r float64) *qtn {
z.r, z.i, z.j, z.k = q.r+r, q.i, q.j, q.k
return z
}
func (z *qtn) addQ(q1, q2 *qtn) *qtn {
z.r, z.i, z.j, z.k = q1.r+q2.r, q1.i+q2.i, q1.j+q2.j, q1.k+q2.k
return z
}
func (z *qtn) mulF(q *qtn, r float64) *qtn {
z.r, z.i, z.j, z.k = q.r*r, q.i*r, q.j*r, q.k*r
return z
}
func (z *qtn) mulQ(q1, q2 *qtn) *qtn {
z.r, z.i, z.j, z.k =
q1.r*q2.r-q1.i*q2.i-q1.j*q2.j-q1.k*q2.k,
q1.r*q2.i+q1.i*q2.r+q1.j*q2.k-q1.k*q2.j,
q1.r*q2.j-q1.i*q2.k+q1.j*q2.r+q1.k*q2.i,
q1.r*q2.k+q1.i*q2.j-q1.j*q2.i+q1.k*q2.r
return z
}
Output:
Inputs q: (1, 2, 3, 4) q1: (2, 3, 4, 5) q2: (3, 4, 5, 6) r: 7 Functions q.norm(): 5.477225575051661 neg(q): (-1, -2, -3, -4) conj(q): (1, -2, -3, -4) addF(q, r): (8, 2, 3, 4) addQ(q1, q2): (5, 7, 9, 11) mulF(q, r): (7, 14, 21, 28) mulQ(q1, q2): (-56, 16, 24, 26) mulQ(q2, q1): (-56, 18, 20, 28)
[edit] Haskell
import Control.Monad
import Control.Arrow
import Data.List
data Quaternion = Q Double Double Double Double
deriving (Show, Ord, Eq)
realQ :: Quaternion -> Double
realQ (Q r _ _ _) = r
imagQ :: Quaternion -> [Double]
imagQ (Q _ i j k) = [i, j, k]
quaternionFromScalar s = Q s 0 0 0
listFromQ (Q a b c d) = [a,b,c,d]
quaternionFromList [a, b, c, d] = Q a b c d
addQ, subQ, mulQ :: Quaternion -> Quaternion -> Quaternion
addQ (Q a b c d) (Q p q r s) = Q (a+p) (b+q) (c+r) (d+s)
subQ (Q a b c d) (Q p q r s) = Q (a-p) (b-q) (c-r) (d-s)
mulQ (Q a b c d) (Q p q r s) =
Q (a*p - b*q - c*r - d*s)
(a*q + b*p + c*s - d*r)
(a*r - b*s + c*p + d*q)
(a*s + b*r - c*q + d*p)
normQ = sqrt. sum. join (zipWith (*)). listFromQ
conjQ, negQ :: Quaternion -> Quaternion
conjQ (Q a b c d) = Q a (-b) (-c) (-d)
negQ (Q a b c d) = Q (-a) (-b) (-c) (-d)
To use with the Examples:
[q,q1,q2] = map quaternionFromList [[1..4],[2..5],[3..6]]
-- a*b == b*a
test :: Quaternion -> Quaternion -> Bool
test a b = a `mulQ` b == b `mulQ` a
Examples:
*Main> mulQ (Q 0 1 0 0) $ mulQ (Q 0 0 1 0) (Q 0 0 0 1) -- i*j*k Q (-1.0) 0.0 0.0 0.0 *Main> test q1 q2 False *Main> mulQ q1 q2 Q (-56.0) 16.0 24.0 26.0 *Main> flip mulQ q1 q2 Q (-56.0) 18.0 20.0 28.0 *Main> imagQ q [2.0,3.0,4.0]
[edit] Icon and Unicon
Using Unicon's class system.
class Quaternion(a, b, c, d)
method norm ()
return sqrt (a*a + b*b + c*c + d*d)
end
method negative ()
return Quaternion(-a, -b, -c, -d)
end
method conjugate ()
return Quaternion(a, -b, -c, -d)
end
method add (n)
if type(n) == "Quaternion__state"
then return Quaternion(a+n.a, b+n.b, c+n.c, d+n.d)
else return Quaternion(a+n, b, c, d)
end
method multiply (n)
if type(n) == "Quaternion__state"
then return Quaternion(a*n.a - b*n.b - c*n.c - d*n.d,
a*n.b + b*n.a + c*n.d - d*n.c,
a*n.c - b*n.d + c*n.a + d*n.b,
a*n.d + b*n.c - c*n.b + d*n.a)
else return Quaternion(a*n, b*n, c*n, d*n)
end
method sign (n)
return if n >= 0 then "+" else "-"
end
method string ()
return ("" || a || sign(b) || abs(b) || "i" || sign(c) || abs(c) || "j" || sign(d) || abs(d) || "k");
end
initially(a, b, c, d)
self.a := if /a then 0 else a
self.b := if /b then 0 else b
self.c := if /c then 0 else c
self.d := if /d then 0 else d
end
To test the above:
procedure main ()
q := Quaternion (1,2,3,4)
q1 := Quaternion (2,3,4,5)
q2 := Quaternion (3,4,5,6)
r := 7
write ("The norm of " || q.string() || " is " || q.norm ())
write ("The negative of " || q.string() || " is " || q.negative().string ())
write ("The conjugate of " || q.string() || " is " || q.conjugate().string ())
write ("Sum of " || q.string() || " and " || r || " is " || q.add(r).string ())
write ("Sum of " || q.string() || " and " || q1.string() || " is " || q.add(q1).string ())
write ("Product of " || q.string() || " and " || r || " is " || q.multiply(r).string ())
write ("Product of " || q.string() || " and " || q1.string() || " is " || q.multiply(q1).string ())
write ("q1*q2 = " || q1.multiply(q2).string ())
write ("q2*q1 = " || q2.multiply(q1).string ())
end
Output:
The norm of 1+2i+3j+4k is 5.477225575 The negative of 1+2i+3j+4k is -1-2i-3j-4k The conjugate of 1+2i+3j+4k is 1-2i-3j-4k Sum of 1+2i+3j+4k and 7 is 8+2i+3j+4k Sum of 1+2i+3j+4k and 2+3i+4j+5k is 3+5i+7j+9k Product of 1+2i+3j+4k and 7 is 7+14i+21j+28k Product of 1+2i+3j+4k and 2+3i+4j+5k is -36+6i+12j+12k q1*q2 = -56+16i+24j+26k q2*q1 = -56+18i+20j+28k
[edit] J
Derived from the j wiki:
NB. utilities
ip=: +/ .* NB. inner product
T=. (_1^#:0 10 9 12)*0 7 16 23 A.=i.4
toQ=: 4&{."1 :[: NB. real scalars -> quaternion
NB. task
norm=: %:@ip~@toQ NB. | y
neg=: -&toQ NB. - y and x - y
conj=: 1 _1 _1 _1 * toQ NB. + y
add=: +&toQ NB. x + y
mul=: (ip T ip ])&toQ NB. x * y
T is a rank 3 tensor which allows us to express quaternion product ab as the inner product ATB if A and B are 4 element vectors representing the quaternions a and b. (Note also that once we have defined mul we no longer need to retain the definition of T, so we define T using =. instead of =:). The value of T is probably more interesting than its definition, so:
T
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 _1 0 0
1 0 0 0
0 0 0 _1
0 0 1 0
0 0 _1 0
0 0 0 1
1 0 0 0
0 _1 0 0
0 0 0 _1
0 0 _1 0
0 1 0 0
1 0 0 0
Example use:
q=: 1 2 3 4
q1=: 2 3 4 5
q2=: 3 4 5 6
r=: 7
norm q
5.47723
neg q
_1 _2 _3 _4
conj q
1 _2 _3 _4
r add q
8 2 3 4
q1 add q2
5 7 9 11
r mul q
7 14 21 28
q1 mul q2
_56 16 24 26
q2 mul q1
_56 18 20 28
[edit] Java
public class Quaternion {
private final double a, b, c, d;
public Quaternion(double a, double b, double c, double d) {
this.a = a;
this.b = b;
this.c = c;
this.d = d;
}
public Quaternion(double r) {
this(r, 0.0, 0.0, 0.0);
}
public double norm() {
return Math.sqrt(a * a + b * b + c * c + d * d);
}
public Quaternion negative() {
return new Quaternion(-a, -b, -c, -d);
}
public Quaternion conjugate() {
return new Quaternion(a, -b, -c, -d);
}
public Quaternion add(double r) {
return new Quaternion(a + r, b, c, d);
}
public static Quaternion add(Quaternion q, double r) {
return q.add(r);
}
public static Quaternion add(double r, Quaternion q) {
return q.add(r);
}
public Quaternion add(Quaternion q) {
return new Quaternion(a + q.a, b + q.b, c + q.c, d + q.d);
}
public static Quaternion add(Quaternion q1, Quaternion q2) {
return q1.add(q2);
}
public Quaternion times(double r) {
return new Quaternion(a * r, b * r, c * r, d * r);
}
public static Quaternion times(Quaternion q, double r) {
return q.times(r);
}
public static Quaternion times(double r, Quaternion q) {
return q.times(r);
}
public Quaternion times(Quaternion q) {
return new Quaternion(
a * q.a - b * q.b - c * q.c - d * q.d,
a * q.b + b * q.a + c * q.d - d * q.c,
a * q.c - b * q.d + c * q.a + d * q.b,
a * q.d + b * q.c - c * q.b + d * q.a
);
}
public static Quaternion times(Quaternion q1, Quaternion q2) {
return q1.times(q2);
}
@Override
public boolean equals(Object obj) {
if (!(obj instanceof Quaternion)) return false;
final Quaternion other = (Quaternion) obj;
if (Double.doubleToLongBits(this.a) != Double.doubleToLongBits(other.a)) return false;
if (Double.doubleToLongBits(this.b) != Double.doubleToLongBits(other.b)) return false;
if (Double.doubleToLongBits(this.c) != Double.doubleToLongBits(other.c)) return false;
if (Double.doubleToLongBits(this.d) != Double.doubleToLongBits(other.d)) return false;
return true;
}
@Override
public String toString() {
return String.format("%.2f + %.2fi + %.2fj + %.2fk", a, b, c, d).replaceAll("\\+ -", "- ");
}
public String toQuadruple() {
return String.format("(%.2f, %.2f, %.2f, %.2f)", a, b, c, d);
}
public static void main(String[] args) {
Quaternion q = new Quaternion(1.0, 2.0, 3.0, 4.0);
Quaternion q1 = new Quaternion(2.0, 3.0, 4.0, 5.0);
Quaternion q2 = new Quaternion(3.0, 4.0, 5.0, 6.0);
double r = 7.0;
System.out.format("q = %s%n", q);
System.out.format("q1 = %s%n", q1);
System.out.format("q2 = %s%n", q2);
System.out.format("r = %.2f%n%n", r);
System.out.format("\u2016q\u2016 = %.2f%n", q.norm());
System.out.format("-q = %s%n", q.negative());
System.out.format("q* = %s%n", q.conjugate());
System.out.format("q + r = %s%n", q.add(r));
System.out.format("q1 + q2 = %s%n", q1.add(q2));
System.out.format("q \u00d7 r = %s%n", q.times(r));
Quaternion q1q2 = q1.times(q2);
Quaternion q2q1 = q2.times(q1);
System.out.format("q1 \u00d7 q2 = %s%n", q1q2);
System.out.format("q2 \u00d7 q1 = %s%n", q2q1);
System.out.format("q1 \u00d7 q2 %s q2 \u00d7 q1%n", (q1q2.equals(q2q1) ? "=" : "\u2260"));
}
}
This outputs:
q = 1.00 + 2.00i + 3.00j + 4.00k q1 = 2.00 + 3.00i + 4.00j + 5.00k q2 = 3.00 + 4.00i + 5.00j + 6.00k r = 7.00 ‖q‖ = 5.48 -q = -1.00 - 2.00i - 3.00j - 4.00k q* = 1.00 - 2.00i - 3.00j - 4.00k q + r = 8.00 + 2.00i + 3.00j + 4.00k q1 + q2 = 5.00 + 7.00i + 9.00j + 11.00k q × r = 7.00 + 14.00i + 21.00j + 28.00k q1 × q2 = -56.00 + 16.00i + 24.00j + 26.00k q2 × q1 = -56.00 + 18.00i + 20.00j + 28.00k q1 × q2 ≠ q2 × q1
[edit] JavaScript 1.8
Runs on Firefox 3+, limited support in other JS engines. More compatible JavaScript deserves its own entry.
