Find the intersection of two lines

From Rosetta Code
Task
Find the intersection of two lines
You are encouraged to solve this task according to the task description, using any language you may know.
Finding the intersection of two lines that are in the same plane is an important topic in collision detection.[1]


Task

Find the point of intersection of two lines in 2D.


The 1st line passes though   (4,0)   and   (6,10) .
The 2nd line passes though   (0,3)   and   (10,7) .

11l[edit]

Translation of: Python
F line_intersect(Ax1, Ay1, Ax2, Ay2, Bx1, By1, Bx2, By2)
   V d = (By2 - By1) * (Ax2 - Ax1) - (Bx2 - Bx1) * (Ay2 - Ay1)
   I d == 0
      R (Float.infinity, Float.infinity)

   V uA = ((Bx2 - Bx1) * (Ay1 - By1) - (By2 - By1) * (Ax1 - Bx1)) / d
   V uB = ((Ax2 - Ax1) * (Ay1 - By1) - (Ay2 - Ay1) * (Ax1 - Bx1)) / d

   I !(uA C 0.0..1.0 & uB C 0.0..1.0)
      R (Float.infinity, Float.infinity)
   V x = Ax1 + uA * (Ax2 - Ax1)
   V y = Ay1 + uA * (Ay2 - Ay1)

   R (x, y)

V (a, b, c, d) = (4.0, 0.0, 6.0, 10.0)
V (e, f, g, h) = (0.0, 3.0, 10.0, 7.0)
V pt = line_intersect(a, b, c, d, e, f, g, h)
print(pt)
Output:
(5, 5)

360 Assembly[edit]

Translation of: Rexx
*        Intersection of two lines   01/03/2019
INTERSEC CSECT                     
         USING  INTERSEC,R13       base register
         B      72(R15)            skip savearea
         DC     17F'0'             savearea
         SAVE   (14,12)            save previous context
         ST     R13,4(R15)         link backward
         ST     R15,8(R13)         link forward
         LR     R13,R15            set addressability
         LE     F0,XA              xa
       IF    CE,F0,EQ,XB THEN      if xa=xb then
         STE    F0,X1                x1=xa
         LE     F0,YA
       IF    CE,F0,EQ,YB THEN        if ya=yb then
         MVI    MSG,C'='               msg='='
       ENDIF    ,                    endif
       ELSE     ,                  else
         MVI    FK1,X'01'            fk1=true
         LE     F0,YB
         SE     F0,YA                yb-ya
         LE     F2,XB
         SE     F2,XA                xb-xa
         DER    F0,F2                /
         STE    F0,K1                k1=(yb-ya)/(xb-xa)
         ME     F0,XA                k1*xa
         LE     F2,YA                ya
         SER    F2,F0                -
         STE    F2,D1                d1=ya-k1*xa
       ENDIF    ,                  endif
         LE     F0,XC
       IF    CE,F0,EQ,XD THEN      if xc=xd then
         STE    F0,X2                x2=xc
         LE     F4,YC                yc
       IF    CE,F4,EQ,YD THEN        if yc=yd then
         MVI    MSG,C'='               msg='='
       ENDIF    ,                    endif
       ELSE     ,                  else
         MVI    FK2,X'01'            fk2=true
         LE     F0,YD
         SE     F0,YC                yd-yc
         LE     F2,XD
         SE     F2,XC                xd-xc
         DER    F0,F2                /
         STE    F0,K2                k2=(yd-yc)/(xd-xc)
         ME     F0,XC                k2*xc
         LE     F2,YC                yc
         SER    F2,F0                -
         STE    F2,D2                d2=yc-k2*xc
       ENDIF    ,                  endif
       IF   CLI,MSG,EQ,C' ' THEN   if msg=' ' then
       IF   CLI,FK1,EQ,X'00' THEN    if not fk1 then
       IF   CLI,FK2,EQ,X'00' THEN      if not fk2 then
         LE     F4,X1
       IF    CE,F4,EQ,X2                 if x1=x2 then
         MVI    MSG,C'='                   msg='='
       ELSE     ,                        else
         MVI    MSG,C'/'                   msg='/'
       ENDIF    ,                        endif
       ELSE     ,                      else
         LE     F0,X1
         STE    F0,X                     x=x1
         LE     F0,K2                    k2
         ME     F0,X                     *x
         AE     F0,D2                    +d2
         STE    F0,Y                     y=k2*x+d2
       ENDIF    ,                      endif
       ELSE     ,                    else
       IF    CLI,FK2,EQ,X'00' THEN     if not fk2 then
         LE     F0,X2
         STE    F0,X                     x=x2
         LE     F0,K1                    k1
         ME     F0,X                     *x
         AE     F0,D1                    +d1
         STE    F0,Y                     y=k1*x+d1
       ELSE     ,                      else
         LE     F4,K1
       IF    CE,F4,EQ,K2 THEN            if k1=k2 then
         LE     F4,D1
       IF    CE,F4,EQ,D2 THEN              if d1=d2 then
         MVI    MSG,C'='                     msg='=';
       ELSE     ,                          else
         MVI    MSG,C'/'                     msg='/';
       ENDIF    ,                          endif
       ELSE     ,                        else
         LE     F0,D2                      d2
         SE     F0,D1                      -d1
         LE     F2,K1                      k1
         SE     F2,K2                      -k2
         DER    F0,F2                      /
         STE    F0,X                       x=(d2-d1)/(k1-k2)
         LE     F0,K1                      k1
         ME     F0,X                       *x
         AE     F0,D1                      +d1
         STE    F0,Y                       y=k1*x+d1
       ENDIF    ,                        endif
       ENDIF    ,                      endif
       ENDIF    ,                    endif
       ENDIF    ,                  endif
       IF   CLI,MSG,EQ,C' ' THEN   if msg=' ' then
         LE     F0,X                 x
         LA     R0,3                 decimal=3
         BAL    R14,FORMATF          format x
         MVC    PG+0(13),0(R1)       output x
         LE     F0,Y                 y
         LA     R0,3                 decimal=3
         BAL    R14,FORMATF          format y
         MVC    PG+13(13),0(R1)      output y
       ENDIF    ,                  endif
         MVC    PG+28(1),MSG       output msg  
         XPRNT  PG,L'PG            print buffer
         L      R13,4(0,R13)       restore previous savearea pointer
         RETURN (14,12),RC=0       restore registers from calling sav
         COPY   plig\$_FORMATF.MLC
XA       DC     E'4.0'             point A
YA       DC     E'0.0'
XB       DC     E'6.0'             point B
YB       DC     E'10.0'
XC       DC     E'0.0'             point C
YC       DC     E'3.0'
XD       DC     E'10.0'            point D
YD       DC     E'7.0'
X        DS     E
Y        DS     E
X1       DS     E
X2       DS     E
K1       DS     E
K2       DS     E
D1       DS     E
D2       DS     E
FK1      DC     X'00'
FK2      DC     X'00'
MSG      DC     C' '
PG       DC     CL80' '
         REGEQU
         END    INTERSEC
Output:
        5.000        5.000

Action![edit]

INCLUDE "D2:REAL.ACT" ;from the Action! Tool Kit

DEFINE REALPTR="CARD"
TYPE PointR=[REALPTR x,y]

PROC Det(REAL POINTER x1,y1,x2,y2,res)
  REAL tmp1,tmp2

  RealMult(x1,y2,tmp1)
  RealMult(y1,x2,tmp2)
  RealSub(tmp1,tmp2,res)
RETURN

BYTE FUNC IsZero(REAL POINTER a)
  CHAR ARRAY s(10)

  StrR(a,s)
  IF s(0)=1 AND s(1)='0 THEN
    RETURN (1)
  FI
RETURN (0)

BYTE FUNC Intersection(PointR POINTER p1,p2,p3,p4,res)
  REAL det1,det2,dx1,dx2,dy1,dy2,nom,denom

  Det(p1.x,p1.y,p2.x,p2.y,det1)
  Det(p3.x,p3.y,p4.x,p4.y,det2)
  RealSub(p1.x,p2.x,dx1)
  RealSub(p1.y,p2.y,dy1)
  RealSub(p3.x,p4.x,dx2)
  RealSub(p3.y,p4.y,dy2)
  Det(dx1,dy1,dx2,dy2,denom)
  
  IF IsZero(denom) THEN
    RETURN (0)
  FI
  
  Det(det1,dx1,det2,dx2,nom)
  RealDiv(nom,denom,res.x)
  Det(det1,dy1,det2,dy2,nom)
  RealDiv(nom,denom,res.y)
RETURN (1)

PROC PrintPoint(PointR POINTER p)
  Print("(") PrintR(p.x)
  Print(",") PrintR(p.y)
  Print(")")
RETURN

PROC PrintLine(PointR POINTER p1,p2)
  PrintPoint(p1)
  Print(" and ")
  PrintPoint(p2)
RETURN

PROC Test(PointR POINTER p1,p2,p3,p4)
  BYTE res
  REAL px,py
  PointR p

  p.x=px p.y=py
  Print("Line 1 points: ")
  PrintLine(p1,p2) PutE()
  Print("Line 2 points: ")
  PrintLine(p3,p4) PutE()

  res=Intersection(p1,p2,p3,p4,p)
  IF res THEN
    Print("Intersection point: ")
    PrintPoint(p) PutE()
  ELSE
    PrintE("There is no intersection")
  FI
  PutE()
RETURN

PROC Main()
  REAL x1,y1,x2,y2,x3,y3,x4,y4,px,py
  PointR p1,p2,p3,p4

  Put(125) PutE() ;clear screen

  p1.x=x1 p1.y=y1
  p2.x=x2 p2.y=y2
  p3.x=x3 p3.y=y3
  p4.x=x4 p4.y=y4

  IntToReal(4,x1) IntToReal(0,y1)
  IntToReal(6,x2) IntToReal(10,y2)
  IntToReal(0,x3) IntToReal(3,y3)
  IntToReal(10,x4) IntToReal(7,y4)
  Test(p1,p2,p3,p4)

  IntToReal(0,x1) IntToReal(0,y1)
  IntToReal(1,x2) IntToReal(1,y2)
  IntToReal(1,x3) IntToReal(2,y3)
  IntToReal(4,x4) IntToReal(5,y4)
  Test(p1,p2,p3,p4)
RETURN
Output:

Screenshot from Atari 8-bit computer

Line 1 points: (4,0) and (6,10)
Line 2 points: (0,3) and (10,7)
Intersection point: (5,5)

Line 1 points: (0,0) and (1,1)
Line 2 points: (1,2) and (4,5)
There is no intersection

Ada[edit]

Works with: Ada version 2005
with Ada.Text_IO;

procedure Intersection_Of_Two_Lines
is
   Do_Not_Intersect : exception;

   type Line is record
      a : Float;
      b : Float;
   end record;

   type Point is record
      x : Float;
      y : Float;
   end record;

   function To_Line(p1, p2 : in Point) return Line
   is
      a : constant Float := (p1.y - p2.y) / (p1.x - p2.x);
      b : constant Float := p1.y - (a * p1.x);
   begin
      return (a,b);
   end To_Line;

   function Intersection(Left, Right : in Line) return Point is
   begin
      if Left.a = Right.a then
         raise Do_Not_Intersect with "The two lines do not intersect.";
      end if;

      declare
         b : constant Float := (Right.b - Left.b) / (Left.a - Right.a);
      begin
         return (b, Left.a * b + Left.b);
      end;
   end Intersection;

   A1 : constant Line := To_Line((4.0, 0.0), (6.0, 10.0));
   A2 : constant Line := To_Line((0.0, 3.0), (10.0, 7.0));
   p : constant Point := Intersection(A1, A2);
begin
   Ada.Text_IO.Put(p.x'Img);
   Ada.Text_IO.Put_Line(p.y'Img);
end Intersection_Of_Two_Lines;
Output:
 5.00000E+00 5.00000E+00

ALGOL 68[edit]

Using "school maths".

