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Find if a point is within a triangle

From Rosetta Code
Task
Find if a point is within a triangle
You are encouraged to solve this task according to the task description, using any language you may know.

Find if a point is within a triangle.


Task
  •   Assume points are on a plane defined by (x, y) real number coordinates.
  •   Given a point P(x, y) and a triangle formed by points A, B, and C, determine if P is within triangle ABC.
  •   You may use any algorithm.
  •   Bonus: explain why the algorithm you chose works.


Related tasks


Also see
  • Discussion of several methods. [[1]]
  • Determine if a point is in a polygon [[2]]
  • Triangle based coordinate systems [[3]]
  • Wolfram entry [[4]]



C[edit]

Translation of: Go
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
 
const double EPS = 0.001;
const double EPS_SQUARE = 0.000001;
 
double side(double x1, double y1, double x2, double y2, double x, double y) {
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1);
}
 
bool naivePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double checkSide1 = side(x1, y1, x2, y2, x, y) >= 0;
double checkSide2 = side(x2, y2, x3, y3, x, y) >= 0;
double checkSide3 = side(x3, y3, x1, y1, x, y) >= 0;
return checkSide1 && checkSide2 && checkSide3;
}
 
bool pointInTriangleBoundingBox(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double xMin = min(x1, min(x2, x3)) - EPS;
double xMax = max(x1, max(x2, x3)) + EPS;
double yMin = min(y1, min(y2, y3)) - EPS;
double yMax = max(y1, max(y2, y3)) + EPS;
return !(x < xMin || xMax < x || y < yMin || yMax < y);
}
 
double distanceSquarePointToSegment(double x1, double y1, double x2, double y2, double x, double y) {
double p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1);
double dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength;
if (dotProduct < 0) {
return (x - x1) * (x - x1) + (y - y1) * (y - y1);
} else if (dotProduct <= 1) {
double p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y);
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength;
} else {
return (x - x2) * (x - x2) + (y - y2) * (y - y2);
}
}
 
bool accuratePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
if (!pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)) {
return false;
}
if (naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
return true;
}
if (distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE) {
return true;
}
return false;
}
 
void printPoint(double x, double y) {
printf("(%f, %f)", x, y);
}
 
void printTriangle(double x1, double y1, double x2, double y2, double x3, double y3) {
printf("Triangle is [");
printPoint(x1, y1);
printf(", ");
printPoint(x2, y2);
printf(", ");
printPoint(x3, y3);
printf("] \n");
}
 
void test(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
printTriangle(x1, y1, x2, y2, x3, y3);
printf("Point ");
printPoint(x, y);
printf(" is within triangle? ");
if (accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
printf("true\n");
} else {
printf("false\n");
}
}
 
int main() {
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 0);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 1);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 3, 1);
printf("\n");
 
test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, 25, 11.11111111111111, 5.414285714285714, 14.349206349206348);
printf("\n");
 
test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, -12.5, 16.666666666666668, 5.414285714285714, 14.349206349206348);
printf("\n");
 
return 0;
}
Output:
Triangle is [(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (0.000000, 0.000000) is within triangle? true
Triangle is [(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (0.000000, 1.000000) is within triangle? true
Triangle is [(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (3.000000, 1.000000) is within triangle? false

Triangle is [(0.100000, 0.111111), (12.500000, 33.333333), (25.000000, 11.111111)]
Point (5.414286, 14.349206) is within triangle? true

Triangle is [(0.100000, 0.111111), (12.500000, 33.333333), (-12.500000, 16.666667)]
Point (5.414286, 14.349206) is within triangle? true

C++[edit]

Translation of: C
#include <iostream>
 
const double EPS = 0.001;
const double EPS_SQUARE = EPS * EPS;
 
double side(double x1, double y1, double x2, double y2, double x, double y) {
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1);
}
 
bool naivePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double checkSide1 = side(x1, y1, x2, y2, x, y) >= 0;
double checkSide2 = side(x2, y2, x3, y3, x, y) >= 0;
double checkSide3 = side(x3, y3, x1, y1, x, y) >= 0;
return checkSide1 && checkSide2 && checkSide3;
}
 
bool pointInTriangleBoundingBox(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double xMin = std::min(x1, std::min(x2, x3)) - EPS;
double xMax = std::max(x1, std::max(x2, x3)) + EPS;
double yMin = std::min(y1, std::min(y2, y3)) - EPS;
double yMax = std::max(y1, std::max(y2, y3)) + EPS;
return !(x < xMin || xMax < x || y < yMin || yMax < y);
}
 
double distanceSquarePointToSegment(double x1, double y1, double x2, double y2, double x, double y) {
double p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1);
double dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength;
if (dotProduct < 0) {
return (x - x1) * (x - x1) + (y - y1) * (y - y1);
} else if (dotProduct <= 1) {
double p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y);
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength;
} else {
return (x - x2) * (x - x2) + (y - y2) * (y - y2);
}
}
 
bool accuratePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
if (!pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)) {
return false;
}
if (naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
return true;
}
if (distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE) {
return true;
}
return false;
}
 
void printPoint(double x, double y) {
std::cout << '(' << x << ", " << y << ')';
}
 
void printTriangle(double x1, double y1, double x2, double y2, double x3, double y3) {
std::cout << "Triangle is [";
printPoint(x1, y1);
std::cout << ", ";
printPoint(x2, y2);
std::cout << ", ";
printPoint(x3, y3);
std::cout << "]\n";
}
 
void test(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
printTriangle(x1, y1, x2, y2, x3, y3);
std::cout << "Point ";
printPoint(x, y);
std::cout << " is within triangle? ";
if (accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
std::cout << "true\n";
} else {
std::cout << "false\n";
}
}
 
int main() {
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 0);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 1);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 3, 1);
std::cout << '\n';
 
test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, 25, 11.11111111111111, 5.414285714285714, 14.349206349206348);
std::cout << '\n';
 
test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, -12.5, 16.666666666666668, 5.414285714285714, 14.349206349206348);
std::cout << '\n';
 
return 0;
}
Output:
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 0) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 1) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3, 1) is within triangle? false