var Quaternion = (function() {
// The Q() function takes an array argument and changes it
// prototype so that it becomes a Quaternion instance. This is
// scoped only for prototype member access.
function Q(a) {
a.__proto__ = proto;
return a;
}
// Actual constructor. This constructor converts its arguments to
// an array, then that array to a Quaternion instance, then
// returns that instance. (using "new" with this constructor is
// optional)
function Quaternion() {
return Q(Array.prototype.slice.call(arguments, 0, 4));
}
// Prototype for all Quaternions
const proto = {
// Inherits from a 4-element Array
__proto__ : [0,0,0,0],
// Properties -- In addition to Array[0..3] access, we
// also define matching a, b, c, and d properties
get a() this[0],
get b() this[1],
get c() this[2],
get d() this[3],
// Methods
norm : function() Math.sqrt(this.map(function(x) x*x).reduce(function(x,y) x+y)),
negate : function() Q(this.map(function(x) -x)),
conjugate : function() Q([ this[0] ].concat(this.slice(1).map(function(x) -x))),
add : function(x) {
if ("number" === typeof x) {
return Q([ this[0] + x ].concat(this.slice(1)));
} else {
return Q(this.map(function(v,i) v+x[i]));
}
},
mul : function(r) {
var q = this;
if ("number" === typeof r) {
return Q(q.map(function(e) e*r));
} else {
return Q([ q[0] * r[0] - q[1] * r[1] - q[2] * r[2] - q[3] * r[3],
q[0] * r[1] + q[1] * r[0] + q[2] * r[3] - q[3] * r[2],
q[0] * r[2] - q[1] * r[3] + q[2] * r[0] + q[3] * r[1],
q[0] * r[3] + q[1] * r[2] - q[2] * r[1] + q[3] * r[0] ]);
}
},
equals : function(q) this.every(function(v,i) v === q[i]),
toString : function() (this[0] + " + " + this[1] + "i + "+this[2] + "j + " + this[3] + "k").replace(/\+ -/g, '- ')
};
Quaternion.prototype = proto;
return Quaternion;
})();
Task/Example Usage:
var q = Quaternion(1,2,3,4);
var q1 = Quaternion(2,3,4,5);
var q2 = Quaternion(3,4,5,6);
var r = 7;
console.log("q = "+q);
console.log("q1 = "+q1);
console.log("q2 = "+q2);
console.log("r = "+r);
console.log("1. q.norm() = "+q.norm());
console.log("2. q.negate() = "+q.negate());
console.log("3. q.conjugate() = "+q.conjugate());
console.log("4. q.add(r) = "+q.add(r));
console.log("5. q1.add(q2) = "+q1.add(q2));
console.log("6. q.mul(r) = "+q.mul(r));
console.log("7.a. q1.mul(q2) = "+q1.mul(q2));
console.log("7.b. q2.mul(q1) = "+q2.mul(q1));
console.log("8. q1.mul(q2) " + (q1.mul(q2).equals(q2.mul(q1)) ? "==" : "!=") + " q2.mul(q1)");
Outputs:
q = 1 + 2i + 3j + 4k q1 = 2 + 3i + 4j + 5k q2 = 3 + 4i + 5j + 6k r = 7 1. q.norm() = 5.477225575051661 2. q.negate() = -1 - 2i - 3j - 4k 3. q.conjugate() = 1 - 2i - 3j - 4k 4. q.add(r) = 8 + 2i + 3j + 4k 5. q1.add(q2) = 5 + 7i + 9j + 11k 6. q.mul(r) = 7 + 14i + 21j + 28k 7.a. q1.mul(q2) = -56 + 16i + 24j + 26k 7.b. q2.mul(q1) = -56 + 18i + 20j + 28k 8. q1.mul(q2) != q2.mul(q1)
[edit] Liberty BASIC
Quaternions saved as a space-separated string of four numbers.
q$ = q$( 1 , 2 , 3 , 4 )
q1$ = q$( 2 , 3 , 4 , 5 )
q2$ = q$( 3 , 4 , 5 , 6 )
real = 7
print "q = " ; q$
print "q1 = " ; q1$
print "q2 = " ; q2$
print "real = " ; real
print "length /norm q = " ; length( q$ ) ' =norm norm of q
print "negative (-q1) = " ; negative$( q1$ ) ' =negative negated q1
print "conjugate q = " ; conjugate$( q$ ) ' conjugate conjugate q
print "real + q = " ; add1$( q$ , real ) ' real +quaternion real +q
print "q + q2 = " ; add2$( q$ , q2$ ) ' sum two quaternions q +q2
print "real * q = " ; multiply1$( q$ , real ) ' real *quaternion real *q
print "q1 * q2 = " ; multiply2$( q1$ , q2$ ) ' product of two quaternions q1 & q2
print "q2 * q1 = " ; multiply2$( q2$ , q1$ ) ' show q1 *q2 <> q2 *q1
end
function q$( r , i , j , k )
q$ = str$( r); " "; str$( i); " "; str$( j); " "; str$( k)
end function
function length( q$ )
r = val( word$( q$ , 1 ) )
i = val( word$( q$ , 2 ) )
j = val( word$( q$ , 3 ) )
k = val( word$( q$ , 4 ) )
length =sqr( r^2 +i^2 +j^2 +k^2)
end function
function multiply1$( q$ , d )
r = val( word$( q$ , 1 ) )
i = val( word$( q$ , 2 ) )
j = val( word$( q$ , 3 ) )
k = val( word$( q$ , 4 ) )
multiply1$ =q$( r*d, i*d, j*d, k*d)
end function
function multiply2$( q$ , b$ )
ar = val( word$( q$ , 1 ) ) 'a1
ai = val( word$( q$ , 2 ) ) 'b1
aj = val( word$( q$ , 3 ) ) 'c1
ak = val( word$( q$ , 4 ) ) 'd1
br = val( word$( b$ , 1 ) ) 'a2
bi = val( word$( b$ , 2 ) ) 'b2
bj = val( word$( b$ , 3 ) ) 'c2
bk = val( word$( b$ , 4 ) ) 'd2
multiply2$ =q$( _
ar *br_
+( 0 -ai) *bi_
+( 0 -aj) *bj_
+( 0 -ak) *bk _
,_
ar *bi_
+ai *br_
+aj *bk_
+( 0 -ak) *bj_
,_
ar *bj_
+( 0 -ai) *bk_
+aj *br_
+ak *bi_
,_
ar *bk_
+ai *bj_
+( 0 -aj) *bi_
+ak *br )
end function
function negative$( q$ )
r = val( word$( q$ , 1 ) )
i = val( word$( q$ , 2 ) )
j = val( word$( q$ , 3 ) )
k = val( word$( q$ , 4 ) )
negative$ =q$( 0-r, 0-i, 0-j, 0-k)
end function
function conjugate$( q$ )
r = val( word$( q$ , 1 ) )
i = val( word$( q$ , 2 ) )
j = val( word$( q$ , 3 ) )
k = val( word$( q$ , 4 ) )
conjugate$ =q$( r, 0-i, 0-j, 0-k)
end function
function add1$( q$ , real )
r = val( word$( q$ , 1 ) )
i = val( word$( q$ , 2 ) )
j = val( word$( q$ , 3 ) )
k = val( word$( q$ , 4 ) )
add1$ =q$( r +real, i, j, k)
end function
function add2$( q$ , b$ )
ar = val( word$( q$ , 1 ) )
ai = val( word$( q$ , 2 ) )
aj = val( word$( q$ , 3 ) )
ak = val( word$( q$ , 4 ) )
br = val( word$( b$ , 1 ) )
bi = val( word$( b$ , 2 ) )
bj = val( word$( b$ , 3 ) )
bk = val( word$( b$ , 4 ) )
add2$ =q$( ar +br, ai +bi, aj +bj, ak +bk)
end function
[edit] Lua
Quaternion = {}
function Quaternion.new( a, b, c, d )
local q = { a = a or 1, b = b or 0, c = c or 0, d = d or 0 }