BEGIN
    # mode to hold a point #
    MODE POINT = STRUCT( REAL x, y );
    # mode to hold a line expressed as y = mx + c #
    MODE LINE  = STRUCT( REAL m, c );
    # returns the line that passes through p1 and p2 #
    PROC find line = ( POINT p1, p2 )LINE:
         IF x OF p1 = x OF p2 THEN
             # the line is vertical                 #
             LINE( 0, x OF p1 )
         ELSE
             # the line is not vertical             #
             REAL m = ( y OF p1 - y OF p2 ) / ( x OF p1 - x OF p2 );
             LINE( m, y OF p1 - ( m * x OF p1 ) )
         FI # find line # ;

    # returns the intersection of two lines - the lines must be distinct and not parallel #
    PRIO INTERSECTION = 5;
    OP   INTERSECTION = ( LINE l1, l2 )POINT:
         BEGIN
             REAL x = ( c OF l2 - c OF l1 ) / ( m OF l1 - m OF l2 );
             POINT( x, ( m OF l1 * x ) + c OF l1 )
         END # INTERSECTION # ;

    # find the intersection of the lines as per the task #
    POINT i = find line( POINT( 4.0, 0.0 ), POINT( 6.0, 10.0 ) )
              INTERSECTION find line( ( 0.0, 3.0 ), ( 10.0, 7.0 ) );
    print( ( fixed( x OF i, -8, 4 ), fixed( y OF i, -8, 4 ), newline ) )

END
Output:
  5.0000  5.0000

APL[edit]

⍝ APL has a powerful operator the « dyadic domino » to solve a system of N linear equations with N unknowns
⍝ We use it first to solve the a and b, defining the 2 lines as y = ax + b, with the x and y of the given points 
⍝ The system of equations for first line will be:
⍝  0 = 4a + b
⍝ 10 = 6a + b
⍝ The two arguments to be passed to the dyadic domino are:
⍝ The (0, 10) vector as the left argument 
⍝ The ( 4  1 ) matrix as the right argument.
⍝     ( 6  1 )
⍝ We will define a solver that will take the matrix of coordinates, one point per row, then massage the argument to extract x,y
⍝ and inject 1, where needed, and return a pair (a, b) of resolved unknowns.
⍝ Applied twice, we will have a, b and a', b' defining the two lines, we need to resolve it in x and y, in order to determine 
⍝ their intersection
⍝ y =  ax + b
⍝ y = a'x + b'
⍝ In order to reuse the same solver, we need to format a little bit the arguments, and change the sign of a and a', therefore
⍝ multiply (a,b) and (a', b') by (-1, 1):
⍝ b  =  -ax + y
⍝ b' = -a'x + y
  A  4 0
  B  6 10
  C  0 3
  D  10 7
  solver  {(,2 ¯1)(2 1),1}
  I  solver 2 2((¯1 1)×solver 2 2A,B),(¯1 1)×solver 2 2C,D
Output:
  I
5 5

Arturo[edit]

Translation of: Go
define :point [x,y][]
define :line [a, b][
    init: [
        this\slope: div this\b\y-this\a\y this\b\x-this\a\x
        this\yInt: this\a\y - this\slope*this\a\x
    ]
]

evalX: function [line, x][
    line\yInt + line\slope * x
]

intersect: function [line1, line2][
    x: div line2\yInt-line1\yInt line1\slope-line2\slope
    y: evalX line1 x

    to :point @[x y]
]

l1: to :line @[to :point [4.0 0.0] to :point [6.0 10.0]]
l2: to :line @[to :point [0.0 3.0] to :point [10.0 7.0]]

print intersect l1 l2
Output:
[x:5.0 y:5.0]

AutoHotkey[edit]

LineIntersectionByPoints(L1, L2){
	x1 := L1[1,1], y1 := L1[1,2]
	x2 := L1[2,1], y2 := L1[2,2]
	x3 := L2[1,1], y3 := L2[1,2]
	x4 := L2[2,1], y4 := L2[2,2]
	return ((x1*y2-y1*x2)*(x3-x4) - (x1-x2)*(x3*y4-y3*x4)) / ((x1-x2)*(y3-y4) - (y1-y2)*(x3-x4)) ", "
	.      ((x1*y2-y1*x2)*(y3-y4) - (y1-y2)*(x3*y4-y3*x4)) / ((x1-x2)*(y3-y4) - (y1-y2)*(x3-x4))
}
Examples:
L1 := [[4,0], [6,10]]
L2 := [[0,3], [10,7]]
MsgBox % LineIntersectionByPoints(L1, L2)
Outputs:
5.000000, 5.000000

AWK[edit]

# syntax: GAWK -f FIND_THE_INTERSECTION_OF_TWO_LINES.AWK
# converted from Ring
BEGIN {
    intersect(4,0,6,10,0,3,10,7)
    exit(0)
}
function intersect(xa,ya,xb,yb,xc,yc,xd,yd,  errors,x,y) {
    printf("the 1st line passes through (%g,%g) and (%g,%g)\n",xa,ya,xb,yb)
    printf("the 2nd line passes through (%g,%g) and (%g,%g)\n",xc,yc,xd,yd)
    if (xb-xa == 0) { print("error: xb-xa=0") ; errors++ }
    if (xd-xc == 0) { print("error: xd-xc=0") ; errors++ }
    if (errors > 0) {
      print("")
      return(0)
    }
    printf("the two lines are:\n")
    printf("yab=%g+x*%g\n",ya-xa*((yb-ya)/(xb-xa)),(yb-ya)/(xb-xa))
    printf("ycd=%g+x*%g\n",yc-xc*((yd-yc)/(xd-xc)),(yd-yc)/(xd-xc))
    x = ((yc-xc*((yd-yc)/(xd-xc)))-(ya-xa*((yb-ya)/(xb-xa))))/(((yb-ya)/(xb-xa))-((yd-yc)/(xd-xc)))
    printf("x=%g\n",x)
    y = ya-xa*((yb-ya)/(xb-xa))+x*((yb-ya)/(xb-xa))
    printf("yab=%g\n",y)
    printf("ycd=%g\n",yc-xc*((yd-yc)/(xd-xc))+x*((yd-yc)/(xd-xc)))
    printf("intersection: %g,%g\n\n",x,y)
    return(1)
}
Output:
the 1st line passes through (4,0) and (6,10)
the 2nd line passes through (0,3) and (10,7)
the two lines are:
yab=-20+x*5
ycd=3+x*0.4
x=5
yab=5
ycd=5
intersection: 5,5

BASIC[edit]

Applesoft BASIC[edit]

 0 A = 1:B = 2: HOME : VTAB 21: HGR : HCOLOR= 3: FOR L = A TO B: READ X1(L),Y1(L),X2(L),Y2(L): HPLOT X1(L),Y1(L) TO X2(L),Y2(L): NEXT : DATA4,0,6,10,0,3,10,7
 1  GOSUB 5: IF NAN THEN  PRINT "THE LINES DO NOT INTERSECT, THEY ARE EITHER PARALLEL OR CO-INCIDENT."
 2  IF  NOT NAN THEN  PRINT "POINT OF INTERSECTION : "X" "Y
 3  PRINT  CHR$ (13)"HIT ANY KEY TO END PROGRAM": IF  NOT NAN THEN  FOR K = 0 TO 1 STEP 0:C = C = 0: HCOLOR= 3 * C: HPLOT X,Y: FOR I = 1 TO 30:K =  PEEK (49152) > 127: NEXT I,K
 4  GET K$: TEXT : END 
 5  FOR L = A TO B:S$(L) = "NAN": IF X1(L) <  > X2(L) THEN S(L) = (Y1(L) - Y2(L)) / (X1(L) - X2(L)):S$(L) =  STR$ (S(L))
 6  NEXT L:NAN = S$(A) = S$(B): IF NAN THEN  RETURN 
 7  IF S$(A) = "NAN" AND S$(B) <  > "NAN" THEN X = X1(A):Y = (X1(A) - X1(B)) * S(B) + Y1(B): RETURN 
 8  IF S$(B) = "NAN" AND S$(A) <  > "NAN" THEN X = X1(B):Y = (X1(B) - X1(A)) * S(A) + Y1(A): RETURN 
 9 X = (S(A) * X1(A) - S(B) * X1(B) + Y1(B) - Y1(A)) / (S(A) - S(B)):Y = S(B) * (X - X1(B)) + Y1(B): RETURN

QBasic[edit]

Works with: QBasic version 1.1
Works with: QuickBasic version 4.5
Works with: BASIC256
Works with: Just BASIC
Works with: Run BASIC
xa = 4: xb = 6: xc = 0: xd = 10
ya = 0: yb = 10: yc = 3: yd = 7
PRINT "The two lines are:"
PRINT "yab ="; (ya - xa * ((yb - ya) / (xb - xa))); "+ x*"; ((yb - ya) / (xb - xa))
PRINT "ycd ="; (yc - xc * ((yd - yc) / (xd - xc))); "+ x*"; ((yd - yc) / (xd - xc))
x = ((yc - xc * ((yd - yc) / (xd - xc))) - (ya - xa * ((yb - ya) / (xb - xa)))) / (((yb - ya) / (xb - xa)) - ((yd - yc) / (xd - xc)))
PRINT "x ="; x
y = ya - xa * ((yb - ya) / (xb - xa)) + x * ((yb - ya) / (xb - xa))
PRINT "yab ="; y
PRINT "ycd ="; (yc - xc * ((yd - yc) / (xd - xc)) + x * ((yd - yc) / (xd - xc)))
PRINT "intersection: ("; x; ","; y; ")"
Output:
The two lines are:
yab =-20 + x* 5
ycd = 3 + x* 0.4
x = 5
yab = 5
ycd = 5
intersection: ( 5, 5 )

BASIC256[edit]

Works with: QBasic
Works with: Just BASIC
Works with: Run BASIC
xa = 4 : xb =  6 : xc = 0 : xd = 10
ya = 0 : yb = 10 : yc = 3 : yd = 7
print "The two lines are:"
print "yab = "; (ya-xa*((yb-ya)/(xb-xa))); " + x*"; ((yb-ya)/(xb-xa))
print "ycd = "; (yc-xc*((yd-yc)/(xd-xc))); " + x*"; ((yd-yc)/(xd-xc))
x = ((yc-xc*((yd-yc)/(xd-xc)))-(ya-xa*((yb-ya)/(xb-xa))))/(((yb-ya)/(xb-xa))-((yd-yc)/(xd-xc)))
print "x = "; x
y = ya-xa*((yb-ya)/(xb-xa))+x*((yb-ya)/(xb-xa))
print "yab = "; y
print "ycd = "; (yc-xc*((yd-yc)/(xd-xc))+x*((yd-yc)/(xd-xc)))
print "intersection: ("; x; ", "; y ; ")"

Run BASIC[edit]

Works with: Just BASIC
Works with: Liberty BASIC
Works with: QBasic
xa = 4: xb = 6: xc = 0: xd = 10
ya = 0: yb = 10: yc = 3: yd = 7
print "The two lines are:"
print "yab = "; (ya - xa * ((yb - ya) / (xb - xa))); "+ x*"; ((yb - ya) / (xb - xa))
print "ycd = "; (yc - xc * ((yd - yc) / (xd - xc))); "+ x*"; ((yd - yc) / (xd - xc))
x = ((yc - xc * ((yd - yc) / (xd - xc))) - (ya - xa * ((yb - ya) / (xb - xa)))) / (((yb - ya) / (xb - xa)) - ((yd - yc) / (xd - xc)))
print "x = "; x
y = ya - xa * ((yb - ya) / (xb - xa)) + x * ((yb - ya) / (xb - xa))
print "yab = "; y
print "ycd = "; (yc - xc * ((yd - yc) / (xd - xc)) + x * ((yd - yc) / (xd - xc)))
print "intersection: ("; x; ","; y; " )"

Sinclair ZX81 BASIC[edit]

Translation of: REXX
(version 1)

Works with 1k of RAM.