Triangle is [(0.1, 0.111111), (12.5, 33.3333), (25, 11.1111)]
Point (5.41429, 14.3492) is within triangle? true

Triangle is [(0.1, 0.111111), (12.5, 33.3333), (-12.5, 16.6667)]
Point (5.41429, 14.3492) is within triangle? true


Common Lisp[edit]

 
; There are different algorithms to solve this problem, such as adding areas, adding angles, etc... but these
; solutions are sensitive to rounding errors intrinsic to float operations. We want to avoid these issues, therefore we
; use the following algorithm which only uses multiplication and subtraction: we consider one side of the triangle
; and see on which side of it is the point P located. We can give +1 if it is on the right hand side, -1 for the
; left side, or 0 if it is on the line. If the point is located on the same side relative to all three sides of the triangle
; then the point is inside of it. This has an added advantage that it can be scaled up to other more complicated figures
; (even concave ones, with some minor modifications).
 
(defun point-inside-triangle (P A B C)
"Is the point P inside the triangle formed by ABC?"
(= (side-of-line P A B)
(side-of-line P B C)
(side-of-line P C A) ))
 
 
; This is the version to include those points which are on one of the sides
(defun point-inside-or-on-triangle (P A B C)
"Is the point P inside the triangle formed by ABC or on one of the sides?"
(apply #'= (remove 0 (list (side-of-line P A B) (side-of-line P B C) (side-of-line P C A)))) )
 
 
(defun side-of-line (P A B)
"Return +1 if it is on the right side, -1 for the left side, or 0 if it is on the line"
; We use the sign of the determinant of vectors (AB,AM), where M(X,Y) is the query point:
; position = sign((Bx - Ax) * (Y - Ay) - (By - Ay) * (X - Ax))
(signum (- (* (- (car B) (car A))
(- (cdr P) (cdr A)) )
(* (- (cdr B) (cdr A))
(- (car P) (car A)) ))))
 
 
Output:
(point-inside-triangle '(0 . 0) '(1.5 . 2.4) '(5.1 . -3.1) '(-3.8 . 1.2))
T

(point-inside-triangle '(0 . 1) '(1.5 . 2.4) '(5.1 . -3.1) '(-3.8 . 1.2))
T

(point-inside-triangle '(3 . 1) '(1.5 . 2.4) '(5.1 . -3.1) '(-3.8 . 1.2))
NIL

(point-inside-triangle '(5.414286 . 14.349206) '(0.1 . 0.111111) '(12.5 . 33.333333) '(25.0 . 11.111111))
T

(point-inside-triangle '(5.414286 . 14.349206) '(0.1 . 0.111111) '(12.5 . 33.333333) '(-12.5 . 16.666667))
NIL

D[edit]

Translation of: C++
import std.algorithm; //.comparison for min and max
import std.stdio;
 
immutable EPS = 0.001;
immutable EPS_SQUARE = EPS * EPS;
 
double side(double x1, double y1, double x2, double y2, double x, double y) {
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1);
}
 
bool naivePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double checkSide1 = side(x1, y1, x2, y2, x, y) >= 0;
double checkSide2 = side(x2, y2, x3, y3, x, y) >= 0;
double checkSide3 = side(x3, y3, x1, y1, x, y) >= 0;
return checkSide1 && checkSide2 && checkSide3;
}
 
bool pointInTriangleBoundingBox(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
double xMin = min(x1, x2, x3) - EPS;
double xMax = max(x1, x2, x3) + EPS;
double yMin = min(y1, y2, y3) - EPS;
double yMax = max(y1, y2, y3) + EPS;
return !(x < xMin || xMax < x || y < yMin || yMax < y);
}
 
double distanceSquarePointToSegment(double x1, double y1, double x2, double y2, double x, double y) {
double p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1);
double dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength;
if (dotProduct < 0) {
return (x - x1) * (x - x1) + (y - y1) * (y - y1);
} else if (dotProduct <= 1) {
double p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y);
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength;
} else {
return (x - x2) * (x - x2) + (y - y2) * (y - y2);
}
}
 
bool accuratePointInTriangle(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
if (!pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)) {
return false;
}
if (naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)) {
return true;
}
if (distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE) {
return true;
}
return false;
}
 
void printPoint(double x, double y) {
write('(', x, ", ", y, ')');
}
 
void printTriangle(double x1, double y1, double x2, double y2, double x3, double y3) {
write("Triangle is [");
printPoint(x1, y1);
write(", ");
printPoint(x2, y2);
write(", ");
printPoint(x3, y3);
writeln(']');
}
 
void test(double x1, double y1, double x2, double y2, double x3, double y3, double x, double y) {
printTriangle(x1, y1, x2, y2, x3, y3);
write("Point ");
printPoint(x, y);
write(" is within triangle? ");
writeln(accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y));
}
 
void main() {
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 0);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 1);
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 3, 1);
writeln;
 
test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, 25, 11.11111111111111, 5.414285714285714, 14.349206349206348);
writeln;
 
test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, -12.5, 16.666666666666668, 5.414285714285714, 14.349206349206348);
writeln;
}
Output:
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 0) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 1) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3, 1) is within triangle? false

Triangle is [(0.1, 0.111111), (12.5, 33.3333), (25, 11.1111)]
Point (5.41429, 14.3492) is within triangle? true

Triangle is [(0.1, 0.111111), (12.5, 33.3333), (-12.5, 16.6667)]
Point (5.41429, 14.3492) is within triangle? true

Factor[edit]

Uses the parametric equations method from [5].