local metatab = {}
setmetatable( q, metatab )
metatab.__add = Quaternion.add
metatab.__sub = Quaternion.sub
metatab.__unm = Quaternion.unm
metatab.__mul = Quaternion.mul
return q
end
function Quaternion.add( p, q )
if type( p ) == "number" then
return Quaternion.new( p+q.a, q.b, q.c, q.d )
elseif type( q ) == "number" then
return Quaternion.new( p.a+q, p.b, p.c, p.d )
else
return Quaternion.new( p.a+q.a, p.b+q.b, p.c+q.c, p.d+q.d )
end
end
function Quaternion.sub( p, q )
if type( p ) == "number" then
return Quaternion.new( p-q.a, q.b, q.c, q.d )
elseif type( q ) == "number" then
return Quaternion.new( p.a-q, p.b, p.c, p.d )
else
return Quaternion.new( p.a-q.a, p.b-q.b, p.c-q.c, p.d-q.d )
end
end
function Quaternion.unm( p )
return Quaternion.new( -p.a, -p.b, -p.c, -p.d )
end
function Quaternion.mul( p, q )
if type( p ) == "number" then
return Quaternion.new( p*q.a, p*q.b, p*q.c, p*q.d )
elseif type( q ) == "number" then
return Quaternion.new( p.a*q, p.b*q, p.c*q, p.d*q )
else
return Quaternion.new( p.a*q.a - p.b*q.b - p.c*q.c - p.d*q.d,
p.a*q.b + p.b*q.a + p.c*q.d - p.d*q.c,
p.a*q.c - p.b*q.d + p.c*q.a + p.d*q.b,
p.a*q.d + p.b*q.c - p.c*q.b + p.d*q.a )
end
end
function Quaternion.conj( p )
return Quaternion.new( p.a, -p.b, -p.c, -p.d )
end
function Quaternion.norm( p )
return math.sqrt( p.a^2 + p.b^2 + p.c^2 + p.d^2 )
end
function Quaternion.print( p )
print( string.format( "%f + %fi + %fj + %fk\n", p.a, p.b, p.c, p.d ) )
end
Examples:
q1 = Quaternion.new( 1, 2, 3, 4 )
q2 = Quaternion.new( 5, 6, 7, 8 )
r = 12
print( "norm(q1) = ", Quaternion.norm( q1 ) )
io.write( "-q1 = " ); Quaternion.print( -q1 )
io.write( "conj(q1) = " ); Quaternion.print( Quaternion.conj( q1 ) )
io.write( "r+q1 = " ); Quaternion.print( r+q1 )
io.write( "q1+r = " ); Quaternion.print( q1+r )
io.write( "r*q1 = " ); Quaternion.print( r*q1 )
io.write( "q1*r = " ); Quaternion.print( q1*r )
io.write( "q1*q2 = " ); Quaternion.print( q1*q2 )
io.write( "q2*q1 = " ); Quaternion.print( q2*q1 )
Output:
norm(q1) = 5.4772255750517
-q1 = -1.000000 -2.000000i -3.000000j -4.000000k
conj(q1) = 1.000000 -2.000000i -3.000000j -4.000000k
r+q1 = 13.000000 + 2.000000i + 3.000000j + 4.000000k
q1+r = 13.000000 + 2.000000i + 3.000000j + 4.000000k
r*q1 = 12.000000 + 24.000000i + 36.000000j + 48.000000k
q1*r = 12.000000 + 24.000000i + 36.000000j + 48.000000k
q1*q2 = -60.000000 + 12.000000i + 30.000000j + 24.000000k
q2*q1 = -60.000000 + 20.000000i + 14.000000j + 32.000000k
[edit] Mathematica
<<Quaternions`
q=Quaternion[1,2,3,4]
q1=Quaternion[2,3,4,5]
q2=Quaternion[3,4,5,6]
r=7
->Quaternion[1,2,3,4]
->Quaternion[2,3,4,5]
->Quaternion[3,4,5,6]
->7
Norm[q]
->30
-q
->Quaternion[-1,-2,-3,-4]
Conjugate[q]
->Quaternion[1,-2,-3,-4]
q1+q2
->Quaternion[5,7,9,11]
q*r
->Quaternion[7,14,21,28]
r*q
->Quaternion[7,14,21,28]
q1**q2
->Quaternion[-56,16,24,26]
q2**q1
->Quaternion[-56,18,20,28]
[edit] OCaml
The implementation as a file q.ml:
type quaternion = float * float * float * float
let q a b c d = (a, b, c, d)
let to_real (r, _, _, _) = r
let imag (_, i, j, k) = (i, j, k)
let quaternion_of_scalar s = (s, 0.0, 0.0, 0.0)
let to_list (a, b, c, d) = [a; b; c; d]
let of_list = function [a; b; c; d] -> (a, b, c, d)
| _ -> invalid_arg "of_list"
let ( + ) = ( +. )
let ( - ) = ( -. )
let ( * ) = ( *. )
let ( / ) = ( /. )
let addr (a, b, c, d) r = (a+r, b, c, d)
let mulr (a, b, c, d) r = (a*r, b*r, c*r, d*r)
let add (a, b, c, d) (p, q, r, s) = (a+p, b+q, c+r, d+s)
let sub (a, b, c, d) (p, q, r, s) = (a-p, b-q, c-r, d-s)
let mul (a, b, c, d) (p, q, r, s) =
( a*p - b*q - c*r - d*s,
a*q + b*p + c*s - d*r,
a*r - b*s + c*p + d*q,
a*s + b*r - c*q + d*p )
let norm2 (a, b, c, d) =
( a * a +
b * b +
c * c +
d * d )
let norm q = sqrt(norm2 q)
let conj (a, b, c, d) = (a, -. b, -. c, -. d)
let neg (a, b, c, d) = (-. a, -. b, -. c, -. d)
let unit ((a, b, c, d) as q) =
let n = norm q in
(a/n, b/n, c/n, d/n)
let reciprocal ((a, b, c, d) as q) =
let n2 = norm2 q in
(a/n2, b/n2, c/n2, d/n2)
and the interface as a file q.mli:
type quaternion = float * float * float * float
val q : float -> float -> float -> float -> quaternion
val to_real : quaternion -> float
val imag : quaternion -> float * float * float
val quaternion_of_scalar : float -> quaternion
val to_list : quaternion -> float list
val of_list : float list -> quaternion
val addr : quaternion -> float -> quaternion
val mulr : quaternion -> float -> quaternion
val add : quaternion -> quaternion -> quaternion
val sub : quaternion -> quaternion -> quaternion
val mul : quaternion -> quaternion -> quaternion
val norm : quaternion -> float
val conj : quaternion -> quaternion
val neg : quaternion -> quaternion
val unit : quaternion -> quaternion
val reciprocal : quaternion -> quaternion
using this module in the interactive interpreter:
$ ocamlc -c q.mli
$ ocamlc -c q.ml
$ ocaml q.cmo
Objective Caml version 3.11.2
# open Q ;;
# let q1 = q 2.0 3.0 4.0 5.0
and q2 = q 3.0 4.0 5.0 6.0 ;;
val q1 : Q.quaternion = (2., 3., 4., 5.)
val q2 : Q.quaternion = (3., 4., 5., 6.)
# (mul q1 q2) <> (mul q2 q1) ;;
- : bool = true
[edit] ooRexx
Note, this example uses operator overloads to perform the math operation. The operator overloads only work if the left-hand-side of the operation is a quaterion instance. Thus something like "7 + q1" would not work because this would get passed to the "+" of the string class. For those situations, the best solution would be an addition method on the .Quaternion class itself that took the appropriate action. I've chosen not to implement those to keep the example shorter.