 10 LET XA=4
 20 LET YA=0
 30 LET XB=6
 40 LET YB=10
 50 LET XC=0
 60 LET YC=3
 70 LET XD=10
 80 LET YD=0
 90 PRINT "THE TWO LINES ARE:"
100 PRINT "YAB=";YA-XA*((YB-YA)/(XB-XA));"+X*";((YB-YA)/(XB-XA))
110 PRINT "YCD=";YC-XC*((YD-YC)/(XD-XC));"+X*";((YD-YC)/(XD-XC))
120 LET X=((YC-XC*((YD-YC)/(XD-XC)))-(YA-XA*((YB-YA)/(XB-XA))))/(((YB-YA)/(XB-XA))-((YD-YC)/(XD-XC)))
130 PRINT "X=";X
140 LET Y=YA-XA*((YB-YA)/(XB-XA))+X*((YB-YA)/(XB-XA))
150 PRINT "YAB=";Y
160 PRINT "YCD=";YC-XC*((YD-YC)/(XD-XC))+X*((YD-YC)/(XD-XC))
170 PRINT "INTERSECTION: ";X;",";Y
Output:
THE TWO LINES ARE:
YAB=-20+X*5
YCD=3+X*0.4
X=5
YAB=5
YCD=5
INTERSECTION: 5,5

True BASIC[edit]

Works with: QBasic
Works with: BASIC256
Works with: Just BASIC
Works with: Run BASIC
LET xa = 4
LET ya = 0
LET xb = 6
LET yb = 10
LET xc = 0
LET yc = 3
LET xd = 10
LET yd = 7
PRINT "The two lines are:"
PRINT "yab ="; (ya-xa*((yb-ya)/(xb-xa))); " + x*"; ((yb-ya)/(xb-xa))
PRINT "ycd ="; (yc-xc*((yd-yc)/(xd-xc))); " + x*"; ((yd-yc)/(xd-xc))
LET x = ((yc-xc*((yd-yc)/(xd-xc)))-(ya-xa*((yb-ya)/(xb-xa))))/(((yb-ya)/(xb-xa))-((yd-yc)/(xd-xc)))
PRINT "x ="; x
LET y = ya-xa*((yb-ya)/(xb-xa))+x*((yb-ya)/(xb-xa))
PRINT "yab ="; y
PRINT "ycd ="; (yc-xc*((yd-yc)/(xd-xc))+x*((yd-yc)/(xd-xc)))
PRINT "intersection: ("; x; ","; y ; " )"
END

Yabasic[edit]

xa = 4: xb = 6: xc = 0: xd = 10
ya = 0: yb = 10: yc = 3: yd = 7
print "The two lines are:"
print "yab = ", (ya - xa * ((yb - ya) / (xb - xa))), " + x*", ((yb - ya) / (xb - xa))
print "ycd = ", (yc - xc * ((yd - yc) / (xd - xc))), " + x*", ((yd - yc) / (xd - xc))
x = ((yc - xc * ((yd - yc) / (xd - xc))) - (ya - xa * ((yb - ya) / (xb - xa)))) / (((yb - ya) / (xb - xa)) - ((yd - yc) / (xd - xc)))
print "x = ", x
y = ya - xa * ((yb - ya) / (xb - xa)) + x * ((yb - ya) / (xb - xa))
print "yab = ", y
print "ycd = ", (yc - xc * ((yd - yc) / (xd - xc)) + x * ((yd - yc) / (xd - xc)))
print "intersection: (", x, ", ", y, ")"

C[edit]

This implementation is generic and considers any two lines in the XY plane and not just the specified example. Usage is printed on incorrect invocation.

#include<stdlib.h>
#include<stdio.h>
#include<math.h>

typedef struct{
	double x,y;
}point;

double lineSlope(point a,point b){
	
	if(a.x-b.x == 0.0)
		return NAN;
	else
		return (a.y-b.y)/(a.x-b.x);
}

point extractPoint(char* str){
	int i,j,start,end,length;
	char* holder;
	point c;
	
	for(i=0;str[i]!=00;i++){
		if(str[i]=='(')
			start = i;
		if(str[i]==','||str[i]==')')
		{
			end = i;
			
			length = end - start;
			
			holder = (char*)malloc(length*sizeof(char));
			
			for(j=0;j<length-1;j++)
				holder[j] = str[start + j + 1];
			holder[j] = 00;
			
			if(str[i]==','){
				start = i;
				c.x = atof(holder);
			}
			else
				c.y = atof(holder);
		}
	}
	
	return c;
}

point intersectionPoint(point a1,point a2,point b1,point b2){
	point c;
	
	double slopeA = lineSlope(a1,a2), slopeB = lineSlope(b1,b2);
	
	if(slopeA==slopeB){
		c.x = NAN;
		c.y = NAN;
	}
	else if(isnan(slopeA) && !isnan(slopeB)){
		c.x = a1.x;
		c.y = (a1.x-b1.x)*slopeB + b1.y;
	}
	else if(isnan(slopeB) && !isnan(slopeA)){
		c.x = b1.x;
		c.y = (b1.x-a1.x)*slopeA + a1.y;
	}
	else{
		c.x = (slopeA*a1.x - slopeB*b1.x + b1.y - a1.y)/(slopeA - slopeB);
		c.y = slopeB*(c.x - b1.x) + b1.y;
	}
	
	return c;
}

int main(int argC,char* argV[])
{
	point c;
	
	if(argC < 5)
		printf("Usage : %s <four points specified as (x,y) separated by a space>",argV[0]);
	else{
		c = intersectionPoint(extractPoint(argV[1]),extractPoint(argV[2]),extractPoint(argV[3]),extractPoint(argV[4]));
		
		if(isnan(c.x))
			printf("The lines do not intersect, they are either parallel or co-incident.");
		else
			printf("Point of intersection : (%lf,%lf)",c.x,c.y);
	}
	
	return 0;
}

Invocation and output:

C:\rosettaCode>lineIntersect.exe (4,0) (6,10) (0,3) (10,7)
Point of intersection : (5.000000,5.000000)

C#[edit]

using System;
using System.Drawing;
public class Program
{
    static PointF FindIntersection(PointF s1, PointF e1, PointF s2, PointF e2) {
        float a1 = e1.Y - s1.Y;
        float b1 = s1.X - e1.X;
        float c1 = a1 * s1.X + b1 * s1.Y;

        float a2 = e2.Y - s2.Y;
        float b2 = s2.X - e2.X;
        float c2 = a2 * s2.X + b2 * s2.Y;

        float delta = a1 * b2 - a2 * b1;
        //If lines are parallel, the result will be (NaN, NaN).
        return delta == 0 ? new PointF(float.NaN, float.NaN)
            : new PointF((b2 * c1 - b1 * c2) / delta, (a1 * c2 - a2 * c1) / delta);
    }

    static void Main() {
        Func<float, float, PointF> p = (x, y) => new PointF(x, y);
        Console.WriteLine(FindIntersection(p(4f, 0f), p(6f, 10f), p(0f, 3f), p(10f, 7f)));
        Console.WriteLine(FindIntersection(p(0f, 0f), p(1f, 1f), p(1f, 2f), p(4f, 5f)));
    }
}
Output:
{X=5, Y=5}
{X=NaN, Y=NaN}

C++[edit]

#include <iostream>
#include <cmath>
#include <cassert>
using namespace std;

/** Calculate determinant of matrix:
	[a b]
	[c d]
*/
inline double Det(double a, double b, double c, double d)
{
	return a*d - b*c;
}

/// Calculate intersection of two lines.
///\return true if found, false if not found or error
bool LineLineIntersect(double x1, double y1, // Line 1 start
	double x2, double y2, // Line 1 end
	double x3, double y3, // Line 2 start
	double x4, double y4, // Line 2 end
	double &ixOut, double &iyOut) // Output 
{
	double detL1 = Det(x1, y1, x2, y2);
	double detL2 = Det(x3, y3, x4, y4);
	double x1mx2 = x1 - x2;
	double x3mx4 = x3 - x4;
	double y1my2 = y1 - y2;
	double y3my4 = y3 - y4;

	double denom = Det(x1mx2, y1my2, x3mx4, y3my4);
	if(denom == 0.0) // Lines don't seem to cross
	{
		ixOut = NAN;
		iyOut = NAN;
		return false;
	}

	double xnom = Det(detL1, x1mx2, detL2, x3mx4);
	double ynom = Det(detL1, y1my2, detL2, y3my4);
	ixOut = xnom / denom;	
	iyOut = ynom / denom;
	if(!isfinite(ixOut) || !isfinite(iyOut)) // Probably a numerical issue
		return false;

	return true; //All OK
}

int main()
{
	// **Simple crossing diagonal lines**

	// Line 1
	double x1=4.0, y1=0.0;
	double x2=6.0, y2=10.0;
	
	// Line 2
	double x3=0.0, y3=3.0;
	double x4=10.0, y4=7.0;

	double ix = -1.0, iy = -1.0;
	bool result = LineLineIntersect(x1, y1, x2, y2, x3, y3, x4, y4, ix, iy);
	cout << "result " <<  result << "," << ix << "," << iy << endl;

	double eps = 1e-6;
	assert(result == true);
	assert(fabs(ix - 5.0) < eps);
	assert(fabs(iy - 5.0) < eps);
        return 0;
}
Output:
result 1,5,5

Clojure[edit]

;; Point is [x y] tuple
(defn compute-line [pt1 pt2]
  (let [[x1 y1] pt1
        [x2 y2] pt2
        m (/ (- y2 y1) (- x2 x1))]
    {:slope  m
     :offset (- y1 (* m x1))}))

(defn intercept [line1 line2]
  (let [x (/ (- (:offset line1) (:offset line2))
             (- (:slope  line2) (:slope  line1)))]
    {:x x
     :y (+ (* (:slope line1) x)
           (:offset line1))}))
Output:
(def line1 (compute-line [4 0] [6 10]))
(def line2 (compute-line [0 3] [10 7]))
line1  ; {:slope 5, :offset -20}
line2  ; {:slope 2/5, :offset 3}

(intercept line1 line2)  ; {:x 5, :y 5}

Common Lisp[edit]

;; Point is [x y] tuple
(defun point-of-intersection (x1 y1 x2 y2 x3 y3 x4 y4)
"Find the point of intersection of the lines defined by the points (x1 y1) (x2 y2) and (x3 y3) (x4 y4)"
  (let* ((dx1 (- x2 x1))
         (dx2 (- x4 x3))
         (dy1 (- y2 y1))
         (dy2 (- y4 y3))
         (den (- (* dy1 dx2) (* dy2 dx1))) )
    (unless (zerop den) 
  	(list (/ (+ (* (- y3 y1) dx1 dx2) (* x1 dy1 dx2) (* -1 x3 dy2 dx1)) den)	  	
       	      (/ (+ (* (+ x3 x1) dy1 dy2) (* -1 y1 dx1 dy2) (* y3 dx2 dy1)) den) ))))
Output:
(point-of-intersection 4 0 6 10 0 3 10 7) => (5 5)


D[edit]

Translation of: Kotlin
import std.stdio;

struct Point {
    real x, y;

    void toString(scope void delegate(const(char)[]) sink) const {
        import std.format;
        sink("{");
        sink.formattedWrite!"%f"(x);
        sink(", ");
        sink.formattedWrite!"%f"(y);
        sink("}");
    }
}

struct Line {
    Point s, e;
}

Point findIntersection(Line l1, Line l2) {
    auto a1 = l1.e.y - l1.s.y;
    auto b1 = l1.s.x - l1.e.x;
    auto c1 = a1 * l1.s.x + b1 * l1.s.y;

    auto a2 = l2.e.y - l2.s.y;
    auto b2 = l2.s.x - l2.e.x;
    auto c2 = a2 * l2.s.x + b2 * l2.s.y;

    auto delta = a1 * b2 - a2 * b1;
    // If lines are parallel, intersection point will contain infinite values
    return Point((b2 * c1 - b1 * c2) / delta, (a1 * c2 - a2 * c1) / delta);
}

void main() {
    auto l1 = Line(Point(4.0, 0.0), Point(6.0, 10.0));
    auto l2 = Line(Point(0f, 3f), Point(10f, 7f));
    writeln(findIntersection(l1, l2));
    l1 = Line(Point(0.0, 0.0), Point(1.0, 1.0));
    l2 = Line(Point(1.0, 2.0), Point(4.0, 5.0));
    writeln(findIntersection(l1, l2));
}
Output:
{5.000000, 5.000000}
{-inf, -inf}