USING: accessors fry io kernel locals math math.order sequences ;
 
TUPLE: point x y ;
C: <point> point
: >point< ( point -- x y ) [ x>> ] [ y>> ] bi ;
 
TUPLE: triangle p1 p2 p3 ;
C: <triangle> triangle
: >triangle< ( triangle -- x1 y1 x2 y2 x3 y3 )
[ p1>> ] [ p2>> ] [ p3>> ] tri [ >point< ] [email protected] ;
 
:: point-in-triangle? ( point triangle -- ? )
point >point< triangle >triangle< :> ( x y x1 y1 x2 y2 x3 y3 )
y2 y3 - x1 * x3 x2 - y1 * + x2 y3 * + y2 x3 * - :> d
y3 y1 - x * x1 x3 - y * + x1 y3 * - y1 x3 * + d / :> t1
y2 y1 - x * x1 x2 - y * + x1 y2 * - y1 x2 * + d neg / :> t2
t1 t2 + :> s
 
t1 t2 [ 0 1 between? ] [email protected] and s 1 <= and ;
 
! Test if it works.
 
20 <iota> dup [ swap <point> ] cartesian-map  ! Make a matrix of points
3 3 <point> 16 10 <point> 10 16 <point> <triangle>  ! Make a triangle
'[ [ _ point-in-triangle? "#" "." ? write ] each nl ] each nl  ! Show points inside the triangle with '#'
 
Output:
....................
....................
....................
...#................
....#...............
.....##.............
.....####...........
......#####.........
......#######.......
.......########.....
.......##########...
........########....
........#######.....
.........#####......
.........####.......
..........##........
..........#.........
....................
....................
....................

Go[edit]

Translation of: Wren
package main
 
import (
"fmt"
"math"
)
 
const EPS = 0.001
const EPS_SQUARE = EPS * EPS
 
func side(x1, y1, x2, y2, x, y float64) float64 {
return (y2-y1)*(x-x1) + (-x2+x1)*(y-y1)
}
 
func naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y float64) bool {
checkSide1 := side(x1, y1, x2, y2, x, y) >= 0
checkSide2 := side(x2, y2, x3, y3, x, y) >= 0
checkSide3 := side(x3, y3, x1, y1, x, y) >= 0
return checkSide1 && checkSide2 && checkSide3
}
 
func pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y float64) bool {
xMin := math.Min(x1, math.Min(x2, x3)) - EPS
xMax := math.Max(x1, math.Max(x2, x3)) + EPS
yMin := math.Min(y1, math.Min(y2, y3)) - EPS
yMax := math.Max(y1, math.Max(y2, y3)) + EPS
return !(x < xMin || xMax < x || y < yMin || yMax < y)
}
 
func distanceSquarePointToSegment(x1, y1, x2, y2, x, y float64) float64 {
p1_p2_squareLength := (x2-x1)*(x2-x1) + (y2-y1)*(y2-y1)
dotProduct := ((x-x1)*(x2-x1) + (y-y1)*(y2-y1)) / p1_p2_squareLength
if dotProduct < 0 {
return (x-x1)*(x-x1) + (y-y1)*(y-y1)
} else if dotProduct <= 1 {
p_p1_squareLength := (x1-x)*(x1-x) + (y1-y)*(y1-y)
return p_p1_squareLength - dotProduct*dotProduct*p1_p2_squareLength
} else {
return (x-x2)*(x-x2) + (y-y2)*(y-y2)
}
}
 
func accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y float64) bool {
if !pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y) {
return false
}
if naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y) {
return true
}
if distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE {
return true
}
if distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE {
return true
}
if distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE {
return true
}
return false
}
 
func main() {
pts := [][2]float64{{0, 0}, {0, 1}, {3, 1}}
tri := [][2]float64{{3.0 / 2, 12.0 / 5}, {51.0 / 10, -31.0 / 10}, {-19.0 / 5, 1.2}}
fmt.Println("Triangle is", tri)
x1, y1 := tri[0][0], tri[0][1]
x2, y2 := tri[1][0], tri[1][1]
x3, y3 := tri[2][0], tri[2][1]
for _, pt := range pts {
x, y := pt[0], pt[1]
within := accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
fmt.Println("Point", pt, "is within triangle?", within)
}
fmt.Println()
tri = [][2]float64{{1.0 / 10, 1.0 / 9}, {100.0 / 8, 100.0 / 3}, {100.0 / 4, 100.0 / 9}}
fmt.Println("Triangle is", tri)
x1, y1 = tri[0][0], tri[0][1]
x2, y2 = tri[1][0], tri[1][1]
x3, y3 = tri[2][0], tri[2][1]
x := x1 + (3.0/7)*(x2-x1)
y := y1 + (3.0/7)*(y2-y1)
pt := [2]float64{x, y}
within := accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
fmt.Println("Point", pt, "is within triangle ?", within)
fmt.Println()
tri = [][2]float64{{1.0 / 10, 1.0 / 9}, {100.0 / 8, 100.0 / 3}, {-100.0 / 8, 100.0 / 6}}
fmt.Println("Triangle is", tri)
x3 = tri[2][0]
y3 = tri[2][1]
within = accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
fmt.Println("Point", pt, "is within triangle ?", within)
}
Output:
Triangle is [[1.5 2.4] [5.1 -3.1] [-3.8 1.2]]
Point [0 0] is within triangle? true
Point [0 1] is within triangle? true
Point [3 1] is within triangle? false

Triangle is [[0.1 0.1111111111111111] [12.5 33.333333333333336] [25 11.11111111111111]]
Point [5.414285714285714 14.349206349206348] is within triangle ? true

