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) .
ALGOL 68
Using "school maths". <lang algol68>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</lang>
- Output:
5.0000 5.0000
AWK
<lang AWK>
- 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)
} </lang>
- 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
Sinclair ZX81 BASIC
(version 1)
Works with 1k of RAM. <lang basic> 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</lang>
- Output:
THE TWO LINES ARE: YAB=-20+X*5 YCD=3+X*0.4 X=5 YAB=5 YCD=5 INTERSECTION: 5,5
C
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. <lang C>
- 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(slopeA==NAN && slopeB!=NAN){ c.x = a1.x; c.y = (a1.x-b1.x)*slopeB + b1.y; } else if(slopeB==NAN && slopeA!=NAN){ c.x = b1.x; c.y = (b1.x-a1.x)*slopeA + b1.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(c.x==NAN) 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; } </lang> Invocation and output:
C:\rosettaCode>lineIntersect.exe (4,0) (6,10) (0,3) (10,7) Point of intersection : (5.000000,5.000000)
C#
<lang csharp>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)));
}
}</lang>
- Output:
{X=5, Y=5}
{X=NaN, Y=NaN}
C++
<lang cpp>#include <iostream>
- include <cmath>
- include <assert.h>
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 xnom = Det(detL1, x1mx2, detL2, x3mx4); double ynom = Det(detL1, y1my2, detL2, y3my4); double denom = Det(x1mx2, y1my2, x3mx4, y3my4); if(denom == 0.0)//Lines don't seem to cross { ixOut = NAN; iyOut = NAN; return false; }
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);
}</lang>
- Output:
result 1,5,5
D
<lang D>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));
}</lang>
- Output:
{5.000000, 5.000000}
{-inf, -inf}
F#
<lang fsharp> (* 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} </lang>
- 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
FreeBASIC
<lang freebasic>' 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</lang>
- Output:
5 5 -1.#IND -1.#IND
Go
<lang go> 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") } } </lang>
- Output:
{5 5}
Haskell
<lang haskell>type Line = (Point, Point) type Point = (Float, Float)
-- INTERSECTION OF TWO LINES -------------------------------------------------- maybeIntersection :: Line -> Line -> Maybe Point maybeIntersection ((ax, ay), (bx, by)) ((px, py), (qx, qy)) =
let (pqDX, abDX) = (px - qx, ax - bx)
(pqDY, abDY) = (py - qy, ay - by)
determinant = abDX * pqDY - abDY * pqDX
f pq ab =
((((ax * by) - (ay * bx)) * pq) -
(((px * qy) - (py * qx)) * ab)) /
determinant
in case determinant of
0 -> Nothing
_ -> Just (f pqDX abDX, f pqDY abDY)
-- TEST ----------------------------------------------------------------------- ab :: Line ab = ((4.0, 0.0), (6.0, 10.0))
pq :: Line pq = ((0.0, 3.0), (10.0, 7.0))
interSection :: Maybe Point interSection = maybeIntersection ab pq
main :: IO () main =
putStrLn $ case interSection of Nothing -> "(parallel lines – no intersection)" Just x -> show x</lang>
- Output:
(5.0,5.0)
J
Solution: <lang j>det=: -/ .* NB. calculate determinant findIntersection=: (det ,."1 [: |: -/"2) %&det -/"2</lang>
Examples: <lang j> 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</lang>
Java
<lang Java>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));
}
}</lang>
- Output:
{5.000000, 5.000000}
{-Infinity, -Infinity}
JavaScript
ES6
<lang JavaScript>(() => {
// INTERSECTION OF TWO LINES ----------------------------------------------
// maybeIntersection :: Line -> Line -> Maybe Point
const maybeIntersection = ([
[ax, ay],
[bx, by]
], [
[px, py],
[qx, qy]
]) => {
const
abDX = ax - bx,
pqDX = px - qx,
abDY = ay - by,
pqDY = py - qy,
determinant = abDX * pqDY - abDY * pqDX;
return determinant !== 0 ? ({
nothing: false,
just: (() => {
const
abD = ax * by - ay * bx,
pqD = px * qy - py * qx;
return ap(
[([pq, ab]) =>
(abD * pq - ab * pqD) / determinant
], [
[pqDX, abDX],
[pqDY, abDY]
]
);
})()
}) : {
nothing: true
};
};
// GENERIC FUNCTIONS ------------------------------------------------------
// A list of functions applied to a list of arguments
// <*> :: [(a -> b)] -> [a] -> [b]
const ap = (fs, xs) => //
[].concat.apply([], fs.map(f => //
[].concat.apply([], xs.map(x => [f(x)]))));
// show :: a -> String const show = x => JSON.stringify(x); //, null, 2);
// TEST -------------------------------------------------------------------
// ab :: Line
const ab = [
[4.0, 0.0],
[6.0, 10.0]
];
// pq :: Line
const pq = [
[0.0, 3.0],
[10.0, 7.0]
];
// intersection :: Maybe Point const intersection = maybeIntersection(ab, pq);
return intersection.nothing ? (
'(Parallel lines – no intersection)'
) : show(intersection.just);
})();</lang>
- Output:
[5,5]
jq
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.