F#[edit]

(*
Find the point of intersection of 2 lines.
Nigel Galloway May 20th., 2017
*)
type Line={a:float;b:float;c:float} member N.toS=sprintf "%.2fx + %.2fy = %.2f" N.a N.b N.c
let intersect (n:Line) g = match (n.a*g.b-g.a*n.b) with
                           |0.0 ->printfn "%s does not intersect %s" n.toS g.toS
                           |ng  ->printfn "%s intersects %s at x=%.2f y=%.2f" n.toS g.toS ((g.b*n.c-n.b*g.c)/ng) ((n.a*g.c-g.a*n.c)/ng)
let fn (i,g) (e,l) = {a=g-l;b=e-i;c=(e-i)*g+(g-l)*i}
intersect (fn (4.0,0.0) (6.0,10.0)) (fn (0.0,3.0) (10.0,7.0))
intersect {a=3.18;b=4.23;c=7.13} {a=6.36;b=8.46;c=9.75}
Output:
-10.00x + 2.00y = -40.00 intersects -4.00x + 10.00y = 30.00 at x=5.00 y=5.00
3.18x + 4.23y = 7.13 does not intersect 6.36x + 8.46y = 9.75

Factor[edit]

Works with: Factor version 0.99 2020-01-23
USING: arrays combinators.extras kernel math
math.matrices.laplace math.vectors prettyprint sequences ;

: det ( pt pt -- x ) 2array determinant ;

: numerator ( x y pt pt quot -- z )
    bi@ swapd [ 2array ] 2bi@ det ; inline

: intersection ( pt pt pt pt -- pt )
    [ [ det ] 2bi@ ]
    [ [ v- ] 2bi@ ] 4bi
    [ [ first ] numerator ]
    [ [ second ] numerator ]
    [ det 2nip ] 4tri
    dup zero? [ 3drop { 0/0. 0/0. } ]
    [ tuck [ / ] 2bi@ 2array ] if ;

{ 4 0 } { 6 10 } { 0 3 } { 10 7 } intersection .
{ 4 0 } { 6 10 } { 0 3 } { 10 7+1/10 } intersection .
{ 0 0 } { 1 1 } { 1 2 } { 4 5 } intersection .
Output:
{ 5 5 }
{ 5+5/459 5+25/459 }
{ NAN: 8000000000000 NAN: 8000000000000 }

Fortran[edit]

Works with: Fortran version 90 and later
program intersect_two_lines
  implicit none
  
  type point
    real::x,y
  end type point
  
  integer, parameter :: n = 4
  type(point)        :: p(n)
  
  p(1)%x = 4; p(1)%y = 0; p(2)%x = 6;  p(2)%y = 10 ! fist line 
  p(3)%x = 0; p(3)%y = 3; p(4)%x = 10; p(4)%y = 7  ! second line
  
  call intersect(p, n)
  
  contains
  
  subroutine intersect(p,m)
  integer, intent(in)       :: m
  type(point), intent(in)   :: p(m)
  integer   :: i
  real      :: a(2), b(2) ! y = a*x + b, for each line
  real      :: x, y       ! intersect point
  real      :: dx,dy      ! working variables
  
  do i = 1, 2
    dx = p(2*i-1)%x - p(2*i)%x
    dy = p(2*i-1)%y - p(2*i)%y
    if( dx == 0.) then    ! in case this line is of the form y = b
        a(i) = 0.
        b(i) = p(2*i-1)%y
    else
        a(i)= dy / dx
        b(i) = p(2*i-1)%y - a(i)*p(2*i-1)%x
    endif
  enddo
  
  if( a(1) - a(2) == 0. ) then
    write(*,*)"lines are not intersecting"
    return
  endif
  
  x = ( b(2) - b(1) ) / ( a(1) - a(2) )
  y = a(1) * x + b(1)
  write(*,*)x,y
  end subroutine intersect
end program intersect_two_lines
Output:
 5.00000000       5.00000000 

FreeBASIC[edit]

' version 16-08-2017
' compile with: fbc -s console
#Define NaN 0 / 0   ' FreeBASIC returns -1.#IND

Type _point_
    As Double x, y
End Type

Function l_l_intersect(s1 As _point_, e1 As _point_, s2 As _point_, e2 As _point_) As _point_

    Dim As Double a1 = e1.y - s1.y
    Dim As Double b1 = s1.x - e1.x
    Dim As Double c1 = a1 * s1.x + b1 * s1.y
    Dim As Double a2 = e2.y - s2.y
    Dim As Double b2 = s2.x - e2.x
    Dim As Double c2 = a2 * s2.x + b2 * s2.y
    Dim As Double det = a1 * b2 - a2 * b1

    If det = 0 Then
        Return Type(NaN, NaN)
    Else
        Return Type((b2 * c1 - b1 * c2) / det, (a1 * c2 - a2 * c1) / det)
    End If

End Function

' ------=< MAIN >=------

Dim As _point_ s1, e1, s2, e2, answer

s1.x = 4.0 : s1.y = 0.0 : e1.x =  6.0 : e1.y = 10.0  ' start and end of first line
s2.x = 0.0 : s2.y = 3.0 : e2.x = 10.0 : e2.y =  7.0  ' start and end of second line
answer = l_l_intersect(s1, e1, s2, e2)
Print answer.x, answer.y

s1.x = 0.0 : s1.y = 0.0 : e1.x =  0.0 : e1.y =  0.0  ' start and end of first line
s2.x = 0.0 : s2.y = 3.0 : e2.x = 10.0 : e2.y =  7.0  ' start and end of second line
answer = l_l_intersect(s1, e1, s2, e2)
Print answer.x, answer.y

' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
Output:
 5              5
-1.#IND       -1.#IND

Frink[edit]

lineIntersection[x1, y1, x2, y2, x3, y3, x4, y4] :=
{
   det = (x1 - x2)(y3 - y4) - (y1 - y2)(x3 - x4)
   if det == 0
      return undef

   t1 = (x1 y2 - y1 x2)
   t2 = (x3 y4 - y3 x4)
   px = (t1 (x3 - x4) - t2 (x1 - x2)) / det
   py = (t1 (y3 - y4) - t2 (y1 - y2)) / det
   return [px, py]
}

println[lineIntersection[4, 0, 6, 10, 0, 3, 10, 7]]
Output:
[5, 5]


Go[edit]

package main

import (
	"fmt"
	"errors"
)

type Point struct {
	x float64
	y float64
}

type Line struct {
	slope float64
	yint float64
}

func CreateLine (a, b Point) Line {
	slope := (b.y-a.y) / (b.x-a.x)
	yint := a.y - slope*a.x
	return Line{slope, yint}
} 

func EvalX (l Line, x float64) float64 {
	return l.slope*x + l.yint
}

func Intersection (l1, l2 Line) (Point, error) {
	if l1.slope == l2.slope {
		return Point{}, errors.New("The lines do not intersect")
	}
	x := (l2.yint-l1.yint) / (l1.slope-l2.slope)
	y := EvalX(l1, x)
	return Point{x, y}, nil
}

func main() {
	l1 := CreateLine(Point{4, 0}, Point{6, 10})
	l2 := CreateLine(Point{0, 3}, Point{10, 7})
	if result, err := Intersection(l1, l2); err == nil {
		fmt.Println(result)
	} else {
		fmt.Println("The lines do not intersect")
	}
}
Output:
{5 5}

Groovy[edit]

Translation of: Java
class Intersection {
    private static class Point {
        double x, y

        Point(double x, double y) {
            this.x = x
            this.y = y
        }

        @Override
        String toString() {
            return "($x, $y)"
        }
    }

    private static class Line {
        Point s, e

        Line(Point s, Point e) {
            this.s = s
            this.e = e
        }
    }

    private static Point findIntersection(Line l1, Line l2) {
        double a1 = l1.e.y - l1.s.y
        double b1 = l1.s.x - l1.e.x
        double c1 = a1 * l1.s.x + b1 * l1.s.y

        double a2 = l2.e.y - l2.s.y
        double b2 = l2.s.x - l2.e.x
        double c2 = a2 * l2.s.x + b2 * l2.s.y

        double delta = a1 * b2 - a2 * b1
        return new Point((b2 * c1 - b1 * c2) / delta, (a1 * c2 - a2 * c1) / delta)
    }

    static void main(String[] args) {
        Line l1 = new Line(new Point(4, 0), new Point(6, 10))
        Line l2 = new Line(new Point(0, 3), new Point(10, 7))
        println(findIntersection(l1, l2))

        l1 = new Line(new Point(0, 0), new Point(1, 1))
        l2 = new Line(new Point(1, 2), new Point(4, 5))
        println(findIntersection(l1, l2))
    }
}
Output:
(5.0, 5.0)
(-Infinity, -Infinity)

Haskell[edit]

type Line = (Point, Point)

type Point = (Float, Float)

intersection :: Line -> Line -> Either String Point
intersection ab pq =
  case determinant of
    0 -> Left "(Parallel lines – no intersection)"
    _ ->
      let [abD, pqD] = (\(a, b) -> diff ([fst, snd] <*> [a, b])) <$> [ab, pq]
          [ix, iy] =
            [\(ab, pq) -> diff [abD, ab, pqD, pq] / determinant] <*>
            [(abDX, pqDX), (abDY, pqDY)]
      in Right (ix, iy)
  where
    delta f x = f (fst x) - f (snd x)
    diff [a, b, c, d] = a * d - b * c
    [abDX, pqDX, abDY, pqDY] = [delta fst, delta snd] <*> [ab, pq]
    determinant = diff [abDX, abDY, pqDX, pqDY]

-- TEST ----------------------------------------------------------------
ab :: Line
ab = ((4.0, 0.0), (6.0, 10.0))

pq :: Line
pq = ((0.0, 3.0), (10.0, 7.0))

interSection :: Either String Point
interSection = intersection ab pq

main :: IO ()
main =
  putStrLn $
  case interSection of
    Left x -> x
    Right x -> show x
Output:
(5.0,5.0)

J[edit]

Translation of: C++

Solution:

det=: -/ .*   NB. calculate determinant
findIntersection=: (det ,."1 [: |: -/"2) %&det -/"2

Examples:

   line1=: 4 0 ,: 6 10
   line2=: 0 3 ,: 10 7
   line3=: 0 3 ,: 10 7.1
   line4=: 0 0 ,: 1 1
   line5=: 1 2 ,: 4 5
   line6=: 1 _1 ,: 4 4
   line7=: 2 5 ,: 3 _2

   findIntersection line1 ,: line2
5 5
   findIntersection line1 ,: line3
5.01089 5.05447
   findIntersection line4 ,: line5
__ __
   findIntersection line6 ,: line7
2.5 1.5

Java[edit]

Translation of: Kotlin
public class Intersection {
    private static class Point {
        double x, y;

        Point(double x, double y) {
            this.x = x;
            this.y = y;
        }

        @Override
        public String toString() {
            return String.format("{%f, %f}", x, y);
        }
    }

    private static class Line {
        Point s, e;