Triangle is [[0.1 0.1111111111111111] [12.5 33.333333333333336] [-12.5 16.666666666666668]]
Point [5.414285714285714 14.349206349206348] is within triangle ? true

Java[edit]

Translation of: Go
import java.util.Objects;
 
public class FindTriangle {
private static final double EPS = 0.001;
private static final double EPS_SQUARE = EPS * EPS;
 
public static class Point {
private final double x, y;
 
public Point(double x, double y) {
this.x = x;
this.y = y;
}
 
public double getX() {
return x;
}
 
public double getY() {
return y;
}
 
@Override
public String toString() {
return String.format("(%f, %f)", x, y);
}
}
 
public static class Triangle {
private final Point p1, p2, p3;
 
public Triangle(Point p1, Point p2, Point p3) {
this.p1 = Objects.requireNonNull(p1);
this.p2 = Objects.requireNonNull(p2);
this.p3 = Objects.requireNonNull(p3);
}
 
public Point getP1() {
return p1;
}
 
public Point getP2() {
return p2;
}
 
public Point getP3() {
return p3;
}
 
private boolean pointInTriangleBoundingBox(Point p) {
var xMin = Math.min(p1.getX(), Math.min(p2.getX(), p3.getX())) - EPS;
var xMax = Math.max(p1.getX(), Math.max(p2.getX(), p3.getX())) + EPS;
var yMin = Math.min(p1.getY(), Math.min(p2.getY(), p3.getY())) - EPS;
var yMax = Math.max(p1.getY(), Math.max(p2.getY(), p3.getY())) + EPS;
return !(p.getX() < xMin || xMax < p.getX() || p.getY() < yMin || yMax < p.getY());
}
 
private static double side(Point p1, Point p2, Point p) {
return (p2.getY() - p1.getY()) * (p.getX() - p1.getX()) + (-p2.getX() + p1.getX()) * (p.getY() - p1.getY());
}
 
private boolean nativePointInTriangle(Point p) {
boolean checkSide1 = side(p1, p2, p) >= 0;
boolean checkSide2 = side(p2, p3, p) >= 0;
boolean checkSide3 = side(p3, p1, p) >= 0;
return checkSide1 && checkSide2 && checkSide3;
}
 
private double distanceSquarePointToSegment(Point p1, Point p2, Point p) {
double p1_p2_squareLength = (p2.getX() - p1.getX()) * (p2.getX() - p1.getX()) + (p2.getY() - p1.getY()) * (p2.getY() - p1.getY());
double dotProduct = ((p.getX() - p1.getX()) * (p2.getX() - p1.getX()) + (p.getY() - p1.getY()) * (p2.getY() - p1.getY())) / p1_p2_squareLength;
if (dotProduct < 0) {
return (p.getX() - p1.getX()) * (p.getX() - p1.getX()) + (p.getY() - p1.getY()) * (p.getY() - p1.getY());
}
if (dotProduct <= 1) {
double p_p1_squareLength = (p1.getX() - p.getX()) * (p1.getX() - p.getX()) + (p1.getY() - p.getY()) * (p1.getY() - p.getY());
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength;
}
return (p.getX() - p2.getX()) * (p.getX() - p2.getX()) + (p.getY() - p2.getY()) * (p.getY() - p2.getY());
}
 
private boolean accuratePointInTriangle(Point p) {
if (!pointInTriangleBoundingBox(p)) {
return false;
}
if (nativePointInTriangle(p)) {
return true;
}
if (distanceSquarePointToSegment(p1, p2, p) <= EPS_SQUARE) {
return true;
}
if (distanceSquarePointToSegment(p2, p3, p) <= EPS_SQUARE) {
return true;
}
return distanceSquarePointToSegment(p3, p1, p) <= EPS_SQUARE;
}
 
public boolean within(Point p) {
Objects.requireNonNull(p);
return accuratePointInTriangle(p);
}
 
@Override
public String toString() {
return String.format("Triangle[%s, %s, %s]", p1, p2, p3);
}
}
 
private static void test(Triangle t, Point p) {
System.out.println(t);
System.out.printf("Point %s is within triangle? %s\n", p, t.within(p));
}
 
public static void main(String[] args) {
var p1 = new Point(1.5, 2.4);
var p2 = new Point(5.1, -3.1);
var p3 = new Point(-3.8, 1.2);
var tri = new Triangle(p1, p2, p3);
test(tri, new Point(0, 0));
test(tri, new Point(0, 1));
test(tri, new Point(3, 1));
System.out.println();
 
p1 = new Point(1.0 / 10, 1.0 / 9);
p2 = new Point(100.0 / 8, 100.0 / 3);
p3 = new Point(100.0 / 4, 100.0 / 9);
tri = new Triangle(p1, p2, p3);
var pt = new Point(p1.getX() + (3.0 / 7) * (p2.getX() - p1.getX()), p1.getY() + (3.0 / 7) * (p2.getY() - p1.getY()));
test(tri, pt);
System.out.println();
 
p3 = new Point(-100.0 / 8, 100.0 / 6);
tri = new Triangle(p1, p2, p3);
test(tri, pt);
}
}
Output:
Triangle[(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (0.000000, 0.000000) is within triangle? true
Triangle[(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (0.000000, 1.000000) is within triangle? true
Triangle[(1.500000, 2.400000), (5.100000, -3.100000), (-3.800000, 1.200000)]
Point (3.000000, 1.000000) is within triangle? false

Triangle[(0.100000, 0.111111), (12.500000, 33.333333), (25.000000, 11.111111)]
Point (5.414286, 14.349206) is within triangle? true

Triangle[(0.100000, 0.111111), (12.500000, 33.333333), (-12.500000, 16.666667)]
Point (5.414286, 14.349206) is within triangle? true

Julia[edit]

Translation of: Python

Using the Wren examples.