q = .quaternion~new(1, 2, 3, 4)
q1 = .quaternion~new(2, 3, 4, 5)
q2 = .quaternion~new(3, 4, 5, 6)
r = 7
say "q =" q
say "q1 =" q1
say "q2 =" q2
say "r =" r
say "norm(q) =" q~norm
say "-q =" (-q)
say "q* =" q~conjugate
say "q + r =" q + r
say "q1 + q2 =" q1 + q2
say "q * r =" q * r
q1q2 = q1 * q2
q2q1 = q2 * q1
say "q1 * q2 =" q1q2
say "q2 * q1 =" q2q1
say "q1 == q1 =" (q1 == q1)
say "q1q2 == q2q1 =" (q1q2 == q2q1)
::class quaternion
::method init
expose r i j k
use strict arg r, i = 0, j = 0, k = 0
-- quaternion instances are immutable, so these are
-- read only attributes
::attribute r GET
::attribute i GET
::attribute j GET
::attribute k GET
::method norm
expose r i j k
return rxcalcsqrt(r * r + i * i + j * j + k * k)
::method invert
expose r i j k
norm = self~norm
return self~class~new(r / norm, i / norm, j / norm, k / norm)
::method negative
expose r i j k
return self~class~new(-r, -i, -j, -k)
::method conjugate
expose r i j k
return self~class~new(r, -i, -j, -k)
::method add
expose r i j k
use strict arg other
if other~isa(.quaternion) then
return self~class~new(r + other~r, i + other~i, j + other~j, k + other~k)
else return self~class~new(r + other, i, j, k)
::method subtract
expose r i j k
use strict arg other
if other~isa(.quaternion) then
return self~class~new(r - other~r, i - other~i, j - other~j, k - other~k)
else return self~class~new(r - other, i, j, k)
::method times
expose r i j k
use strict arg other
if other~isa(.quaternion) then
return self~class~new(r * other~r - i * other~i - j * other~j - k * other~k, -
r * other~i + i * other~r + j * other~k - k * other~j, -
r * other~j - i * other~k + j * other~r + k * other~i, -
r * other~k + i * other~j - j * other~i + k * other~r)
else return self~class~new(r * other, i * other, j * other, k * other)
::method divide
use strict arg other
-- this is easier if everything is a quaternion
if \other~isA(.quaternion) then other = .quaternion~new(other)
-- division is multiplication with the inversion
return self * other~invert
::method "=="
expose r i j k
use strict arg other
if \other~isa(.quaternion) then return .false
-- Note: these are numeric comparisons, so we're using the "="
-- method so those are handled correctly
return r = other~r & i = other~i & j = other~j & k = other~k
::method "\=="
use strict arg other
return \self~"\=="(other)
::method "="
-- this is equivalent of "=="
forward message("==")
::method "\="
-- this is equivalent of "\=="
forward message("\==")
::method "<>"
-- this is equivalent of "\=="
forward message("\==")
::method "><"
-- this is equivalent of "\=="
forward message("\==")
-- some operator overrides -- these only work if the left-hand-side of the
-- subexpression is a quaternion
::method "*"
forward message("TIMES")
::method "/"
forward message("DIVIDE")
::method "-"
-- need to check if this is a prefix minus or a subtract
if arg() == 0 then
forward message("NEGATIVE")
else
forward message("SUBTRACT")
::method "+"
-- need to check if this is a prefix plus or an addition
if arg() == 0 then
return self -- we can return this copy since it is immutable
else
forward message("ADD")
::method string
expose r i j k
return r self~formatnumber(i)"i" self~formatnumber(j)"j" self~formatnumber(k)"k"
::method formatnumber private
use arg value
if value > 0 then return "+" value
else return "-" value~abs
-- override hashcode for collection class hash uses
::method hashCode
expose r i j k
return r~hashcode~bitxor(i~hashcode)~bitxor(j~hashcode)~bitxor(k~hashcode)
::requires rxmath LIBRARY
q = 1 + 2i + 3j + 4k q1 = 2 + 3i + 4j + 5k q2 = 3 + 4i + 5j + 6k r = 7 norm(q) = 5.47722558 -q = -1 - 2i - 3j - 4k q* = 1 - 2i - 3j - 4k q + r = 8 + 2i + 3j + 4k q1 + q2 = 5 + 7i + 9j + 11k q * r = 7 + 14i + 21j + 28k q1 * q2 = -56 + 16i + 24j + 26k q2 * q1 = -56 + 18i + 20j + 28k q1 == q1 = 1 q1q2 == q2q1 = 0
[edit] PARI/GP
Here is a simple solution in GP. I think it's possible to implement this type directly in Pari by abusing t_COMPLEX, but I haven't attempted this.
q.norm={
if(type(q) != "t_VEC" || #q != 4, error("incorrect type"));
sqrt(q[1]^2+q[2]^2+q[3]^2+q[4]^2)
};
q.conj={
if(type(q) != "t_VEC" || #q != 4, error("incorrect type"));
-[-q[1],q[2],q[3],q[4]]
};
q.add={
if(type(q) != "t_VEC" || #q != 4, error("incorrect type"));
x->if(type(x) == "t_INT" || type(x) == t_REAL,
[q[1]+x,q[2],q[3],q[4]]
,
if(type(x) == "t_VEC" && #x == 4,
q+x
,
error("incorrect type")
)
)
};
q.mult={
if(type(q) != "t_VEC" || #q != 4, error("incorrect type"));
x->if(type(x) == "t_INT" || type(x) == t_REAL,
x*q
,
if(type(x) == "t_VEC" && #x == 4,
[q[1]*x[1] - q[2]*x[2] - q[3]*x[3] - q[4]*x[4],
q[1]*x[2] + q[2]*x[1] + q[3]*x[4] - q[4]*x[3],
q[1]*x[3] - q[2]*x[4] + q[3]*x[1] + q[4]*x[2],
q[1]*x[4] + q[2]*x[3] - q[3]*x[2] + q[4]*x[1]]
,
error("incorrect type")
)
)
};
Usage:
r=7;q=[1,2,3,4];q1=[2,3,4,5];q2=[3,4,5,6];
q.norm
-q
q.conj
q.add(r)
q1.add(q2)
q1.add(q2) \\ or q1+q2
q.mult(r) \\ or r*q or q*r
q1.mult(q2)
q1.mult(q2) != q2.mult(q1)
[edit] Pascal
The Delphi example also works with FreePascal.
[edit] Perl
package Quaternion;
use List::Util 'reduce';
use List::MoreUtils 'pairwise';
sub make {
my $cls = shift;
if (@_ == 1) { return bless [ @_, 0, 0, 0 ] }
elsif (@_ == 4) { return bless [ @_ ] }
else { die "Bad number of components: @_" }
}
sub _abs { sqrt reduce { $a + $b * $b } @{ +shift } }
sub _neg { bless [ map(-$_, @{+shift}) ] }
sub _str { "(@{+shift})" }
sub _add {
my ($x, $y) = @_;
$y = [ $y, 0, 0, 0 ] unless ref $y;
bless [ pairwise { $a + $b } @$x, @$y ]
}
sub _sub {
my ($x, $y, $swap) = @_;
$y = [ $y, 0, 0, 0 ] unless ref $y;
my @x = pairwise { $a - $b } @$x, @$y;
if ($swap) { $_ = -$_ for @x }
bless \@x;
}
sub _mul {
my ($x, $y) = @_;
if (!ref $y) { return bless [ map($_ * $y, @$x) ] }
my ($a1, $b1, $c1, $d1) = @$x;
my ($a2, $b2, $c2, $d2) = @$y;
bless [ $a1 * $a2 - $b1 * $b2 - $c1 * $c2 - $d1 * $d2,
$a1 * $b2 + $b1 * $a2 + $c1 * $d2 - $d1 * $c2,
$a1 * $c2 - $b1 * $d2 + $c1 * $a2 + $d1 * $b2,
$a1 * $d2 + $b1 * $c2 - $c1 * $b2 + $d1 * $a2]
}
sub conjugate {
my @a = map { -$_ } @{$_[0]};
$a[0] = $_[0][0];
bless \@a
}
use overload (
'""' => \&_str,
'+' => \&_add,
'-' => \&_sub,
'*' => \&_mul,
'neg' => \&_neg,
'abs' => \&_abs,
);
package main;
my $a = Quaternion->make(1, 2, 3, 4);
my $b = Quaternion->make(1, 1, 1, 1);
print "a = $a\n";
print "b = $b\n";
print "|a| = ", abs($a), "\n";
print "-a = ", -$a, "\n";
print "a + 1 = ", $a + 1, "\n";
print "a + b = ", $a + $b, "\n";
print "a - b = ", $a - $b, "\n";
print "a conjugate is ", $a->conjugate, "\n";
print "a * b = ", $a * $b, "\n";
print "b * a = ", $b * $a, "\n";
[edit] Perl 6
class Qu {Output:
has Real ( $.r, $.i, $.j, $.k );
multi method new ( Real $r, Real $i, Real $j, Real $k ) {
self.bless: *, :$r, :$i, :$j, :$k;
}
method Str ( ) { "$.r + {$.i}i + {$.j}j + {$.k}k" }
method reals ( ) { $.r, $.i, $.j, $.k }
method conj ( ) { self.new: $.r, -$.i, -$.j, -$.k }
method norm ( ) { sqrt [+] self.reals X** 2 }
}
multi sub infix:<eqv> ( Qu $a, Qu $b ) { [and] $a.reals Z== $b.reals }
multi sub infix:<+> ( Qu $a, Real $b ) { $a.new: $b+$a.r, $a.i, $a.j, $a.k }
multi sub infix:<+> ( Real $a, Qu $b ) { $b.new: $a+$b.r, $b.i, $b.j, $b.k }
multi sub infix:<+> ( Qu $a, Qu $b ) { $a.new: |( $a.reals Z+ $b.reals ) }
multi sub prefix:<-> ( Qu $a ) { $a.new: |( $a.reals X* -1 ) }
multi sub infix:<*> ( Qu $a, Real $b ) { $a.new: |( $a.reals X* $b ) }
multi sub infix:<*> ( Real $a, Qu $b ) { $b.new: |( $b.reals X* $a ) }
multi sub infix:<*> ( Qu $a, Qu $b ) {
my @a_rijk = $a.reals;
my ( $r, $i, $j, $k ) = $b.reals;
return $a.new: ( [+] @a_rijk Z* $r, -$i, -$j, -$k ), # real
( [+] @a_rijk Z* $i, $r, $k, -$j ), # i
( [+] @a_rijk Z* $j, -$k, $r, $i ), # j
( [+] @a_rijk Z* $k, $j, -$i, $r ); # k
}
my Qu $q .= new: 1, 2, 3, 4;
my Qu $q1 .= new: 2, 3, 4, 5;
my Qu $q2 .= new: 3, 4, 5, 6;
my $r = 7;
say "1) q norm = {$q.norm}";
say "2) -q = {-$q}";
say "3) q conj = {$q.conj}";
say "4) q + r = {$q + $r}";
say "5) q1 + q2 = {$q1 + $q2}";
say "6) q * r = {$q * $r}";
say "7) q1 * q2 = {$q1 * $q2}";
say "8) q1q2 { $q1 * $q2 eqv $q2 * $q1 ?? '==' !! '!=' } q2q1";
1) q norm = 5.47722557505166 2) -q = -1 + -2i + -3j + -4k 3) q conj = 1 + -2i + -3j + -4k 4) q + r = 8 + 2i + 3j + 4k 5) q1 + q2 = 5 + 7i + 9j + 11k 6) q * r = 7 + 14i + 21j + 28k 7) q1 * q2 = -56 + 16i + 24j + 26k 8) q1q2 != q2q1
[edit] PicoLisp
(scl 6)
(def 'quatCopy copy)
(de quatNorm (Q)
(sqrt (sum * Q Q)) )
(de quatNeg (Q)
(mapcar - Q) )
(de quatConj (Q)
(cons (car Q) (mapcar - (cdr Q))) )
(de quatAddR (Q R)
(cons (+ R (car Q)) (cdr Q)) )
(de quatAdd (Q1 Q2)
(mapcar + Q1 Q2) )
(de quatMulR (Q R)
(mapcar */ (mapcar * Q (circ R)) (1.0 .)) )
(de quatMul (Q1 Q2)
(mapcar
'((Ops I)
(sum '((Op R I) (Op (*/ R (get Q2 I) 1.0))) Ops Q1 I) )
'((+ - - -) (+ + + -) (+ - + +) (+ + - +))
'((1 2 3 4) (2 1 4 3) (3 4 1 2) (4 3 2 1)) ) )
(de quatFmt (Q)
(mapcar '((R S) (pack (format R *Scl) S))
Q
'(" + " "i + " "j + " "k") ) )
Test:
(setq
Q (1.0 2.0 3.0 4.0)
Q1 (2.0 3.0 4.0 5.0)
Q2 (3.0 4.0 5.0 6.0)
R 7.0 )
(prinl "R = " (format R *Scl))
(prinl "Q = " (quatFmt Q))
(prinl "Q1 = " (quatFmt Q1))
(prinl "Q2 = " (quatFmt Q2))
(prinl)
(prinl "norm(Q) = " (format (quatNorm Q) *Scl))
(prinl "norm(Q1) = " (format (quatNorm Q1) *Scl))
(prinl "norm(Q2) = " (format (quatNorm Q2) *Scl))
(prinl "neg(Q) = " (quatFmt (quatNeg Q)))
(prinl "conj(Q) = " (quatFmt (quatConj Q)))
(prinl "Q + R = " (quatFmt (quatAddR Q R)))
(prinl "Q1 + Q2 = " (quatFmt (quatAdd Q1 Q2)))
(prinl "Q * R = " (quatFmt (quatMulR Q R)))
(prinl "Q1 * Q2 = " (quatFmt (quatMul Q1 Q2)))
(prinl "Q2 * Q1 = " (quatFmt (quatMul Q2 Q1)))
(prinl (if (= (quatMul Q1 Q2) (quatMul Q2 Q1)) "Equal" "Not equal"))
Output:
R = 7.000000 Q = 1.000000 + 2.000000i + 3.000000j + 4.000000k Q1 = 2.000000 + 3.000000i + 4.000000j + 5.000000k Q2 = 3.000000 + 4.000000i + 5.000000j + 6.000000k norm(Q) = 5.477225 norm(Q1) = 7.348469 norm(Q2) = 9.273618 neg(Q) = -1.000000 + -2.000000i + -3.000000j + -4.000000k conj(Q) = 1.000000 + -2.000000i + -3.000000j + -4.000000k Q + R = 8.000000 + 2.000000i + 3.000000j + 4.000000k Q1 + Q2 = 5.000000 + 7.000000i + 9.000000j + 11.000000k Q * R = 7.000000 + 14.000000i + 21.000000j + 28.000000k Q1 * Q2 = -56.000000 + 16.000000i + 24.000000j + 26.000000k Q2 * Q1 = -56.000000 + 18.000000i + 20.000000j + 28.000000k Not equal
[edit] Prolog
% A quaternion is represented as a complex term qx/4
add(qx(R0,I0,J0,K0), qx(R1,I1,J1,K1), qx(R,I,J,K)) :-
!, R is R0+R1, I is I0+I1, J is J0+J1, K is K0+K1.
add(qx(R0,I,J,K), F, qx(R,I,J,K)) :-
number(F), !, R is R0 + F.
add(F, qx(R0,I,J,K), Qx) :-
add(qx(R0,I,J,K), F, Qx).
mul(qx(R0,I0,J0,K0), qx(R1,I1,J1,K1), qx(R,I,J,K)) :- !,
R is R0*R1 - I0*I1 - J0*J1 - K0*K1,
I is R0*I1 + I0*R1 + J0*K1 - K0*J1,
J is R0*J1 - I0*K1 + J0*R1 + K0*I1,
K is R0*K1 + I0*J1 - J0*I1 + K0*R1.
mul(qx(R0,I0,J0,K0), F, qx(R,I,J,K)) :-
number(F), !, R is R0*F, I is I0*F, J is J0*F, K is K0*F.
mul(F, qx(R0,I0,J0,K0), Qx) :-
mul(qx(R0,I0,J0,K0),F,Qx).
abs(qx(R,I,J,K), Norm) :-
Norm is sqrt(R*R+I*I+J*J+K*K).
negate(qx(Ri,Ii,Ji,Ki),qx(R,I,J,K)) :-
R is -Ri, I is -Ii, J is -Ji, K is -Ki.
conjugate(qx(R,Ii,Ji,Ki),qx(R,I,J,K)) :-
I is -Ii, J is -Ji, K is -Ki.
Test:
data(q, qx(1,2,3,4)).
data(q1, qx(2,3,4,5)).
data(q2, qx(3,4,5,6)).
data(r, 7).
test :- data(Name, qx(A,B,C,D)), abs(qx(A,B,C,D), Norm),
writef('abs(%w) is %w\n', [Name, Norm]), fail.
test :- data(q, Qx), negate(Qx, Nqx),
writef('negate(%w) is %w\n', [q, Nqx]), fail.
test :- data(q, Qx), conjugate(Qx, Nqx),
writef('conjugate(%w) is %w\n', [q, Nqx]), fail.
test :- data(q1, Q1), data(q2, Q2), add(Q1, Q2, Qx),
writef('q1+q2 is %w\n', [Qx]), fail.
test :- data(q1, Q1), data(q2, Q2), add(Q2, Q1, Qx),
writef('q2+q1 is %w\n', [Qx]), fail.
test :- data(q, Qx), data(r, R), mul(Qx, R, Nqx),
writef('q*r is %w\n', [Nqx]), fail.
test :- data(q, Qx), data(r, R), mul(R, Qx, Nqx),
writef('r*q is %w\n', [Nqx]), fail.
test :- data(q1, Q1), data(q2, Q2), mul(Q1, Q2, Qx),
writef('q1*q2 is %w\n', [Qx]), fail.
test :- data(q1, Q1), data(q2, Q2), mul(Q2, Q1, Qx),
writef('q2*q1 is %w\n', [Qx]), fail.
test.
Output:
?- test. abs(q) is 5.477225575051661 abs(q1) is 7.3484692283495345 abs(q2) is 9.273618495495704 negate(q) is qx(-1,-2,-3,-4) conjugate(q) is qx(1,-2,-3,-4) q1+q2 is qx(5,7,9,11) q2+q1 is qx(5,7,9,11) q*r is qx(7,14,21,28) r*q is qx(7,14,21,28) q1*q2 is qx(-56,16,24,26) q2*q1 is qx(-56,18,20,28)
[edit] PureBasic
Structure Quaternion
a.f
b.f
c.f
d.f
EndStructure
Procedure.f QNorm(*x.Quaternion)
ProcedureReturn Sqr(Pow(*x\a, 2) + Pow(*x\b, 2) + Pow(*x\c, 2) + Pow(*x\d, 2))
EndProcedure
;If supplied, the result is returned in the quaternion structure *res,
;otherwise a new quaternion is created. A pointer to the result is returned.
Procedure QNeg(*x.Quaternion, *res.Quaternion = 0)
If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf
If *res
*res\a = -*x\a
*res\b = -*x\b
*res\c = -*x\c
*res\d = -*x\d
EndIf
ProcedureReturn *res
EndProcedure
Procedure QConj(*x.Quaternion, *res.Quaternion = 0)
If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf
If *res
*res\a = *x\a
*res\b = -*x\b
*res\c = -*x\c
*res\d = -*x\d
EndIf
ProcedureReturn *res
EndProcedure
Procedure QAddReal(r.f, *x.Quaternion, *res.Quaternion = 0)
If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf
If *res
*res\a = *x\a + r
*res\b = *x\b
*res\c = *x\c
*res\d = *x\d
EndIf
ProcedureReturn *res
EndProcedure
Procedure QAddQuaternion(*x.Quaternion, *y.Quaternion, *res.Quaternion = 0)
If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf
If *res
*res\a = *x\a + *y\a
*res\b = *x\b + *y\b
*res\c = *x\c + *y\c
*res\d = *x\d + *y\d
EndIf
ProcedureReturn *res
EndProcedure
Procedure QMulReal_and_Quaternion(r.f, *x.Quaternion, *res.Quaternion = 0)
If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf
If *res
*res\a = *x\a * r
*res\b = *x\b * r
*res\c = *x\c * r
*res\d = *x\d * r
EndIf
ProcedureReturn *res
EndProcedure
Procedure QMulQuaternion(*x.Quaternion, *y.Quaternion, *res.Quaternion = 0)
If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf
If *res
*res\a = *x\a * *y\a - *x\b * *y\b - *x\c * *y\c - *x\d * *y\d
*res\b = *x\a * *y\b + *x\b * *y\a + *x\c * *y\d - *x\d * *y\c
*res\c = *x\a * *y\c - *x\b * *y\d + *x\c * *y\a + *x\d * *y\b
*res\d = *x\a * *y\d + *x\b * *y\c - *x\c * *y\b + *x\d * *y\a
EndIf
ProcedureReturn *res
EndProcedure
Procedure Q_areEqual(*x.Quaternion, *y.Quaternion)
If (*x\a <> *y\a) Or (*x\b <> *y\b) Or (*x\c <> *y\c) Or (*x\d <> *y\d)
ProcedureReturn 0 ;false
EndIf
ProcedureReturn 1 ;true
EndProcedure
Implementation & test
Procedure.s ShowQ(*x.Quaternion, NN = 0)
ProcedureReturn "{" + StrF(*x\a, NN) + "," + StrF(*x\b, NN) + "," + StrF(*x\c, NN) + "," + StrF(*x\d, NN) + "}"
EndProcedure
If OpenConsole()
Define.Quaternion Q0, Q1, Q2, res, res2
Define.f r = 7
Q0\a = 1: Q0\b = 2: Q0\c = 3: Q0\d = 4
Q1\a = 2: Q1\b = 3: Q1\c = 4: Q1\d = 5
Q2\a = 3: Q2\b = 4: Q2\c = 5: Q2\d = 6
PrintN("Q0 = " + ShowQ(Q0, 0))
PrintN("Q1 = " + ShowQ(Q1, 0))
PrintN("Q2 = " + ShowQ(Q2, 0))
PrintN("Normal of Q0 = " + StrF(QNorm(Q0)))
PrintN("Neg(Q0) = " + ShowQ(QNeg(Q0, res)))
PrintN("Conj(Q0) = " + ShowQ(QConj(Q0, res)))
PrintN("r + Q0 = " + ShowQ(QAddReal(r, Q0, res)))
PrintN("Q0 + Q1 = " + ShowQ(QAddQuaternion(Q0, Q1, res)))
PrintN("Q1 + Q2 = " + ShowQ(QAddQuaternion(Q1, Q2, res)))
PrintN("Q1 * Q2 = " + ShowQ(QMulQuaternion(Q1, Q2, res)))
PrintN("Q2 * Q1 = " + ShowQ(QMulQuaternion(Q2, Q1, res2)))
Print( "Q1 * Q2"): If Q_areEqual(res, res2): Print(" = "): Else: Print(" <> "): EndIf: Print( "Q2 * Q1")
Print(#CRLF$ + #CRLF$ + "Press ENTER to exit"): Input()
CloseConsole()
EndIf
Result
Q0 = {1,2,3,4}
Q1 = {2,3,4,5}
Q2 = {3,4,5,6}
Normal of Q0 = 5.4772257805
Neg(Q0) = {-1,-2,-3,-4}
Conj(Q0) = {1,-2,-3,-4}
r + Q0 = {8,2,3,4}
Q0 + Q1 = {3,5,7,9}
Q1 + Q2 = {5,7,9,11}
Q1 * Q2 = {-56,16,24,26}
Q2 * Q1 = {-56,18,20,28}
Q1 * Q2 <> Q2 * Q1
[edit] Python
This example extends Pythons namedtuples to add extra functionality.
from collections import namedtuple
import math
class Q(namedtuple('Quaternion', 'real, i, j, k')):
'Quaternion type: Q(real=0.0, i=0.0, j=0.0, k=0.0)'
__slots__ = ()
def __new__(_cls, real=0.0, i=0.0, j=0.0, k=0.0):
'Defaults all parts of quaternion to zero'
return super().__new__(_cls, float(real), float(i), float(j), float(k))
def conjugate(self):
return Q(self.real, -self.i, -self.j, -self.k)
def _norm2(self):
return sum( x*x for x in self)
def norm(self):
return math.sqrt(self._norm2())
def reciprocal(self):
n2 = self._norm2()
return Q(*(x / n2 for x in self.conjugate()))
def __str__(self):
'Shorter form of Quaternion as string'
return 'Q(%g, %g, %g, %g)' % self
def __neg__(self):
return Q(-self.real, -self.i, -self.j, -self.k)
def __add__(self, other):
if type(other) == Q:
return Q( *(s+o for s,o in zip(self, other)) )
try:
f = float(other)
except:
return NotImplemented
return Q(self.real + f, self.i, self.j, self.k)
def __radd__(self, other):
return Q.__add__(self, other)
def __mul__(self, other):
if type(other) == Q:
a1,b1,c1,d1 = self
a2,b2,c2,d2 = other
return Q(
a1*a2 - b1*b2 - c1*c2 - d1*d2,
a1*b2 + b1*a2 + c1*d2 - d1*c2,
a1*c2 - b1*d2 + c1*a2 + d1*b2,
a1*d2 + b1*c2 - c1*b2 + d1*a2 )
try:
f = float(other)
except:
return NotImplemented
return Q(self.real * f, self.i * f, self.j * f, self.k * f)
def __rmul__(self, other):
return Q.__mul__(self, other)
def __truediv__(self, other):
if type(other) == Q:
return self.__mul__(other.reciprocal())
try:
f = float(other)
except:
return NotImplemented
return Q(self.real / f, self.i / f, self.j / f, self.k / f)
def __rtruediv__(self, other):
return other * self.reciprocal()
__div__, __rdiv__ = __truediv__, __rtruediv__
Quaternion = Q
q = Q(1, 2, 3, 4)
q1 = Q(2, 3, 4, 5)
q2 = Q(3, 4, 5, 6)
r = 7
Continued shell session Run the above with the -i flag to python on the command line, or run with idle then continue in the shell as follows:
>>> q
Quaternion(real=1.0, i=2.0, j=3.0, k=4.0)
>>> q1
Quaternion(real=2.0, i=3.0, j=4.0, k=5.0)
>>> q2
Quaternion(real=3.0, i=4.0, j=5.0, k=6.0)
>>> r
7
>>> q.norm()
5.477225575051661
>>> q1.norm()
7.3484692283495345
>>> q2.norm()
9.273618495495704
>>> -q
Quaternion(real=-1.0, i=-2.0, j=-3.0, k=-4.0)
>>> q.conjugate()
Quaternion(real=1.0, i=-2.0, j=-3.0, k=-4.0)
>>> r + q
Quaternion(real=8.0, i=2.0, j=3.0, k=4.0)
>>> q + r
Quaternion(real=8.0, i=2.0, j=3.0, k=4.0)
>>> q1 + q2
Quaternion(real=5.0, i=7.0, j=9.0, k=11.0)
>>> q2 + q1
Quaternion(real=5.0, i=7.0, j=9.0, k=11.0)
>>> q * r
Quaternion(real=7.0, i=14.0, j=21.0, k=28.0)
>>> r * q
Quaternion(real=7.0, i=14.0, j=21.0, k=28.0)
>>> q1 * q2
Quaternion(real=-56.0, i=16.0, j=24.0, k=26.0)
>>> q2 * q1
Quaternion(real=-56.0, i=18.0, j=20.0, k=28.0)
>>> assert q1 * q2 != q2 * q1
>>>
>>> i, j, k = Q(0,1,0,0), Q(0,0,1,0), Q(0,0,0,1)
>>> i*i
Quaternion(real=-1.0, i=0.0, j=0.0, k=0.0)
>>> j*j
Quaternion(real=-1.0, i=0.0, j=0.0, k=0.0)
>>> k*k
Quaternion(real=-1.0, i=0.0, j=0.0, k=0.0)
>>> i*j*k
Quaternion(real=-1.0, i=0.0, j=0.0, k=0.0)
>>> q1 / q2
Quaternion(real=0.7906976744186047, i=0.023255813953488358, j=-2.7755575615628914e-17, k=0.046511627906976744)
>>> q1 / q2 * q2
Quaternion(real=2.0000000000000004, i=3.0000000000000004, j=4.000000000000001, k=5.000000000000001)
>>> q2 * q1 / q2
Quaternion(real=2.0, i=3.465116279069768, j=3.906976744186047, k=4.767441860465116)
>>> q1.reciprocal() * q1
Quaternion(real=0.9999999999999999, i=0.0, j=0.0, k=0.0)
>>> q1 * q1.reciprocal()
Quaternion(real=0.9999999999999999, i=0.0, j=0.0, k=0.0)
>>>
[edit] R
Using the quaternions package.
library(quaternions)
q <- Q(1, 2, 3, 4)
q1 <- Q(2, 3, 4, 5)
q2 <- Q(3, 4, 5, 6)
r <- 7.0
display <- function(x){
e <- deparse(substitute(x))
res <- if(class(x) == "Q") paste(x$r, "+", x$i, "i+", x$j, "j+", x$k, "k", sep = "") else x
cat(noquote(paste(c(e, " = ", res, "\n"), collapse="")))
invisible(res)
}
display(norm(q))
display(-q)
display(Conj(q))
display(r + q)
display(q1 + q2)
display(r*q)
display(q*r)
if(display(q1*q2) == display(q2*q1)) cat("q1*q2 == q2*q1\n") else cat("q1*q2 != q2*q1\n")
## norm(q) = 5.47722557505166
## -q = -1+-2i+-3j+-4k
## Conj(q) = 1+-2i+-3j+-4k
## r + q = 8+2i+3j+4k
## q1 + q2 = 5+7i+9j+11k
## r * q = 7+14i+21j+28k
## q * r = 7+14i+21j+28k
## q1 * q2 = -56+16i+24j+26k
## q2 * q1 = -56+18i+20j+28k
## q1*q2 != q2*q1
[edit] Racket
#lang racket
(struct quaternion (a b c d)
#:transparent)
(define-match-expander quaternion:
(λ (stx)
(syntax-case stx ()
[(_ a b c d)
#'(or (quaternion a b c d)
(and a (app (λ(_) 0) b) (app (λ(_) 0) c) (app (λ(_) 0) d)))])))
(define (norm q)
(match q
[(quaternion: a b c d)
(sqrt (+ (sqr a) (sqr b) (sqr c) (sqr d)))]))
(define (negate q)
(match q
[(quaternion: a b c d)
(quaternion (- a) (- b) (- c) (- d))]))
(define (conjugate q)
(match q
[(quaternion: a b c d)
(quaternion a (- b) (- c) (- d))]))
(define (add q1 q2 . q-rest)
(let ((ans (match* (q1 q2)
[((quaternion: a1 b1 c1 d1) (quaternion: a2 b2 c2 d2))
(quaternion (+ a1 a2) (+ b1 b2) (+ c1 c2) (+ d1 d2))])))
(if (empty? q-rest)
ans
(apply add (cons ans q-rest)))))
(define (multiply q1 q2 . q-rest)
(let ((ans (match* (q1 q2)
[((quaternion: a1 b1 c1 d1) (quaternion: a2 b2 c2 d2))
(quaternion (- (* a1 a2) (* b1 b2) (* c1 c2) (* d1 d2))
(+ (* a1 b2) (* b1 a2) (* c1 d2) (- (* d1 c2)))
(+ (* a1 c2) (- (* b1 d2)) (* c1 a2) (* d1 b2))
(+ (* a1 d2) (* b1 c2) (- (* c1 b2)) (* d1 a2)))])))
(if (empty? q-rest)
ans
(apply multiply (cons ans q-rest)))))
;; Tests
(module+ main
(define i (quaternion 0 1 0 0))
(define j (quaternion 0 0 1 0))
(define k (quaternion 0 0 0 1))
(displayln (multiply i j k))
(newline)
(define q (quaternion 1 2 3 4))
(define q1 (quaternion 2 3 4 5))
(define q2 (quaternion 3 4 5 6))
(define r 7)
(for ([quat (list q q1 q2)])
(displayln quat)
(displayln (norm quat))
(displayln (negate quat))
(displayln (conjugate quat))
(newline))
(add r q)
(add q1 q2)
(multiply r q)
(newline)
(multiply q1 q2)
(multiply q2 q1)
(equal? (multiply q1 q2)
(multiply q2 q1)))
Output:
#(struct:quaternion -1 0 0 0) #(struct:quaternion 1 2 3 4) 5.477225575051661 #(struct:quaternion -1 -2 -3 -4) #(struct:quaternion 1 -2 -3 -4) #(struct:quaternion 2 3 4 5) 7.3484692283495345 #(struct:quaternion -2 -3 -4 -5) #(struct:quaternion 2 -3 -4 -5) #(struct:quaternion 3 4 5 6) 9.273618495495704 #(struct:quaternion -3 -4 -5 -6) #(struct:quaternion 3 -4 -5 -6) (quaternion 8 2 3 4) (quaternion 5 7 9 11) (quaternion 7 14 21 28) (quaternion -56 16 24 26) (quaternion -56 18 20 28) #f
[edit] REXX
The REXX language has no native quaternion support, but subroutines can be easily written.
/*REXX program to perform simple operations of quaternion type numbers.*/
q = 1 2 3 4 ; q1 = 2 3 4 5
r = 7 ; q2 = 3 4 5 6
call quatShow q , 'q'
call quatShow q1 , 'q1'
call quatShow q2 , 'q2'
call quatShow r , 'r'
call quatShow quatNorm(q) , 'norm q' , "task 1:"
call quatShow quatNeg(q) , 'negative q' , "task 2:"
call quatShow quatConj(q) , 'conjugate q' , "task 3:"
call quatShow quatAdd( r, q ) , 'addition r+q' , "task 4:"
call quatShow quatAdd(q1, q2 ) , 'addition q1+q2' , "task 5:"
call quatShow quatMul( q, r ) , 'multiplication q*r' , "task 6:"
call quatShow quatMul(q1, q2 ) , 'multiplication q1*q2' , "task 7:"
call quatShow quatMul(q2, q1 ) , 'multiplication q2*q1' , "task 8:"
exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────QUATADD─────────────────────────────*/
quatAdd: procedure; parse arg x,y; call quatXY 2
return x.1+y.1 x.2+y.2 x.3+y.3 x.4+y.4
/*──────────────────────────────────QUATCONJ────────────────────────────*/
quatConj: procedure; parse arg x; call quatXY
return x.1 (-x.2) (-x.3) (-x.4)
/*──────────────────────────────────QUATMUL─────────────────────────────*/
quatMul: procedure; parse arg x,y; call quatXY y
return x.1*y.1-x.2*y.2-x.3*y.3-x.4*y.4 x.1*y.2+x.2*y.1+x.3*y.4-x.4*y.3,
x.1*y.3-x.2*y.4+x.3*y.1+x.4*y.2 x.1*y.4+x.2*y.3-x.3*y.2+x.4*y.1
/*──────────────────────────────────QUATNEG─────────────────────────────*/
quatNeg: procedure; parse arg x; call quatXY
return -x.1 (-x.2) (-x.3) (-x.4)
/*──────────────────────────────────QUATNORM────────────────────────────*/
quatNorm: procedure; parse arg x; call quatXY
return sqrt(x.1**2 + x.2**2 + x.3**2 + x.4**2)
/*──────────────────────────────────QUATSHOW────────────────────────────*/
quatShow: procedure; parse arg x; call quatXY; quat=
do m=1 for 4; _=x.m; if _==0 then iterate; if _ >=0 then _='+'_
if m\==1 then _=_||substr('~ijk',m,1) ; quat=strip(quat || _,,'+')
end /*m*/
say left(arg(3),9) right(arg(2),20) ' ──► ' quat
return quat
/*──────────────────────────────────QUATXY──────────────────────────────*/
quatXY: do n=1 for 4; x.n=word(word(x,n) 0,1)/1; end /*n*/
if arg()==1 then do m=1 for 4; y.m=word(word(y,m) 0,1)/1; end /*m*/
return
/*──────────────────────────────────SQRT subroutine─────────────────────*/
sqrt: procedure;parse arg x;if x=0 then return 0;d=digits();numeric digits 11
m.=11;numeric form;p=d+d%4+2;parse value format(x,2,1,,0) 'E0' with g 'E' _ .
g=g*.5'E'_%2; do j=0 while p>9; m.j=p; p=p%2+1; end; do k=j+5 to 0 by -1
if m.k>11 then numeric digits m.k;g=.5*(g+x/g);end;numeric digits d;return g/1
output when using the default input
q ──► 1+2i+3j+4k
q1 ──► 2+3i+4j+5k
q2 ──► 3+4i+5j+6k
r ──► 7
task 1: norm q ──► 5.47722558
task 2: negative q ──► -1-2i-3j-4k
task 3: conjugate q ──► 1-2i-3j-4k
task 4: addition r+q ──► 8+2i+3j+4k
task 5: addition q1+q2 ──► 5+7i+9j+11k
task 6: multiplication q*r ──► 7+14i+21j+28k
task 7: multiplication q1*q2 ──► -56+16i+24j+26k
task 8: multiplication q2*q1 ──► -56+18i+20j+28k
[edit] Ruby
require 'matrix' # For Vector#norm
class Quaternion
def initialize(*parts)
raise "Invalid number of quaternion parts" unless parts.length == 4
@parts, @vector = parts, Vector[*parts]
end
def to_a; @parts; end
def to_s; "Quaternion#{to_a.to_s}" end
def complex_parts; [Complex(*to_a[0..1]), Complex(*to_a[2..3])]; end
def zip(other); to_a.zip(other.to_a); end
def real; @parts.first; end
def imag; @parts[1..3]; end
def conj; Quaternion.new(real, *imag.map(&:-@)); end
def norm; @vector.norm; end # Or: Math.sqrt(to_a.reduce { |sum, e| sum + e**2 }) # In Rails: Math.sqrt(to_a.sum { e**2 })
def ==(other); to_a == other.to_a end
def -@; Quaternion.new(*to_a.map(&:-@)); end
def -(other); self + -other; end
def +(other)
case other
when Numeric
Quaternion.new(real + other, *imag)
when Quaternion
Quaternion.new(*zip(other).map { |x,y| x + y }) # In Rails: zip(other).map(&:sum) # Or: (vector + other.vector).to_a
end
end
def *(other)
case other
when Numeric
Quaternion.new(*to_a.map { |x| x * other }) # Or: (vector * other).to_a
when Quaternion
# Multiplication of quaternions in C x C space. See "Cayley-Dickson construction".
a, b, c, d = *complex_parts, *other.complex_parts
x, y = a*c - d.conj*b, a*d + b*c.conj
Quaternion.new(x.real, x.imag, y.real, y.imag)
end
end
# Coerce is called by Ruby to return a compatible type/receiver when the called method/operation does not accept a Quaternion
def coerce(other)
case other
when Numeric then [Scalar.new(other), self]
else raise TypeError, "#{other.class} can't be coerced into #{self.class}"
end
end
class Scalar
def initialize(val); @val = val; end
def +(other); other + @val; end
def *(other); other * @val; end
def -(other); Quaternion.new(@val, 0, 0, 0) - other; end
end
end
IRB session
irb(main):001:0> require 'quaternion'
=> true
irb(main):002:0> q = Quaternion.new(1,2,3,4)
=> Quaternion[1, 2, 3, 4]
irb(main):003:0> q1 = Quaternion.new(2,3,4,5)
=> Quaternion[2, 3, 4, 5]
irb(main):004:0> q2 = Quaternion.new(3,4,5,6)
=> Quaternion[3, 4, 5, 6]
irb(main):005:0> r = 7
=> 7
irb(main):006:0> q.norm
=> 5.477225575051661
irb(main):007:0> q1.norm
=> 7.3484692283495345
irb(main):008:0> q2.norm
=> 9.273618495495704
irb(main):009:0> -q
=> Quaternion[-1, -2, -3, -4]
irb(main):010:0> q.conj
=> Quaternion[1, -2, -3, -4]
irb(main):011:0> q1 + q2
=> Quaternion[5, 7, 9, 11]
irb(main):012:0> q2 + q1
=> Quaternion[5, 7, 9, 11]
irb(main):013:0> q + r
=> Quaternion[8, 2, 3, 4]
irb(main):014:0> r + q
=> Quaternion[8, 2, 3, 4]
irb(main):015:0> q * r
=> Quaternion[7, 14, 21, 28]
irb(main):016:0> r * q
=> Quaternion[7, 14, 21, 28]
irb(main):017:0> q1 * q2
=> Quaternion[-56, 16, 24, 26]
irb(main):018:0> q2 * q1
=> Quaternion[-56, 18, 20, 28]
irb(main):019:0> q1 * q2 != q2 * q1
=> true
[edit] Scala
case class Quaternion(re:Double =0.0, i:Double =0.0, j:Double =0.0, k:Double =0.0) {
lazy val im=(i, j, k)
private lazy val norm2=re*re + i*i + j*j + k*k
lazy val norm=math.sqrt(norm2)
def negative=new Quaternion(-re, -i, -j, -k)
def conjugate=new Quaternion(re, -i, -j, -k)
def reciprocal=new Quaternion(re/norm2, -i/norm2, -j/norm2, -k/norm2)
def +(q:Quaternion)=new Quaternion(re+q.re, i+q.i, j+q.j, k+q.k)
def -(q:Quaternion)=new Quaternion(re-q.re, i-q.i, j-q.j, k-q.k)
def *(q:Quaternion)=new Quaternion(
re*q.re - i*q.i - j*q.j - k*q.k,
re*q.i + i*q.re + j*q.k - k*q.j,
re*q.j - i*q.k + j*q.re + k*q.i,
re*q.k + i*q.j - j*q.i + k*q.re
)
def /(q:Quaternion)=this*q.reciprocal
def unary_- = negative
def unary_~ = conjugate
override def equals(x:Any):Boolean=x match {
case Quaternion(re, i, j, k) => (Double.doubleToLongBits(this.re)==Double.doubleToLongBits(re)) &&
Double.doubleToLongBits(this.i)==Double.doubleToLongBits(i) &&
Double.doubleToLongBits(this.j)==Double.doubleToLongBits(j) &&
Double.doubleToLongBits(this.k)==Double.doubleToLongBits(k)
case _ => false
}
override def toString()="Q(%.2f, %.2fi, %.2fj, %.2fk)".formatLocal(Locale.ENGLISH, re,i,j,k)
}
object Quaternion {
implicit def number2Quaternion[T <% Number](n:T):Quaternion = apply(n.doubleValue)
}
Demonstration:
val q0=Quaternion(1.0, 2.0, 3.0, 4.0);
val q1=Quaternion(2.0, 3.0, 4.0, 5.0);
val q2=Quaternion(3.0, 4.0, 5.0, 6.0);
val r=7;
println("q0 = "+ q0)
println("q1 = "+ q1)
println("q2 = "+ q2)
println("r = "+ r)
println()
println("q0.re = "+ q0.re)
println("q0.im = "+ q0.im)
println("q0.norm = "+ q0.norm)
println("q0.negative = "+ q0.negative)
println("-q0 = "+ -q0)
println("q0.conjugate = "+ q0.conjugate)
println("~q0 = "+ ~q0)
println("q1+q2 = "+ (q1+q2))
println("q2+q1 = "+ (q2+q1))
println("q1+r = "+ (q1+r))
println("r+q1 = "+ (r+q1))
println("q1-q2 = "+ (q1-q2))
println("q2-q1 = "+ (q2-q1))
println("q1-r = "+ (q1-r))
println("r-q1 = "+ (r-q1))
println("q1*q2 = "+ q1*q2)
println("q2*q1 = "+ q2*q1)
println("q1*r = "+ q1*r)
println("r*q1 = "+ r*q1)
println("(q1*q2)!=(q2*q1) = "+ ((q1*q2)!=(q2*q1)))
println("q1/q2 = "+ q1/q2)
println("q2/q1 = "+ q2/q1)
println("q1/r = "+ q1/r)
println("r/q1 = "+ r/q1)
Output:
q0 = Q(1.00, 2.00i, 3.00j, 4.00k) q1 = Q(2.00, 3.00i, 4.00j, 5.00k) q2 = Q(3.00, 4.00i, 5.00j, 6.00k) r = 7 q0.re = 1.0 q0.im = (2.0,3.0,4.0) q0.norm = 5.477225575051661 q0.negative = Q(-1.00, -2.00i, -3.00j, -4.00k) -q0 = Q(-1.00, -2.00i, -3.00j, -4.00k) q0.conjugate = Q(1.00, -2.00i, -3.00j, -4.00k) ~q0 = Q(1.00, -2.00i, -3.00j, -4.00k) q1+q2 = Q(5.00, 7.00i, 9.00j, 11.00k) q2+q1 = Q(5.00, 7.00i, 9.00j, 11.00k) q1+r = Q(9.00, 3.00i, 4.00j, 5.00k) r+q1 = Q(9.00, 3.00i, 4.00j, 5.00k) q1-q2 = Q(-1.00, -1.00i, -1.00j, -1.00k) q2-q1 = Q(1.00, 1.00i, 1.00j, 1.00k) q1-r = Q(-5.00, 3.00i, 4.00j, 5.00k) r-q1 = Q(5.00, -3.00i, -4.00j, -5.00k) q1*q2 = Q(-56.00, 16.00i, 24.00j, 26.00k) q2*q1 = Q(-56.00, 18.00i, 20.00j, 28.00k) q1*r = Q(14.00, 21.00i, 28.00j, 35.00k) r*q1 = Q(14.00, 21.00i, 28.00j, 35.00k) (q1*q2)!=(q2*q1) = true q1/q2 = Q(0.79, 0.02i, -0.00j, 0.05k) q2/q1 = Q(1.26, -0.04i, 0.00j, -0.07k) q1/r = Q(0.29, 0.43i, 0.57j, 0.71k) r/q1 = Q(0.26, -0.39i, -0.52j, -0.65k)
[edit] Tcl
orpackage require TclOO
# Support class that provides C++-like RAII lifetimes
oo::class create RAII-support {
constructor {} {
upvar 1 { end } end
lappend end [self]
trace add variable end unset [namespace code {my destroy}]
}
destructor {
catch {
upvar 1 { end } end
trace remove variable end unset [namespace code {my destroy}]
}
}
method return {{level 1}} {
incr level
upvar 1 { end } end
upvar $level { end } parent
trace remove variable end unset [namespace code {my destroy}]
lappend parent [self]
trace add variable parent unset [namespace code {my destroy}]
return -level $level [self]
}
}
# Class of quaternions
oo::class create Q {
superclass RAII-support
variable R I J K
constructor {{real 0} {i 0} {j 0} {k 0}} {
next
namespace import ::tcl::mathfunc::* ::tcl::mathop::*
variable R [double $real] I [double $i] J [double $j] K [double $k]
}
self method return args {
[my new {*}$args] return 2
}
method p {} {
return "Q($R,$I,$J,$K)"
}
method values {} {
list $R $I $J $K
}
method Norm {} {
+ [* $R $R] [* $I $I] [* $J $J] [* $K $K]
}
method conjugate {} {
Q return $R [- $I] [- $J] [- $K]
}
method norm {} {
sqrt [my Norm]
}
method unit {} {
set n [my norm]
Q return [/ $R $n] [/ $I $n] [/ $J $n] [/ $K $n]
}
method reciprocal {} {
set n2 [my Norm]
Q return [/ $R $n2] [/ $I $n2] [/ $J $n2] [/ $K $n2]
}
method - {{q ""}} {
if {[llength [info level 0]] == 2} {
Q return [- $R] [- $I] [- $J] [- $K]
}
[my + [$q -]] return
}
method + q {
if {[info object isa object $q]} {
lassign [$q values] real i j k
Q return [+ $R $real] [+ $I $i] [+ $J $j] [+ $K $k]
}
Q return [+ $R [double $q]] $I $J $K
}
method * q {
if {[info object isa object $q]} {
lassign [my values] a1 b1 c1 d1
lassign [$q values] a2 b2 c2 d2
Q return [expr {$a1*$a2 - $b1*$b2 - $c1*$c2 - $d1*$d2}] \
[expr {$a1*$b2 + $b1*$a2 + $c1*$d2 - $d1*$c2}] \
[expr {$a1*$c2 - $b1*$d2 + $c1*$a2 + $d1*$b2}] \
[expr {$a1*$d2 + $b1*$c2 - $c1*$b2 + $d1*$a2}]
}
set f [double $q]
Q return [* $R $f] [* $I $f] [* $J $f] [* $K $f]
}
method == q {
expr {
[info object isa object $q]
&& [info object isa typeof $q [self class]]
&& [my values] eq [$q values]
}
}
export - + * ==
}
Demonstration code:
set q [Q new 1 2 3 4]
set q1 [Q new 2 3 4 5]
set q2 [Q new 3 4 5 6]
set r 7
puts "q = [$q p]"
puts "q1 = [$q1 p]"
puts "q2 = [$q2 p]"
puts "r = $r"
puts "q norm = [$q norm]"
puts "q1 norm = [$q1 norm]"
puts "q2 norm = [$q2 norm]"
puts "-q = [[$q -] p]"
puts "q conj = [[$q conjugate] p]"
puts "q + r = [[$q + $r] p]"
# Real numbers are not objects, so no extending operations for them
puts "q1 + q2 = [[$q1 + $q2] p]"
puts "q2 + q1 = [[$q2 + $q1] p]"
puts "q * r = [[$q * $r] p]"
puts "q1 * q2 = [[$q1 * $q2] p]"
puts "q2 * q1 = [[$q2 * $q1] p]"
puts "equal(q1*q2, q2*q1) = [[$q1 * $q2] == [$q2 * $q1]]"
Output:
q = Q(1.0,2.0,3.0,4.0) q1 = Q(2.0,3.0,4.0,5.0) q2 = Q(3.0,4.0,5.0,6.0) r = 7 q norm = 5.477225575051661 q1 norm = 7.3484692283495345 q2 norm = 9.273618495495704 -q = Q(-1.0,-2.0,-3.0,-4.0) q conj = Q(1.0,-2.0,-3.0,-4.0) q + r = Q(8.0,2.0,3.0,4.0) q1 + q2 = Q(5.0,7.0,9.0,11.0) q2 + q1 = Q(5.0,7.0,9.0,11.0) q * r = Q(7.0,14.0,21.0,28.0) q1 * q2 = Q(-56.0,16.0,24.0,26.0) q2 * q1 = Q(-56.0,18.0,20.0,28.0) equal(q1*q2, q2*q1) = 0
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