        Line(Point s, Point e) {
            this.s = s;
            this.e = e;
        }
    }

    private static Point findIntersection(Line l1, Line l2) {
        double a1 = l1.e.y - l1.s.y;
        double b1 = l1.s.x - l1.e.x;
        double c1 = a1 * l1.s.x + b1 * l1.s.y;

        double a2 = l2.e.y - l2.s.y;
        double b2 = l2.s.x - l2.e.x;
        double c2 = a2 * l2.s.x + b2 * l2.s.y;

        double delta = a1 * b2 - a2 * b1;
        return new Point((b2 * c1 - b1 * c2) / delta, (a1 * c2 - a2 * c1) / delta);
    }

    public static void main(String[] args) {
        Line l1 = new Line(new Point(4, 0), new Point(6, 10));
        Line l2 = new Line(new Point(0, 3), new Point(10, 7));
        System.out.println(findIntersection(l1, l2));

        l1 = new Line(new Point(0, 0), new Point(1, 1));
        l2 = new Line(new Point(1, 2), new Point(4, 5));
        System.out.println(findIntersection(l1, l2));
    }
}
Output:
{5.000000, 5.000000}
{-Infinity, -Infinity}

JavaScript[edit]

Translation of: Haskell

ES6[edit]

(() => {
    'use strict';
    // INTERSECTION OF TWO LINES ----------------------------------------------

    // intersection :: Line -> Line -> Either String (Float, Float)
    const intersection = (ab, pq) => {
        const
            delta = f => x => f(fst(x)) - f(snd(x)),
            [abDX, pqDX, abDY, pqDY] = apList(
                [delta(fst), delta(snd)], [ab, pq]
            ),
            determinant = abDX * pqDY - abDY * pqDX;

        return determinant !== 0 ? Right((() => {
            const [abD, pqD] = map(
                ([a, b]) => fst(a) * snd(b) - fst(b) * snd(a),
                [ab, pq]
            );
            return apList(
                [([pq, ab]) =>
                    (abD * pq - ab * pqD) / determinant
                ], [
                    [pqDX, abDX],
                    [pqDY, abDY]
                ]
            );
        })()) : Left('(Parallel lines – no intersection)');
    };

    // GENERIC FUNCTIONS ------------------------------------------------------

    // Left :: a -> Either a b
    const Left = x => ({
        type: 'Either',
        Left: x
    });

    // Right :: b -> Either a b
    const Right = x => ({
        type: 'Either',
        Right: x
    });

    // A list of functions applied to a list of arguments
    // <*> :: [(a -> b)] -> [a] -> [b]
    const apList = (fs, xs) => //
        [].concat.apply([], fs.map(f => //
            [].concat.apply([], xs.map(x => [f(x)]))));

    // fst :: (a, b) -> a
    const fst = tpl => tpl[0];

    // map :: (a -> b) -> [a] -> [b]
    const map = (f, xs) => xs.map(f);

    // snd :: (a, b) -> b
    const snd = tpl => tpl[1];

    // show :: a -> String
    const show = x => JSON.stringify(x); //, null, 2);


    // TEST --------------------------------------------------

    // lrIntersection ::Either String Point
    const lrIntersection = intersection([
        [4.0, 0.0],
        [6.0, 10.0]
    ], [
        [0.0, 3.0],
        [10.0, 7.0]
    ]);
    return show(lrIntersection.Left || lrIntersection.Right);
})();
Output:
[5,5]

jq[edit]

The implementation closely follows the zkl entry but uses the JSON array [x,y] to represent the point (x,y), and an array [P1,P2] to represent the line through points P1 and P2. Array destructuring is used for simplicity.

# determinant of 2x2 matrix
def det(a;b;c;d): a*d - b*c ;

# Input: an array representing a line (L1)
# Output: the intersection of L1 and L2 unless the lines are judged to be parallel
# This implementation uses "destructuring" to assign local variables
def lineIntersection(L2):
  .    as [[$ax,$ay], [$bx,$by]]
  | L2 as [[$cx,$cy], [$dx,$dy]]
  | {detAB: det($ax;$ay; $bx;$by),
     detCD: det($cx;$cy; $dx;$dy),
     abDx: ($ax - $bx),
     cdDx: ($cx - $dx),
     abDy: ($ay - $by),
     cdDy: ($cy - $dy)}
  | . + {xnom:  det(.detAB;.abDx;.detCD;.cdDx),
         ynom:  det(.detAB;.abDy;.detCD;.cdDy),
         denom: det(.abDx; .abDy;.cdDx; .cdDy) }
  | if (.denom|length < 10e-6)  # length/0 emits the absolute value
    then error("lineIntersect: parallel lines")
    else [.xnom/.denom, .ynom/.denom]
    end ;

Example:

[[4.0, 0.0], [6.0,10.0]] | lineIntersection([[0.0, 3.0], [10.0, 7.0]])
Output:
[5,5]

Julia[edit]

Works with: Julia version 0.6
Translation of: Kotlin
struct Point{T}
    x::T
    y::T
end

struct Line{T}
    s::Point{T}
    e::Point{T}
end

function intersection(l1::Line{T}, l2::Line{T}) where T<:Real
    a1 = l1.e.y - l1.s.y
    b1 = l1.s.x - l1.e.x
    c1 = a1 * l1.s.x + b1 * l1.s.y

    a2 = l2.e.y - l2.s.y
    b2 = l2.s.x - l2.e.x
    c2 = a2 * l2.s.x + b2 * l2.s.y

    Δ = a1 * b2 - a2 * b1
    # If lines are parallel, intersection point will contain infinite values
    return Point((b2 * c1 - b1 * c2) / Δ, (a1 * c2 - a2 * c1) / Δ)
end

l1 = Line(Point{Float64}(4, 0), Point{Float64}(6, 10))
l2 = Line(Point{Float64}(0, 3), Point{Float64}(10, 7))
println(intersection(l1, l2))

l1 = Line(Point{Float64}(0, 0), Point{Float64}(1, 1))
l2 = Line(Point{Float64}(1, 2), Point{Float64}(4, 5))
println(intersection(l1, l2))
Output:
Point{Float64}(5.0, 5.0)
Point{Float64}(-Inf, -Inf)

GeometryTypes library version[edit]

using GeometryTypes

a = LineSegment(Point2f0(4, 0), Point2f0(6, 10))
b = LineSegment(Point2f0(0, 3), Point2f0(10, 7))
@show intersects(a, b)   # --> intersects(a, b) = (true, Float32[5.0, 5.0])

Kotlin[edit]

Translation of: C#
// version 1.1.2

class PointF(val x: Float, val y: Float) {
    override fun toString() = "{$x, $y}"
}

class LineF(val s: PointF, val e: PointF)

fun findIntersection(l1: LineF, l2: LineF): PointF {
    val a1 = l1.e.y - l1.s.y
    val b1 = l1.s.x - l1.e.x
    val c1 = a1 * l1.s.x + b1 * l1.s.y

    val a2 = l2.e.y - l2.s.y
    val b2 = l2.s.x - l2.e.x
    val c2 = a2 * l2.s.x + b2 * l2.s.y

    val delta = a1 * b2 - a2 * b1
    // If lines are parallel, intersection point will contain infinite values
    return PointF((b2 * c1 - b1 * c2) / delta, (a1 * c2 - a2 * c1) / delta)
}

fun main(args: Array<String>) {
    var l1 = LineF(PointF(4f, 0f), PointF(6f, 10f))
    var l2 = LineF(PointF(0f, 3f), PointF(10f, 7f))
    println(findIntersection(l1, l2))
    l1 = LineF(PointF(0f, 0f), PointF(1f, 1f))
    l2 = LineF(PointF(1f, 2f), PointF(4f, 5f))
    println(findIntersection(l1, l2))
}
Output:
{5.0, 5.0}
{-Infinity, -Infinity}

Lua[edit]

Translation of: C#
function intersection (s1, e1, s2, e2)
  local d = (s1.x - e1.x) * (s2.y - e2.y) - (s1.y - e1.y) * (s2.x - e2.x)
  local a = s1.x * e1.y - s1.y * e1.x
  local b = s2.x * e2.y - s2.y * e2.x
  local x = (a * (s2.x - e2.x) - (s1.x - e1.x) * b) / d
  local y = (a * (s2.y - e2.y) - (s1.y - e1.y) * b) / d
  return x, y
end

local line1start, line1end = {x = 4, y = 0}, {x = 6, y = 10}
local line2start, line2end = {x = 0, y = 3}, {x = 10, y = 7}
print(intersection(line1start, line1end, line2start, line2end))
Output:
5       5

M2000 Interpreter[edit]

Module Lineintersection (lineAtuple, lineBtuple) {
	class line {
	private:
		slop, k
	public:
		function f(x) {
			=x*.slop-.k
		}
		function intersection(b as line) {
			if b.slop==.slop then
				=(,)
			else
				x1=(.k-b.k)/(.slop-b.slop)
				=(x1, .f(x1))
			end if
		}
	Class:
		module line {
			read  x1, y1, x2, y2
			if x1==x2 then error "wrong input"
			if x1>x2 then swap x1,x2 : swap y1, y2
			.slop<=(y1-y2)/(x1-x2)
			.k<=x1*.slop-y1
		}
	}
	M=line(!lineAtuple)
	N=line(!lineBtuple)
	Print M.intersection(N)
}
Lineintersection (4,0,6,10), (0,3,10,7)  ' print   5  5
Output:
   5  5

Maple[edit]

with(geometry):
line(L1, [point(A,[4,0]), point(B,[6,10])]):
line(L2, [point(C,[0,3]), point(E,[10,7])]):
coordinates(intersection(x,L1,L2));
Out:
[5, 5]

Mathematica/Wolfram Language[edit]

RegionIntersection[
    InfiniteLine[{{4, 0}, {6, 10}}], 
    InfiniteLine[{{0, 3}, {10, 7}}]
]
Output:
Point[{5, 5}]

MATLAB[edit]

function cross=intersection(line1,line2)
    a=polyfit(line1(:,1),line1(:,2),1);
    b=polyfit(line2(:,1),line2(:,2),1);
    cross=[a(1) -1; b(1) -1]\[-a(2);-b(2)];
end
Output:
line1=[4 0; 6 10]; line2=[0 3; 10 7]; cross=intersection(line1,line2)
cross =

    5.0000
    5.0000

Modula-2[edit]

MODULE LineIntersection;
FROM RealStr IMPORT RealToStr;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;

TYPE
    Point = RECORD
        x,y : REAL;
    END;

PROCEDURE PrintPoint(p : Point);
VAR buf : ARRAY[0..31] OF CHAR;
BEGIN
    WriteString("{");
    RealToStr(p.x, buf);
    WriteString(buf);
    WriteString(", ");
    RealToStr(p.y, buf);
    WriteString(buf);
    WriteString("}");
END PrintPoint;

TYPE
    Line = RECORD
        s,e : Point;
    END;

PROCEDURE FindIntersection(l1,l2 : Line) : Point;
VAR a1,b1,c1,a2,b2,c2,delta : REAL;
BEGIN
    a1 := l1.e.y - l1.s.y;
    b1 := l1.s.x - l1.e.x;
    c1 := a1 * l1.s.x + b1 * l1.s.y;

    a2 := l2.e.y - l2.s.y;
    b2 := l2.s.x - l2.e.x;
    c2 := a2 * l2.s.x + b2 * l2.s.y;

    delta := a1 * b2 - a2 * b1;
    RETURN Point{(b2 * c1 - b1 * c2) / delta, (a1 * c2 - a2 * c1) / delta};
END FindIntersection;

VAR
    l1,l2 : Line;
    result : Point;
BEGIN
    l1 := Line{{4.0,0.0}, {6.0,10.0}};
    l2 := Line{{0.0,3.0}, {10.0,7.0}};
    PrintPoint(FindIntersection(l1,l2));
    WriteLn;

    l1 := Line{{0.0,0.0}, {1.0,1.0}};
    l2 := Line{{1.0,2.0}, {4.0,5.0}};
    PrintPoint(FindIntersection(l1,l2));
    WriteLn;

    ReadChar;
END LineIntersection.