Point(x, y) = [x, y]
Triangle(a, b, c) = [a, b, c]
LEzero(x) = x < 0 || isapprox(x, 0, atol=0.00000001)
GEzero(x) = x > 0 || isapprox(x, 0, atol=0.00000001)
 
""" Determine which side of plane cut by line (p2, p3) p1 is on """
side(p1, p2, p3) = (p1[1] - p3[1]) * (p2[2] - p3[2]) - (p2[1] - p3[1]) * (p1[2] - p3[2])
 
"""
Determine if point is within triangle formed by points p1, p2, p3.
If so, the point will be on the same side of each of the half planes
defined by vectors p1p2, p2p3, and p3p1. Each z is positive if outside,
negative if inside such a plane. All should be positive or all negative
if point is within the triangle.
"""
function iswithin(point, p1, p2, p3)
z1 = side(point, p1, p2)
z2 = side(point, p2, p3)
z3 = side(point, p3, p1)
notanyneg = GEzero(z1) && GEzero(z2) && GEzero(z3)
notanypos = LEzero(z1) && LEzero(z2) && LEzero(z3)
return notanyneg || notanypos
end
 
const POINTS = [Point(0 // 1, 0 // 1), Point(0 // 1, 1 // 1), Point(3 // 1, 1 // 1),
Point(1 // 10 + (3 // 7) * (100 // 8 - 1 // 10), 1 // 9 + (3 // 7) * (100 // 3 - 1 // 9)),
Point(3 // 2, 12 // 5), Point(51 // 100, -31 // 100), Point(-19 // 50, 6 // 5),
Point(1 // 10, 1 // 9), Point(25 / 2, 100 // 3), Point(25, 100 // 9),
Point(-25 // 2, 50 // 3)
]
 
const TRI = [
Triangle(POINTS[5], POINTS[6], POINTS[7]),
Triangle(POINTS[8], POINTS[9], POINTS[10]),
Triangle(POINTS[8], POINTS[9], POINTS[11])
]
 
for tri in TRI
pstring(pt) = "[$(Float32(pt[1])), $(Float32(pt[2]))]"
println("\nUsing triangle [", join([pstring(x) for x in tri], ", "), "]:")
a, b, c = tri[1], tri[2], tri[3]
for p in POINTS[1:4]
isornot = iswithin(p, a, b, c) ? "is" : "is not"
println("Point $(pstring(p)) $isornot within the triangle.")
end
end
 
Output:
Using triangle [[1.5, 2.4], [0.51, -0.31], [-0.38, 1.2]]:
Point [0.0, 0.0] is not within the triangle.
Point [0.0, 1.0] is within the triangle.
Point [3.0, 1.0] is not within the triangle.
Point [5.4142857, 14.349206] is not within the triangle.

Using triangle [[0.1, 0.11111111], [12.5, 33.333332], [25.0, 11.111111]]:
Point [0.0, 0.0] is not within the triangle.
Point [0.0, 1.0] is not within the triangle.
Point [3.0, 1.0] is not within the triangle.
Point [5.4142857, 14.349206] is within the triangle.

Using triangle [[0.1, 0.11111111], [12.5, 33.333332], [-12.5, 16.666666]]:
Point [0.0, 0.0] is not within the triangle.
Point [0.0, 1.0] is within the triangle.
Point [3.0, 1.0] is not within the triangle.
Point [5.4142857, 14.349206] is within the triangle.

Kotlin[edit]

Translation of: Java
import kotlin.math.max
import kotlin.math.min
 
private const val EPS = 0.001
private const val EPS_SQUARE = EPS * EPS
 
private fun test(t: Triangle, p: Point) {
println(t)
println("Point $p is within triangle ? ${t.within(p)}")
}
 
fun main() {
var p1 = Point(1.5, 2.4)
var p2 = Point(5.1, -3.1)
var p3 = Point(-3.8, 1.2)
var tri = Triangle(p1, p2, p3)
test(tri, Point(0.0, 0.0))
test(tri, Point(0.0, 1.0))
test(tri, Point(3.0, 1.0))
println()
p1 = Point(1.0 / 10, 1.0 / 9)
p2 = Point(100.0 / 8, 100.0 / 3)
p3 = Point(100.0 / 4, 100.0 / 9)
tri = Triangle(p1, p2, p3)
val pt = Point(p1.x + 3.0 / 7 * (p2.x - p1.x), p1.y + 3.0 / 7 * (p2.y - p1.y))
test(tri, pt)
println()
p3 = Point(-100.0 / 8, 100.0 / 6)
tri = Triangle(p1, p2, p3)
test(tri, pt)
}
 
class Point(val x: Double, val y: Double) {
override fun toString(): String {
return "($x, $y)"
}
}
 
class Triangle(private val p1: Point, private val p2: Point, private val p3: Point) {
private fun pointInTriangleBoundingBox(p: Point): Boolean {
val xMin = min(p1.x, min(p2.x, p3.x)) - EPS
val xMax = max(p1.x, max(p2.x, p3.x)) + EPS
val yMin = min(p1.y, min(p2.y, p3.y)) - EPS
val yMax = max(p1.y, max(p2.y, p3.y)) + EPS
return !(p.x < xMin || xMax < p.x || p.y < yMin || yMax < p.y)
}
 
private fun nativePointInTriangle(p: Point): Boolean {
val checkSide1 = side(p1, p2, p) >= 0
val checkSide2 = side(p2, p3, p) >= 0
val checkSide3 = side(p3, p1, p) >= 0
return checkSide1 && checkSide2 && checkSide3
}
 
private fun distanceSquarePointToSegment(p1: Point, p2: Point, p: Point): Double {
val p1P2SquareLength = (p2.x - p1.x) * (p2.x - p1.x) + (p2.y - p1.y) * (p2.y - p1.y)
val dotProduct = ((p.x - p1.x) * (p2.x - p1.x) + (p.y - p1.y) * (p2.y - p1.y)) / p1P2SquareLength
if (dotProduct < 0) {
return (p.x - p1.x) * (p.x - p1.x) + (p.y - p1.y) * (p.y - p1.y)
}
if (dotProduct <= 1) {
val pP1SquareLength = (p1.x - p.x) * (p1.x - p.x) + (p1.y - p.y) * (p1.y - p.y)
return pP1SquareLength - dotProduct * dotProduct * p1P2SquareLength
}
return (p.x - p2.x) * (p.x - p2.x) + (p.y - p2.y) * (p.y - p2.y)
}
 
private fun accuratePointInTriangle(p: Point): Boolean {
if (!pointInTriangleBoundingBox(p)) {
return false
}
if (nativePointInTriangle(p)) {
return true
}
if (distanceSquarePointToSegment(p1, p2, p) <= EPS_SQUARE) {
return true
}
return if (distanceSquarePointToSegment(p2, p3, p) <= EPS_SQUARE) {
true
} else distanceSquarePointToSegment(p3, p1, p) <= EPS_SQUARE
}
 
fun within(p: Point): Boolean {
return accuratePointInTriangle(p)
}
 
override fun toString(): String {
return "Triangle[$p1, $p2, $p3]"
}
 
companion object {
private fun side(p1: Point, p2: Point, p: Point): Double {
return (p2.y - p1.y) * (p.x - p1.x) + (-p2.x + p1.x) * (p.y - p1.y)
}
}
}
Output:
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0.0, 0.0) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0.0, 1.0) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3.0, 1.0) is within triangle ? false

Triangle[(0.1, 0.1111111111111111), (12.5, 33.333333333333336), (25.0, 11.11111111111111)]
Point (5.414285714285714, 14.349206349206348) is within triangle ? true

Triangle[(0.1, 0.1111111111111111), (12.5, 33.333333333333336), (-12.5, 16.666666666666668)]
Point (5.414285714285714, 14.349206349206348) is within triangle ? true

Lua[edit]

Translation of: C++
EPS = 0.001
EPS_SQUARE = EPS * EPS
 
function side(x1, y1, x2, y2, x, y)
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1)
end
 
function naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
local checkSide1 = side(x1, y1, x2, y2, x, y) >= 0
local checkSide2 = side(x2, y2, x3, y3, x, y) >= 0
local checkSide3 = side(x3, y3, x1, y1, x, y) >= 0
return checkSide1 and checkSide2 and checkSide3
end
 
function pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)
local xMin = math.min(x1, x2, x3) - EPS
local xMax = math.max(x1, x2, x3) + EPS
local yMin = math.min(y1, y2, y3) - EPS
local yMax = math.max(y1, y2, y3) + EPS
return not (x < xMin or xMax < x or y < yMin or yMax < y)
end
 
function distanceSquarePointToSegment(x1, y1, x2, y2, x, y)
local p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1)
local dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength
if dotProduct < 0 then
return (x - x1) * (x - x1) + (y - y1) * (y - y1)
end
if dotProduct <= 1 then
local p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y)
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength
end
return (x - x2) * (x - x2) + (y - y2) * (y - y2)
end
 
function accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
if not pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y) then
return false
end
if naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y) then
return true
end
if distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE then
return true
end
if distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE then
return true
end
if distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE then
return true
end
return false
end
 
function printPoint(x, y)
io.write('('..x..", "..y..')')
end
 
function printTriangle(x1, y1, x2, y2, x3, y3)
io.write("Triangle is [")
printPoint(x1, y1)
io.write(", ")
printPoint(x2, y2)
io.write(", ")
printPoint(x3, y3)
print("]")
end
 
function test(x1, y1, x2, y2, x3, y3, x, y)
printTriangle(x1, y1, x2, y2, x3, y3)
io.write("Point ")
printPoint(x, y)
print(" is within triangle? " .. tostring(accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)))
end
 
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 0)
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 0, 1)
test(1.5, 2.4, 5.1, -3.1, -3.8, 1.2, 3, 1)
print()
 
test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, 25, 11.11111111111111, 5.414285714285714, 14.349206349206348)
print()
 
test(0.1, 0.1111111111111111, 12.5, 33.333333333333336, -12.5, 16.666666666666668, 5.414285714285714, 14.349206349206348)
print()
Output:
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 0) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0, 1) is within triangle? true
Triangle is [(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3, 1) is within triangle? false

Triangle is [(0.1, 0.11111111111111), (12.5, 33.333333333333), (25, 11.111111111111)]
Point (5.4142857142857, 14.349206349206) is within triangle? true

Triangle is [(0.1, 0.11111111111111), (12.5, 33.333333333333), (-12.5, 16.666666666667)]
Point (5.4142857142857, 14.349206349206) is within triangle? true

Mathematica/Wolfram Language[edit]

RegionMember[Polygon[{{1, 2}, {3, 1}, {2, 4}}], {2, 2}]
Output:
True

Nim[edit]

Translation of: Kotlin
import strformat
 
const
Eps = 0.001
Eps2 = Eps * Eps
 
type
Point = tuple[x, y: float]
Triangle = object
p1, p2, p3: Point
 
 
func initTriangle(p1, p2, p3: Point): Triangle =
Triangle(p1: p1, p2: p2, p3: p3)
 
func side(p1, p2, p: Point): float =
(p2.y - p1.y) * (p.x - p1.x) + (-p2.x + p1.x) * (p.y - p1.y)
 
 
func distanceSquarePointToSegment(p1, p2, p: Point): float =
let p1P2SquareLength = (p2.x - p1.x) * (p2.x - p1.x) + (p2.y - p1.y) * (p2.y - p1.y)
let dotProduct = ((p.x - p1.x) * (p2.x - p1.x) + (p.y - p1.y) * (p2.y - p1.y)) / p1P2SquareLength
if dotProduct < 0:
return (p.x - p1.x) * (p.x - p1.x) + (p.y - p1.y) * (p.y - p1.y)
if dotProduct <= 1:
let pP1SquareLength = (p1.x - p.x) * (p1.x - p.x) + (p1.y - p.y) * (p1.y - p.y)
return pP1SquareLength - dotProduct * dotProduct * p1P2SquareLength
result = (p.x - p2.x) * (p.x - p2.x) + (p.y - p2.y) * (p.y - p2.y)
 
 
func pointInTriangleBoundingBox(t: Triangle; p: Point): bool =
let xMin = min(t.p1.x, min(t.p2.x, t.p3.x)) - EPS
let xMax = max(t.p1.x, max(t.p2.x, t.p3.x)) + EPS
let yMin = min(t.p1.y, min(t.p2.y, t.p3.y)) - EPS
let yMax = max(t.p1.y, max(t.p2.y, t.p3.y)) + EPS
result = p.x in xMin..xMax and p.y in yMin..yMax
 
 
func nativePointInTriangle(t: Triangle; p: Point): bool =
let checkSide1 = side(t.p1, t.p2, p) >= 0
let checkSide2 = side(t.p2, t.p3, p) >= 0
let checkSide3 = side(t.p3, t.p1, p) >= 0
result = checkSide1 and checkSide2 and checkSide3
 
 
func accuratePointInTriangle(t: Triangle; p: Point): bool =
if not t.pointInTriangleBoundingBox(p):
return false
if t.nativePointInTriangle(p):
return true
if distanceSquarePointToSegment(t.p1, t.p2, p) <= Eps2 or
distanceSquarePointToSegment(t.p3, t.p1, p) <= Eps2:
return true
 
 
func `$`(p: Point): string = &"({p.x}, {p.y})"
 
func `$`(t: Triangle): string = &"Triangle[{t.p1}, {t.p2}, {t.p3}]"
 
func contains(t: Triangle; p: Point): bool = t.accuratePointInTriangle(p)
 
 
when isMainModule:
 
proc test(t: Triangle; p: Point) =
echo t
echo &"Point {p} is within triangle ? {p in t}"
 
var p1: Point = (1.5, 2.4)
var p2: Point = (5.1, -3.1)
var p3: Point = (-3.8, 1.2)
var tri = initTriangle(p1, p2, p3)
test(tri, (0.0, 0.0))
test(tri, (0.0, 1.0))
test(tri, (3.0, 1.0))
echo()
p1 = (1 / 10, 1 / 9)
p2 = (100 / 8, 100 / 3)
p3 = (100 / 4, 100 / 9)
tri = initTriangle(p1, p2, p3)
let pt = (p1.x + 3.0 / 7 * (p2.x - p1.x), p1.y + 3.0 / 7 * (p2.y - p1.y))
test(tri, pt)
echo()
p3 = (-100 / 8, 100 / 6)
tri = initTriangle(p1, p2, p3)
test(tri, pt)
Output:
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0.0, 0.0) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (0.0, 1.0) is within triangle ? true
Triangle[(1.5, 2.4), (5.1, -3.1), (-3.8, 1.2)]
Point (3.0, 1.0) is within triangle ? false

Triangle[(0.1, 0.1111111111111111), (12.5, 33.33333333333334), (25.0, 11.11111111111111)]
Point (5.414285714285714, 14.34920634920635) is within triangle ? true

Triangle[(0.1, 0.1111111111111111), (12.5, 33.33333333333334), (-12.5, 16.66666666666667)]
Point (5.414285714285714, 14.34920634920635) is within triangle ? true

Perl[edit]

Translate the Java program at this blog post and data set is taken from the Raku entry.

# 20201123 added Perl programming solution
 
use strict;
use warnings;
 
use List::AllUtils qw(min max natatime);
use constant EPSILON => 0.001;
use constant EPSILON_SQUARE => EPSILON*EPSILON;
 
sub side {
my ($x1, $y1, $x2, $y2, $x, $y) = @_;
return ($y2 - $y1)*($x - $x1) + (-$x2 + $x1)*($y - $y1);
}
 
sub naivePointInTriangle {
my ($x1, $y1, $x2, $y2, $x3, $y3, $x, $y) = @_;
my $checkSide1 = side($x1, $y1, $x2, $y2, $x, $y) >= 0 ;
my $checkSide2 = side($x2, $y2, $x3, $y3, $x, $y) >= 0 ;
my $checkSide3 = side($x3, $y3, $x1, $y1, $x, $y) >= 0 ;
return $checkSide1 && $checkSide2 && $checkSide3 || 0 ;
}
 
sub pointInTriangleBoundingBox {
my ($x1, $y1, $x2, $y2, $x3, $y3, $x, $y) = @_;
my $xMin = min($x1, min($x2, $x3)) - EPSILON;
my $xMax = max($x1, max($x2, $x3)) + EPSILON;
my $yMin = min($y1, min($y2, $y3)) - EPSILON;
my $yMax = max($y1, max($y2, $y3)) + EPSILON;
( $x < $xMin || $xMax < $x || $y < $yMin || $yMax < $y ) ? 0 : 1
}
 
sub distanceSquarePointToSegment {
my ($x1, $y1, $x2, $y2, $x, $y) = @_;
my $p1_p2_squareLength = ($x2 - $x1)**2 + ($y2 - $y1)**2;
my $dotProduct = ($x-$x1)*($x2-$x1)+($y-$y1)*($y2-$y1) ;
if ( $dotProduct < 0 ) {
return ($x - $x1)**2 + ($y - $y1)**2;
} elsif ( $dotProduct <= $p1_p2_squareLength ) {
my $p_p1_squareLength = ($x1 - $x)**2 + ($y1 - $y)**2;
return $p_p1_squareLength - $dotProduct**2 / $p1_p2_squareLength;
} else {
return ($x - $x2)**2 + ($y - $y2)**2;
}
}
 
sub accuratePointInTriangle {
my ($x1, $y1, $x2, $y2, $x3, $y3, $x, $y) = @_;
return 0 unless pointInTriangleBoundingBox($x1,$y1,$x2,$y2,$x3,$y3,$x,$y);
return 1 if ( naivePointInTriangle($x1, $y1, $x2, $y2, $x3, $y3, $x, $y)
or distanceSquarePointToSegment($x1, $y1, $x2, $y2, $x, $y) <= EPSILON_SQUARE
or distanceSquarePointToSegment($x2, $y2, $x3, $y3, $x, $y) <= EPSILON_SQUARE
or distanceSquarePointToSegment($x3, $y3, $x1, $y1, $x, $y) <= EPSILON_SQUARE);
return 0
}
 
my @DATA = (1.5, 2.4, 5.1, -3.1, -3.8, 0.5);
 
for my $point ( [0,0] , [0,1] ,[3,1] ) {
print "Point (", join(',',@$point), ") is within triangle ";
my $iter = natatime 2, @DATA;
while ( my @vertex = $iter->()) { print '(',join(',',@vertex),') ' }
print ': ',naivePointInTriangle (@DATA, @$point) ? 'True' : 'False', "\n" ;
}
Output:
Point (0,0) is within triangle (1.5,2.4) (5.1,-3.1) (-3.8,0.5) : True
Point (0,1) is within triangle (1.5,2.4) (5.1,-3.1) (-3.8,0.5) : True
Point (3,1) is within triangle (1.5,2.4) (5.1,-3.1) (-3.8,0.5) : False

Phix[edit]

using convex_hull[edit]

Using convex_hull() from Convex_hull#Phix

constant p0 = {0,0},
p1 = {0,1},
p2 = {3,1},
triangle = {{3/2, 12/5}, {51/10, -31/10}, {-19/5, 1/2}}
function inside(sequence p) return sort(convex_hull({p}&triangle))==sort(triangle) end function
printf(1,"Point %v is with triangle %v?:%t\n",{p0,triangle,inside(p0)})
printf(1,"Point %v is with triangle %v?:%t\n",{p1,triangle,inside(p1)})
printf(1,"Point %v is with triangle %v?:%t\n",{p2,triangle,inside(p2)})
Output:
Point {0,0} is with triangle {{1.5,2.4},{5.1,-3.1},{-3.8,0.5}}?:true
Point {0,1} is with triangle {{1.5,2.4},{5.1,-3.1},{-3.8,0.5}}?:true
Point {3,1} is with triangle {{1.5,2.4},{5.1,-3.1},{-3.8,0.5}}?:false

trans python[edit]

(using the same p0/p1/p2/triangle constants from above, same output)

function side(sequence p1, p2, p3)
-- which side of plane cut by line (p2, p3) is p1 on?
atom {x1, y1} = p1,
{x2, y2} = p2,
{x3, y3} = p3
return (x1 - x3) * (y2 - y3) - (x2 - x3) * (y1 - y3)
end function
 
function iswithin(sequence point, triangle)
--
-- Determine if point is within triangle.
-- If so, the point will be on the same side of each of the half planes
-- defined by vectors p1p2, p2p3, and p3p1. zval is positive if outside,
-- negative if inside such a plane. All should be positive or all negative
-- if point is within the triangle.
--
sequence {pt1, pt2, pt3} = triangle
atom zval1 = side(point, pt1, pt2),
zval2 = side(point, pt2, pt3),
zval3 = side(point, pt3, pt1)
bool notanyneg = zval1 >= 0 and zval2 >= 0 and zval3 >= 0,
notanypos = zval1 <= 0 and zval2 <= 0 and zval3 <= 0
return notanyneg or notanypos
end function
 
printf(1,"point %v is with triangle %v?:%t\n",{p0,triangle,iswithin(p0,triangle)})
printf(1,"point %v is with triangle %v?:%t\n",{p1,triangle,iswithin(p1,triangle)})
printf(1,"point %v is with triangle %v?:%t\n",{p2,triangle,iswithin(p2,triangle)})

Python[edit]

 
""" find if point is in a triangle """
 
from sympy.geometry import Point, Triangle
 
def sign(pt1, pt2, pt3):
""" which side of plane cut by line (pt2, pt3) is pt1 on? """
return (pt1.x - pt3.x) * (pt2.y - pt3.y) - (pt2.x - pt3.x) * (pt1.y - pt3.y)
 
 
def iswithin(point, pt1, pt2, pt3):
"""
Determine if point is within triangle formed by points p1, p2, p3.
If so, the point will be on the same side of each of the half planes
defined by vectors p1p2, p2p3, and p3p1. zval is positive if outside,
negative if inside such a plane. All should be positive or all negative
if point is within the triangle.
"""

zval1 = sign(point, pt1, pt2)
zval2 = sign(point, pt2, pt3)
zval3 = sign(point, pt3, pt1)
notanyneg = zval1 >= 0 and zval2 >= 0 and zval3 >= 0
notanypos = zval1 <= 0 and zval2 <= 0 and zval3 <= 0
return notanyneg or notanypos
 
if __name__ == "__main__":
POINTS = [Point(0, 0)]
TRI = Triangle(Point(1.5, 2.4), Point(5.1, -3.1), