Quaternion type: Difference between revisions
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# The negative of a quaternion:<br><code>=(-a, -b, -c, -d)</code> |
# The negative of a quaternion:<br><code>=(-a, -b, -c, -d)</code> |
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# The conjugate of a quaternion:<br><code>=( a, -b, -c, -d)</code> |
# The conjugate of a quaternion:<br><code>=( a, -b, -c, -d)</code> |
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# Addition of a real number r and a quaternion q:<br><code>r + q = q + r = (a+r, b, c, d) |
# Addition of a real number r and a quaternion q:<br><code>r + q = q + r = (a+r, b, c, d)</code> |
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# Addition of two quaternions:<br><code>q1 + q2 = (a1+a2, b1+b2, c1+c2, d1+d2) |
# Addition of two quaternions:<br><code>q1 + q2 = (a1+a2, b1+b2, c1+c2, d1+d2)</code> |
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# Multiplication of a real number and a quaternion:<br><code>qr = rq = (ar, br, cr, dr)</code> |
# Multiplication of a real number and a quaternion:<br><code>qr = rq = (ar, br, cr, dr)</code> |
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# Multiplication of two quaternions q1 and q2 is given by:<br><code>( a1a2 − b1b2 − c1c2 − d1d2,</code><br><code>a1b2 + b1a2 + c1d2 − d1c2,</code><br><code>a1c2 − b1d2 + c1a2 + d1b2,</code><br><code>a1d2 + b1c2 − c1b2 + d1a2 )</code> |
# Multiplication of two quaternions q1 and q2 is given by:<br><code>( a1a2 − b1b2 − c1c2 − d1d2,</code><br><code>a1b2 + b1a2 + c1d2 − d1c2,</code><br><code>a1c2 − b1d2 + c1a2 + d1b2,</code><br><code>a1d2 + b1c2 − c1b2 + d1a2 )</code> |
Revision as of 22:23, 2 August 2010
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.
python
This example extends Pythons namedtuples to add extra functionality. <lang python>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)
Quaternion = Q
q = Q(1, 2, 3, 4) q1 = Q(2, 3, 4, 5) q2 = Q(3, 4, 5, 6) r = 7</lang>
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: <lang python>>>> 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) >>> </lang>