Nim[edit]

Translation of: Go
type
  Line = tuple
    slope: float
    yInt: float
  Point = tuple
    x: float
    y: float

func createLine(a, b: Point): Line =
  result.slope = (b.y - a.y) / (b.x - a.x)
  result.yInt = a.y - result.slope * a.x

func evalX(line: Line, x: float): float =
  line.slope * x + line.yInt

func intersection(line1, line2: Line): Point =
  let x = (line2.yInt - line1.yInt) / (line1.slope - line2.slope)
  let y = evalX(line1, x)
  (x, y)

var line1 = createLine((4.0, 0.0), (6.0, 10.0))
var line2 = createLine((0.0, 3.0), (10.0, 7.0))
echo intersection(line1, line2)
line1 = createLine((0.0, 0.0), (1.0, 1.0))
line2 = createLine((1.0, 2.0), (4.0, 5.0))
echo intersection(line1, line2)
Output:
(x: 5.0, y: 5.0)
(x: inf, y: inf)

Perl[edit]

Translation of: C#

If warning are enabled the second print will issue a warning since we are trying to print out an undef

sub intersect {
  my ($x1, $y1, $x2, $y2, $x3, $y3, $x4, $y4) = @_;
  my $a1 = $y2 - $y1;
  my $b1 = $x1 - $x2;
  my $c1 = $a1 * $x1 + $b1 * $y1;
  my $a2 = $y4 - $y3;
  my $b2 = $x3 - $x4;
  my $c2 = $a2 * $x3 + $b2 * $y3;
  my $delta = $a1 * $b2 - $a2 * $b1;
  return (undef, undef) if $delta == 0;
  # If delta is 0, i.e. lines are parallel then the below will fail
  my $ix = ($b2 * $c1 - $b1 * $c2) / $delta;
  my $iy = ($a1 * $c2 - $a2 * $c1) / $delta;
  return ($ix, $iy);
}

my ($ix, $iy) = intersect(4, 0, 6, 10, 0, 3, 10, 7);
print "$ix $iy\n";
($ix, $iy) = intersect(0, 0, 1, 1, 1, 2, 4, 5);
print "$ix $iy\n";

Phix[edit]

Library: Phix/online

You can run this online here.

with javascript_semantics
enum X, Y
 
function abc(sequence s,e)
-- yeilds {a,b,c}, corresponding to ax+by=c
    atom a = e[Y]-s[Y], b = s[X]-e[X], c = a*s[X]+b*s[Y]
    return {a,b,c}
end function
 
procedure intersect(sequence s1, e1, s2, e2)
    atom {a1,b1,c1} = abc(s1,e1),
         {a2,b2,c2} = abc(s2,e2),
         delta = a1*b2 - a2*b1,
         x = b2*c1 - b1*c2,
         y = a1*c2 - a2*c1
    ?iff(delta=0?"parallel lines/do not intersect"
                :{x/delta, y/delta})
end procedure
 
intersect({4,0},{6,10},{0,3},{10,7})        -- {5,5}
intersect({4,0},{6,10},{0,3},{10,7.1})      -- {5.010893246,5.054466231}
intersect({0,0},{0,0},{0,3},{10,7})         -- "parallel lines/do not intersect"
intersect({0,0},{1,1},{1,2},{4,5})          -- "parallel lines/do not intersect"
intersect({1,-1},{4,4},{2,5},{3,-2})        -- {2.5,1.5}

Processing[edit]

void setup() {
  // test lineIntersect() with visual and textual output
  float lineA[] = {4, 0, 6, 10};  // try 4, 0, 6, 4
  float lineB[] = {0, 3, 10, 7};  // for non intersecting test
  PVector pt = lineInstersect(lineA[0], lineA[1], lineA[2], lineA[3], 
                              lineB[0], lineB[1], lineB[2], lineB[3]);
  scale(9);
  line(lineA[0], lineA[1], lineA[2], lineA[3]);
  line(lineB[0], lineB[1], lineB[2], lineB[3]);
  if (pt != null) {
    stroke(255);
    point(pt.x, pt.y);
    println(pt.x, pt.y);
  } else {
    println("No point");
  }
}

PVector lineInstersect(float Ax1, float Ay1, float Ax2, float Ay2, 
  float  Bx1, float By1, float Bx2, float By2) {
  // returns null if there is no intersection
  float uA, uB;
  float d = ((By2 - By1) * (Ax2 - Ax1) - (Bx2 - Bx1) * (Ay2 - Ay1));
  if (d != 0) {
    uA = ((Bx2 - Bx1) * (Ay1 - By1) - (By2 - By1) * (Ax1 - Bx1)) / d;         
    uB = ((Ax2 - Ax1) * (Ay1 - By1) - (Ay2 - Ay1) * (Ax1 - Bx1)) / d;
  } else {
    return null;
  }
  if (0 > uA || uA > 1 || 0 > uB || uB > 1) {
    return null;
  }
  float x = Ax1 + uA * (Ax2 - Ax1);
  float y = Ay1 + uA * (Ay2 - Ay1);
  return new PVector(x, y);
}

Processing Python mode[edit]

from __future__ import division

def setup():
    """ test line_intersect() with visual and textual output """
    (a, b), (c, d) = (4, 0), (6, 10)  # try (4, 0), (6, 4)
    (e, f), (g, h) = (0, 3), (10, 7)  # for non intersecting test
    pt = line_instersect(a, b, c, d, e, f, g, h)
    scale(9)
    line(a, b, c, d)
    line(e, f, g, h)
    if pt:
        x, y = pt
        stroke(255)
        point(x, y)
    println(pt)  # prints x, y coordinates or 'None'

def line_instersect(Ax1, Ay1, Ax2, Ay2, Bx1, By1, Bx2, By2):
    """ returns a (x, y) tuple or None if there is no intersection """
    d = (By2 - By1) * (Ax2 - Ax1) - (Bx2 - Bx1) * (Ay2 - Ay1)
    if d:
        uA = ((Bx2 - Bx1) * (Ay1 - By1) - (By2 - By1) * (Ax1 - Bx1)) / d
        uB = ((Ax2 - Ax1) * (Ay1 - By1) - (Ay2 - Ay1) * (Ax1 - Bx1)) / d
    else:
        return
    if not(0 <= uA <= 1 and 0 <= uB <= 1):
        return
    x = Ax1 + uA * (Ax2 - Ax1)
    y = Ay1 + uA * (Ay2 - Ay1)
    return x, y

Python[edit]

Find the intersection without importing third-party libraries.

def line_intersect(Ax1, Ay1, Ax2, Ay2, Bx1, By1, Bx2, By2):
    """ returns a (x, y) tuple or None if there is no intersection """
    d = (By2 - By1) * (Ax2 - Ax1) - (Bx2 - Bx1) * (Ay2 - Ay1)
    if d:
        uA = ((Bx2 - Bx1) * (Ay1 - By1) - (By2 - By1) * (Ax1 - Bx1)) / d
        uB = ((Ax2 - Ax1) * (Ay1 - By1) - (Ay2 - Ay1) * (Ax1 - Bx1)) / d
    else:
        return
    if not(0 <= uA <= 1 and 0 <= uB <= 1):
        return
    x = Ax1 + uA * (Ax2 - Ax1)
    y = Ay1 + uA * (Ay2 - Ay1)
    
    return x, y

if __name__ == '__main__':
    (a, b), (c, d) = (4, 0), (6, 10)  # try (4, 0), (6, 4)
    (e, f), (g, h) = (0, 3), (10, 7)  # for non intersecting test
    pt = line_intersect(a, b, c, d, e, f, g, h)
    print(pt)
Output:
(5.0, 5.0)


Or, labelling the moving parts a little more, and returning a composable option value containing either a message (in the absence of an intersection), or a pair of coordinates:

Works with: Python version 3.7
'''The intersection of two lines.'''

from itertools import product


# intersection :: Line -> Line -> Either String Point
def intersection(ab):
    '''Either the point at which the lines ab and pq
       intersect, or a message string indicating that
       they are parallel and have no intersection.'''
    def delta(f):
        return lambda x: f(fst(x)) - f(snd(x))

    def prodDiff(abcd):
        [a, b, c, d] = abcd
        return (a * d) - (b * c)

    def go(pq):
        [abDX, pqDX, abDY, pqDY] = apList(
            [delta(fst), delta(snd)]
        )([ab, pq])
        determinant = prodDiff([abDX, abDY, pqDX, pqDY])

        def point():
            [abD, pqD] = map(
                lambda xy: prodDiff(
                    apList([fst, snd])([fst(xy), snd(xy)])
                ), [ab, pq]
            )
            return apList(
                [lambda abpq: prodDiff(
                    [abD, fst(abpq), pqD, snd(abpq)]) / determinant]
            )(
                [(abDX, pqDX), (abDY, pqDY)]
            )
        return Right(point()) if 0 != determinant else Left(
            '( Parallel lines - no intersection )'
        )

    return lambda pq: bindLR(go(pq))(
        lambda xs: Right((fst(xs), snd(xs)))
    )


# --------------------------TEST---------------------------
# main :: IO()
def main():
    '''Test'''

    # Left(message - no intersection) or Right(point)
    # lrPoint :: Either String Point
    lrPoint = intersection(
        ((4.0, 0.0), (6.0, 10.0))
    )(
        ((0.0, 3.0), (10.0, 7.0))
    )
    print(
        lrPoint['Left'] or lrPoint['Right']
    )


# --------------------GENERIC FUNCTIONS--------------------

# Left :: a -> Either a b
def Left(x):
    '''Constructor for an empty Either (option type) value
       with an associated string.'''
    return {'type': 'Either', 'Right': None, 'Left': x}


# Right :: b -> Either a b
def Right(x):
    '''Constructor for a populated Either (option type) value'''
    return {'type': 'Either', 'Left': None, 'Right': x}


# apList (<*>) :: [(a -> b)] -> [a] -> [b]
def apList(fs):
    '''The application of each of a list of functions,
       to each of a list of values.
    '''
    def go(fx):
        f, x = fx
        return f(x)
    return lambda xs: [
        go(x) for x
        in product(fs, xs)
    ]


# bindLR (>>=) :: Either a -> (a -> Either b) -> Either b
def bindLR(m):
    '''Either monad injection operator.
       Two computations sequentially composed,
       with any value produced by the first
       passed as an argument to the second.'''
    return lambda mf: (
        mf(m.get('Right')) if None is m.get('Left') else m
    )


# fst :: (a, b) -> a
def fst(tpl):
    '''First member of a pair.'''
    return tpl[0]


# snd :: (a, b) -> b
def snd(tpl):
    '''Second member of a pair.'''
    return tpl[1]


# MAIN ---
if __name__ == '__main__':
    main()
Output:
(5.0, 5.0)
Library: Shapely

Find the intersection by importing the external Shapely library.

from shapely.geometry import LineString

if __name__ == "__main__":
    line1 = LineString([(4, 0), (6, 10)])
    line2 = LineString([(0, 3), (10, 7)])
    print(line1.intersection(line2))
Output:
POINT (5 5)

Racket[edit]

Translation of: C++
#lang racket/base
(define (det a b c d) (- (* a d) (* b c))) ; determinant

(define (line-intersect ax ay bx by cx cy dx dy) ; --> (values x y)
  (let* ((det.ab (det ax ay bx by))
         (det.cd (det cx cy dx dy))
         (abΔx (- ax bx))
         (cdΔx (- cx dx))
         (abΔy (- ay by))
         (cdΔy (- cy dy))
         (xnom (det det.ab abΔx det.cd cdΔx))
         (ynom (det det.ab abΔy det.cd cdΔy))
         (denom (det abΔx abΔy cdΔx cdΔy)))
    (when (zero? denom)
      (error 'line-intersect "parallel lines"))
    (values (/ xnom denom) (/ ynom denom))))

(module+ test (line-intersect 4 0 6 10
                              0 3 10 7))
Output:
5
5

Raku[edit]

(formerly Perl 6)

Works with: Rakudo version 2016.11
Translation of: zkl
sub intersection (Real $ax, Real $ay, Real $bx, Real $by,
                  Real $cx, Real $cy, Real $dx, Real $dy ) {

    sub term:<|AB|> { determinate($ax, $ay, $bx, $by) }
    sub term:<|CD|> { determinate($cx, $cy, $dx, $dy) }

    my $ΔxAB = $ax - $bx;
    my $ΔyAB = $ay - $by;
    my $ΔxCD = $cx - $dx;
    my $ΔyCD = $cy - $dy;

    my $x-numerator = determinate( |AB|, $ΔxAB, |CD|, $ΔxCD );
    my $y-numerator = determinate( |AB|, $ΔyAB, |CD|, $ΔyCD );
    my $denominator = determinate( $ΔxAB, $ΔyAB, $ΔxCD, $ΔyCD );

    return 'Lines are parallel' if $denominator == 0;

    ($x-numerator/$denominator, $y-numerator/$denominator);
}

sub determinate ( Real $a, Real $b, Real $c, Real $d ) { $a * $d - $b * $c }

# TESTING
say 'Intersection point: ', intersection( 4,0, 6,10, 0,3, 10,7 );
say 'Intersection point: ', intersection( 4,0, 6,10, 0,3, 10,7.1 );
say 'Intersection point: ', intersection( 0,0, 1,1, 1,2, 4,5 );
Output:
Intersection point: (5 5)
Intersection point: (5.010893 5.054466)
Intersection point: Lines are parallel

REXX[edit]

version 1[edit]

Naive implementation. To be improved for parallel lines and degenerate lines such as y=5 or x=8.