<lang jq># 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 - $dy)}
| . + {xnom: det(.detAB;.abDx;.detCD;.cdDx),
ynom: det(.detAB;.abDy;.detCD;.cdDy),
denom: det(.abDx; .abDy;.cdDx; .cdDy) }
| if (.denom|length < 10e-6) # length/0 emits the absolute value
then error("lineIntersect: parallel lines")
else [.xnom/.denom, .ynom/.denom]
end ;</lang>
Example: <lang jq>[[4.0, 0.0], [6.0,10.0]] | lineIntersection([[0.0, 3.0], [10.0, 7.0]])</lang>
- Output:
<lang jq>[5,5]</lang>
Julia
<lang julia>struct Point{T}
x::T y::T
end
struct Line{T}
s::Point{T}
e::Point{T}
end
function intersection(l1::Line{T}, l2::Line{T}) where T<:Real
a1 = l1.e.y - l1.s.y b1 = l1.s.x - l1.e.x c1 = a1 * l1.s.x + b1 * l1.s.y
a2 = l2.e.y - l2.s.y b2 = l2.s.x - l2.e.x c2 = a2 * l2.s.x + b2 * l2.s.y
Δ = a1 * b2 - a2 * b1 # If lines are parallel, intersection point will contain infinite values return Point((b2 * c1 - b1 * c2) / Δ, (a1 * c2 - a2 * c1) / Δ)
end
l1 = Line(Point{Float64}(4, 0), Point{Float64}(6, 10)) l2 = Line(Point{Float64}(0, 3), Point{Float64}(10, 7)) println(intersection(l1, l2))
l1 = Line(Point{Float64}(0, 0), Point{Float64}(1, 1)) l2 = Line(Point{Float64}(1, 2), Point{Float64}(4, 5)) println(intersection(l1, l2))</lang>
- Output:
Point{Float64}(5.0, 5.0)
Point{Float64}(-Inf, -Inf)
Kotlin
<lang kotlin>// version 1.1.2
class PointF(val x: Float, val y: Float) {
override fun toString() = "{$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))
}</lang>
- Output:
{5.0, 5.0}
{-Infinity, -Infinity}
Lua
<lang lua>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))</lang>
- Output:
5 5
Maple
<lang Maple>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));</lang>
- Out:
[5, 5]
MATLAB
<lang MATLAB> 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 </lang>
- Output:
line1=[4 0; 6 10]; line2=[0 3; 10 7]; cross=intersection(line1,line2)
cross =
5.0000
5.0000
Modula-2
<lang modula2>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.</lang>
Perl
If warning are enabled the second print will issue a warning since we are trying to print out an undef
<lang perl> 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"; </lang>
Perl 6
<lang perl6>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 );</lang>
- Output:
Intersection point: (5 5) Intersection point: (5.010893 5.054466) Intersection point: Lines are parallel
Python
<lang python>from __future__ import print_function from shapely.geometry import LineString
if __name__=="__main__": line1 = LineString([(4.0,0.0), (6.0,10.0)]) line2 = LineString([(0.0,3.0), (10.0,7.0)]) print (line1.intersection(line2))</lang>
- Output:
POINT (5 5)
Racket
<lang racket>#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))</lang>
- Output:
5 5
REXX
version 1
Naive implementation. To be improved for parallel lines and degenerate lines such as y=5 or x=8. <lang rexx>/* 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')'</lang>
- 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
complete implementation taking care of all possibilities.
Variables are named after the Austrian notation for a straight line: y=k*x+d
<lang>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</lang>
- 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
<lang ring>
- 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 </lang> Output:
the two lines are: yab=-20+x*5 ycd=3+x*0.4 x=5 yab=5 ycd=5 intersection: 5,5
Ruby
<lang ruby>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)}." </lang>
- Output:
Line y = 5.0x + -20.0 intersects line y = 0.4x + 3.0 at #<struct Point x=5.0, y=5.0>.
</lang>
Scala
<lang Scala>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))
}</lang>
- Output:
See it in running in your browser by (JavaScript)
or by Scastie (JVM).
Sidef
<lang ruby>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))</lang>
- Output:
Intersection point: [5, 5] Intersection point: [2300/459, 2320/459] Intersection point: lines are parallel
zkl
<lang zkl>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</lang> <lang zkl>lineIntersect(4.0,0.0, 6.0,10.0, 0.0,3.0, 10.0,7.0).println();</lang>
- Output:
L(5,5)
References