/* REXX */
Parse Value '(4.0,0.0)'  With '(' xa ',' ya ')'
Parse Value '(6.0,10.0)' With '(' xb ',' yb ')'
Parse Value '(0.0,3.0)'  With '(' xc ',' yc ')'
Parse Value '(10.0,7.0)' With '(' xd ',' yd ')'

Say 'The two lines are:'
Say 'yab='ya-xa*((yb-ya)/(xb-xa))'+x*'||((yb-ya)/(xb-xa))
Say 'ycd='yc-xc*((yd-yc)/(xd-xc))'+x*'||((yd-yc)/(xd-xc))

x=((yc-xc*((yd-yc)/(xd-xc)))-(ya-xa*((yb-ya)/(xb-xa))))/,
                         (((yb-ya)/(xb-xa))-((yd-yc)/(xd-xc)))
Say 'x='||x
        y=ya-xa*((yb-ya)/(xb-xa))+x*((yb-ya)/(xb-xa))
Say 'yab='y
Say 'ycd='yc-xc*((yd-yc)/(xd-xc))+x*((yd-yc)/(xd-xc))
Say 'Intersection: ('||x','y')'
Output:
The two lines are:
yab=-20.0+x*5
ycd=3.0+x*0.4
x=5
yab=5.0
ycd=5.0
Intersection: (5,5.0)

version 2[edit]

complete implementation taking care of all possibilities.
Variables are named after the Austrian notation for a straight line: y=k*x+d

say ggx1('4.0 0.0 6.0 10.0 0.0 3.0 10.0 7.0')
say ggx1('0.0 0.0 0.0 10.0 0.0 3.0 10.0 7.0')
say ggx1('0.0 0.0 0.0 10.0 0.0 3.0 10.0 7.0')
say ggx1('0.0 0.0 0.0  1.0 1.0 0.0  1.0 7.0')
say ggx1('0.0 0.0 0.0  0.0 0.0 3.0 10.0 7.0')
say ggx1('0.0 0.0 3.0  3.0 0.0 0.0  6.0 6.0')
say ggx1('0.0 0.0 3.0  3.0 0.0 1.0  6.0 7.0')
Exit

ggx1: Procedure
/*---------------------------------------------------------------------
* find the intersection of the lines AB and CD
*--------------------------------------------------------------------*/
Parse Arg xa  ya  xb  yb   xc  yc  xd   yd
Say 'A=('xa'/'ya') B=('||xb'/'yb') C=('||xc'/'yc') D=('||xd'/'yd')'
res=''
If xa=xb Then Do                    /* AB is a vertical line         */
  k1='*'                            /* slope is infinite             */
  x1=xa                             /* intersection's x is xa        */
  If ya=yb Then                     /* coordinates are equal         */
    res='Points A and B are identical' /* special case               */
  End
Else Do                             /* AB is not a vertical line     */
  k1=(yb-ya)/(xb-xa)                /* compute the slope of AB       */
  d1=ya-k1*xa                /* and its intersection with the y-axis */
  End
If xc=xd Then Do                    /* CD is a vertical line         */
  k2='*'                            /* slope is infinite             */
  x2=xc                             /* intersection's x is xc        */
  If yc=yd Then                     /* coordinates are equal         */
    res='Points C and D are identical' /* special case               */
  End
Else Do                             /* CD is not a vertical line     */
  k2=(yd-yc)/(xd-xc)                /* compute the slope of CD       */
  d2=yc-k2*xc                /* and its intersection with the y-axis */
  End

If res='' Then Do                   /* no special case so far        */
  If k1='*' Then Do                 /* AB is vertical                */
    If k2='*' Then Do               /* CD is vertical                */
      If x1=x2 Then                 /* and they are identical        */
        res='Lines AB and CD are identical'
      Else                          /* not identical                 */
        res='Lines AB and CD are parallel'
      End
    Else Do
      x=x1                          /* x is taken from AB            */
      y=k2*x+d2                     /* y is computed from CD         */
      End
    End
  Else Do                           /* AB is not verical             */
    If k2='*' Then Do               /* CD is vertical                */
      x=x2                          /* x is taken from CD            */
      y=k1*x+d1                     /* y is computed from AB         */
      End
    Else Do                         /* AB and CD are not vertical    */
      If k1=k2 Then Do              /* identical slope               */
        If d1=d2 Then               /* same intersection with x=0    */
          res='Lines AB and CD are identical'
        Else                        /* otherwise                     */
          res='Lines AB and CD are parallel'
        End
      Else Do                       /* finally the normal case       */
        x=(d2-d1)/(k1-k2)           /* compute x                     */
        y=k1*x+d1                   /* and y                         */
        End
      End
    End
  End
  If res='' Then                    /* not any special case          */
    res='Intersection is ('||x'/'y')'  /* that's the result          */
  Return '  -->' res
Output:
A=(4.0/0.0) B=(6.0/10.0) C=(0.0/3.0) D=(10.0/7.0)
  --> Intersection is (5/5.0)
A=(0.0/0.0) B=(0.0/10.0) C=(0.0/3.0) D=(10.0/7.0)
  --> Intersection is (0.0/3.0)
A=(0.0/0.0) B=(0.0/10.0) C=(0.0/3.0) D=(10.0/7.0)
  --> Intersection is (0.0/3.0)
A=(0.0/0.0) B=(0.0/1.0) C=(1.0/0.0) D=(1.0/7.0)
  --> Lines AB and CD are parallel
A=(0.0/0.0) B=(0.0/0.0) C=(0.0/3.0) D=(10.0/7.0)
  --> Points A and B are identical
A=(0.0/0.0) B=(3.0/3.0) C=(0.0/0.0) D=(6.0/6.0)
  --> Lines AB and CD are identical
A=(0.0/0.0) B=(3.0/3.0) C=(0.0/1.0) D=(6.0/7.0)
  --> Lines AB and CD are parallel

Ring[edit]

# Project : Find the intersection of two lines

xa=4
ya=0
xb=6
yb=10
xc=0
yc=3
xd=10
yd=7
see "the two lines are:" + nl
see "yab=" + (ya-xa*((yb-ya)/(xb-xa))) + "+x*" + ((yb-ya)/(xb-xa)) + nl
see "ycd=" + (yc-xc*((yd-yc)/(xd-xc))) + "+x*" + ((yd-yc)/(xd-xc)) + nl
x=((yc-xc*((yd-yc)/(xd-xc)))-(ya-xa*((yb-ya)/(xb-xa))))/(((yb-ya)/(xb-xa))-((yd-yc)/(xd-xc)))
see "x=" + x + nl
y=ya-xa*((yb-ya)/(xb-xa))+x*((yb-ya)/(xb-xa))
see "yab=" + y + nl
see "ycd=" + (yc-xc*((yd-yc)/(xd-xc))+x*((yd-yc)/(xd-xc))) + nl
see "intersection: " + x + "," + y + nl

Output:

the two lines are:
yab=-20+x*5
ycd=3+x*0.4
x=5
yab=5
ycd=5
intersection: 5,5

Ruby[edit]

Point = Struct.new(:x, :y)

class Line
  attr_reader :a, :b

  def initialize(point1, point2)
    @a = (point1.y - point2.y).fdiv(point1.x - point2.x)
    @b = point1.y - @a*point1.x
  end

  def intersect(other)
    return nil if @a == other.a
    x = (other.b - @b).fdiv(@a - other.a)
    y = @a*x + @b
    Point.new(x,y)
  end

  def to_s
    "y = #{@a}x + #{@b}"
  end

end

l1 = Line.new(Point.new(4, 0), Point.new(6, 10))
l2 = Line.new(Point.new(0, 3), Point.new(10, 7))

puts "Line #{l1} intersects line #{l2} at #{l1.intersect(l2)}."
Output:
Line y = 5.0x + -20.0 intersects line y = 0.4x + 3.0 at #<struct Point x=5.0, y=5.0>.

Rust[edit]

#[derive(Copy, Clone, Debug)]
struct Point {
    x: f64,
    y: f64,
}

impl Point {
    pub fn new(x: f64, y: f64) -> Self {
        Point { x, y }
    }
}

#[derive(Copy, Clone, Debug)]
struct Line(Point, Point);

impl Line {
    pub fn intersect(self, other: Self) -> Option<Point> {
        let a1 = self.1.y - self.0.y;
        let b1 = self.0.x - self.1.x;
        let c1 = a1 * self.0.x + b1 * self.0.y;

        let a2 = other.1.y - other.0.y;
        let b2 = other.0.x - other.1.x;
        let c2 = a2 * other.0.x + b2 * other.0.y;

        let delta = a1 * b2 - a2 * b1;

        if delta == 0.0 {
            return None;
        }

        Some(Point {
            x: (b2 * c1 - b1 * c2) / delta,
            y: (a1 * c2 - a2 * c1) / delta,
        })
    }
}

fn main() {
    let l1 = Line(Point::new(4.0, 0.0), Point::new(6.0, 10.0));
    let l2 = Line(Point::new(0.0, 3.0), Point::new(10.0, 7.0));
    println!("{:?}", l1.intersect(l2));

    let l1 = Line(Point::new(0.0, 0.0), Point::new(1.0, 1.0));
    let l2 = Line(Point::new(1.0, 2.0), Point::new(4.0, 5.0));
    println!("{:?}", l1.intersect(l2));
}
Output:
Some(Point { x: 5.0, y: 5.0 })
None

Scala[edit]

object Intersection extends App {
  val (l1, l2) = (LineF(PointF(4, 0), PointF(6, 10)), LineF(PointF(0, 3), PointF(10, 7)))

  def findIntersection(l1: LineF, l2: LineF): PointF = {
    val a1 = l1.e.y - l1.s.y
    val b1 = l1.s.x - l1.e.x
    val c1 = a1 * l1.s.x + b1 * l1.s.y

    val a2 = l2.e.y - l2.s.y
    val b2 = l2.s.x - l2.e.x
    val c2 = a2 * l2.s.x + b2 * l2.s.y

    val delta = a1 * b2 - a2 * b1
    // If lines are parallel, intersection point will contain infinite values
    PointF((b2 * c1 - b1 * c2) / delta, (a1 * c2 - a2 * c1) / delta)
  }

  def l01 = LineF(PointF(0f, 0f), PointF(1f, 1f))
  def l02 = LineF(PointF(1f, 2f), PointF(4f, 5f))

  case class PointF(x: Float, y: Float) {
    override def toString = s"{$x, $y}"
  }

  case class LineF(s: PointF, e: PointF)

  println(findIntersection(l1, l2))
  println(findIntersection(l01, l02))

}
Output:
See it in running in your browser by (JavaScript)

or by Scastie (JVM).

Sidef[edit]

Translation of: Raku
func det(a, b, c, d) { a*d - b*c }

func intersection(ax, ay, bx, by,
                  cx, cy, dx, dy) {

    var detAB = det(ax,ay, bx,by)
    var detCD = det(cx,cy, dx,dy)

    var ΔxAB = (ax - bx)
    var ΔyAB = (ay - by)
    var ΔxCD = (cx - dx)
    var ΔyCD = (cy - dy)

    var x_numerator = det(detAB, ΔxAB, detCD, ΔxCD)
    var y_numerator = det(detAB, ΔyAB, detCD, ΔyCD)
    var denominator = det( ΔxAB, ΔyAB,  ΔxCD, ΔyCD)

    denominator == 0 && return 'lines are parallel'
    [x_numerator / denominator, y_numerator / denominator]
}

say ('Intersection point: ', intersection(4,0, 6,10, 0,3, 10,7))
say ('Intersection point: ', intersection(4,0, 6,10, 0,3, 10,7.1))
say ('Intersection point: ', intersection(0,0, 1,1, 1,2, 4,5))
Output:
Intersection point: [5, 5]
Intersection point: [2300/459, 2320/459]
Intersection point: lines are parallel

Swift[edit]

struct Point {
  var x: Double
  var y: Double
}

struct Line {
  var p1: Point
  var p2: Point

  var slope: Double {
    guard p1.x - p2.x != 0.0 else { return .nan }

    return (p1.y-p2.y) / (p1.x-p2.x)
  }

  func intersection(of other: Line) -> Point? {
    let ourSlope = slope
    let theirSlope = other.slope

    guard ourSlope != theirSlope else { return nil }

    if ourSlope.isNaN && !theirSlope.isNaN {
      return Point(x: p1.x, y: (p1.x - other.p1.x) * theirSlope + other.p1.y)
    } else if theirSlope.isNaN && !ourSlope.isNaN {
      return Point(x: other.p1.x, y: (other.p1.x - p1.x) * ourSlope + p1.y)
    } else {
      let x = (ourSlope*p1.x - theirSlope*other.p1.x + other.p1.y - p1.y) / (ourSlope - theirSlope)
      return Point(x: x, y: theirSlope*(x - other.p1.x) + other.p1.y)
    }
  }
}

let l1 = Line(p1: Point(x: 4.0, y: 0.0), p2: Point(x: 6.0, y: 10.0))
let l2 = Line(p1: Point(x: 0.0, y: 3.0), p2: Point(x: 10.0, y: 7.0))

print("Intersection at : \(l1.intersection(of: l2)!)")
Output:
Intersection at : Point(x: 5.0, y: 5.0)

TI-83 BASIC[edit]

Works with: TI-83 BASIC version TI-84Plus 2.55MP
Translation of: Rexx

Simple version:

[[4,0][6,10][0,3][10,7]]→[A]
([A](2,2)-[A](1,2))/([A](2,1)-[A](1,1))→B
[A](1,2)-[A](1,1)*B→A
([A](4,2)-[A](3,2))/([A](4,1)-[A](3,1))→D
[A](3,2)-[A](3,1)*D→C
(C-A)/(B-D)→X
A+X*B→Y
C+X*D→Z
Disp {X,Y}
Output:
       {5 5}

Full version:

[[4,0][6,10][0,3][10,7]]→[A]
{4,2}→Dim([B])
0→M
If [A](1,1)=[A](2,1)
Then
  [A](1,1)→[B](3,1)
  If [A](1,2)=[A](2,2):1→M
Else
  1→[B](4,1)
  ([A](2,2)-[A](1,2))/([A](2,1)-[A](1,1))→[B](1,1)
  [A](1,2)-[B](1,1)*[A](1,1)→[B](2,1)
End
If [A](3,1)=[A](4,1) 
Then
  [A](3,1)→[B](3,2)
  If [A](3,2)=[A](4,2):2→M
Else
  1→[B](4,2)
  ([A](4,2)-[A](3,2))/([A](4,1)-[A](3,1))→[B](1,2)
  [A](3,2)-[B](1,2)*[A](3,1)→[B](2,2)
End
If M=0 Then
  If [B](4,1)=0
  Then 
    If [B](4,2)=0
    Then 
      If [B](3,1)=[B](3,2) 
      Then:3→M
      Else:4→M
      End
    Else
      [B](3,1)→X
      [B](1,2)*X+[B](2,2)→Y
    End
  Else
    If [B](4,2)=0
    Then
      [B](3,2)→X
      [B](1,1)*X+[B](2,1)→Y
    Else
      If [B](1,1)=[B](1,2)
      Then 
        If [B](2,1)=[B](2,2)
        Then:5→M
        Else:6→M
        End
      Else
        ([B](2,2)-[B](2,1))/([B](1,1)-[B](1,2))→X
        [B](1,1)*X+[B](2,1)→Y
      End
    End
  End
End
Disp {X,Y,M}
Output:
       {5 5}

Visual Basic[edit]

Works with: Visual Basic version 5
Works with: Visual Basic version 6
Works with: VBA version 6.5
Works with: VBA version 7.1
Option Explicit

Public Type Point
  x As Double
  y As Double
  invalid As Boolean
End Type

Public Type Line
  s As Point
  e As Point
End Type
  
Public Function GetIntersectionPoint(L1 As Line, L2 As Line) As Point
Dim a1 As Double
Dim b1 As Double
Dim c1 As Double
Dim a2 As Double
Dim b2 As Double
Dim c2 As Double
Dim det As Double

  a1 = L1.e.y - L1.s.y
  b1 = L1.s.x - L1.e.x
  c1 = a1 * L1.s.x + b1 * L1.s.y
  a2 = L2.e.y - L2.s.y
  b2 = L2.s.x - L2.e.x
  c2 = a2 * L2.s.x + b2 * L2.s.y
  det = a1 * b2 - a2 * b1
  
  If det Then
    With GetIntersectionPoint
      .x = (b2 * c1 - b1 * c2) / det
      .y = (a1 * c2 - a2 * c1) / det
    End With
  Else
    GetIntersectionPoint.invalid = True
  End If
End Function

Sub Main()
Dim ln1 As Line
Dim ln2 As Line
Dim ip As Point
  
  ln1.s.x = 4
  ln1.s.y = 0
  ln1.e.x = 6
  ln1.e.y = 10
  ln2.s.x = 0
  ln2.s.y = 3
  ln2.e.x = 10
  ln2.e.y = 7
  ip = GetIntersectionPoint(ln1, ln2)
  Debug.Assert Not ip.invalid
  Debug.Assert ip.x = 5 And ip.y = 5
  
  LSet ln2.s = ln2.e
  ip = GetIntersectionPoint(ln1, ln2)
  Debug.Assert ip.invalid
  
  LSet ln2 = ln1
  ip = GetIntersectionPoint(ln1, ln2)
  Debug.Assert ip.invalid
  
End Sub

Visual Basic .NET[edit]

Translation of: C#
Imports System.Drawing

Module Module1

    Function FindIntersection(s1 As PointF, e1 As PointF, s2 As PointF, e2 As PointF) As PointF
        Dim a1 = e1.Y - s1.Y
        Dim b1 = s1.X - e1.X
        Dim c1 = a1 * s1.X + b1 * s1.Y

        Dim a2 = e2.Y - s2.Y
        Dim b2 = s2.X - e2.X
        Dim c2 = a2 * s2.X + b2 * s2.Y

        Dim delta = a1 * b2 - a2 * b1

        'If lines are parallel, the result will be (NaN, NaN).
        Return If(delta = 0, New PointF(Single.NaN, Single.NaN), New PointF((b2 * c1 - b1 * c2) / delta, (a1 * c2 - a2 * c1) / delta))
    End Function

    Sub Main()
        Dim p = Function(x As Single, y As Single) New PointF(x, y)
        Console.WriteLine(FindIntersection(p(4.0F, 0F), p(6.0F, 10.0F), p(0F, 3.0F), p(10.0F, 7.0F)))
        Console.WriteLine(FindIntersection(p(0F, 0F), p(1.0F, 1.0F), p(1.0F, 2.0F), p(4.0F, 5.0F)))
    End Sub

End Module
Output:
{X=5, Y=5}
{X=NaN, Y=NaN}

Wren[edit]

Translation of: Kotlin
class Point {
    construct new(x, y) {
        _x = x
        _y = y
    }

    x { _x }
    y { _y }

    toString { "(%(_x), %(_y))" }
}

class Line {
    construct new(s, e) {
        _s = s
        _e = e
    }

    s { _s }
    e { _e }
}

var findIntersection = Fn.new { |l1, l2|
    var a1 = l1.e.y - l1.s.y
    var b1 = l1.s.x - l1.e.x
    var c1 = a1*l1.s.x + b1*l1.s.y

    var a2 = l2.e.y - l2.s.y
    var b2 = l2.s.x - l2.e.x
    var c2 = a2*l2.s.x + b2*l2.s.y

    var delta = a1*b2 - a2*b1
    // if lines are parallel, intersection point will contain infinite values
    return Point.new((b2*c1 - b1*c2)/delta, (a1*c2 - a2*c1)/delta)
}

var l1 = Line.new(Point.new(4, 0), Point.new(6, 10))
var l2 = Line.new(Point.new(0, 3), Point.new(10, 7))
System.print(findIntersection.call(l1, l2))
l1 = Line.new(Point.new(0, 0), Point.new(1, 1))
l2 = Line.new(Point.new(1, 2), Point.new(4, 5))
System.print(findIntersection.call(l1, l2))
Output:
(5, 5)
(-infinity, -infinity)

XPL0[edit]

func real Det; real A0, B0, A1, B1;
return A0*B1 - A1*B0;

func Cramer; real A0, B0, C0, A1, B1, C1;
real Denom;
[Denom:= Det(A0, B0, A1, B1);
RlOut(0, Det(C0, B0, C1, B1) / Denom);
RlOut(0, Det(A0, C0, A1, C1) / Denom);
];

real L0, L1, M0, M1;
[L0:= [[ 4.,  0.], [ 6., 10.]];
 L1:= [[ 0.,  3.], [10.,  7.]];
M0:= (L0(1,1) - L0(0,1)) / (L0(1,0) - L0(0,0));
M1:= (L1(1,1) - L1(0,1)) / (L1(1,0) - L1(0,0));
Cramer(M0, -1., M0*L0(0,0)-L0(0,1), M1, -1., M1*L1(0,0)-L1(0,1));
]
Output:
    5.00000    5.00000

zkl[edit]

Translation of: C++
fcn lineIntersect(ax,ay, bx,by,   cx,cy, dx,dy){	// --> (x,y)
   detAB,detCD := det(ax,ay, bx,by), det(cx,cy, dx,dy);
   abDx,cdDx := ax - bx, cx - dx;	// delta x
   abDy,cdDy := ay - by, cy - dy;	// delta y

   xnom,ynom := det(detAB,abDx, detCD,cdDx), det(detAB,abDy, detCD,cdDy);
   denom     := det(abDx,abDy, cdDx,cdDy);
   if(denom.closeTo(0.0, 0.0001))
      throw(Exception.MathError("lineIntersect: Parallel lines"));

   return(xnom/denom, ynom/denom);
}
fcn det(a,b,c,d){ a*d - b*c }	// determinant
lineIntersect(4.0,0.0, 6.0,10.0,  0.0,3.0, 10.0,7.0).println();
Output:
L(5,5)

References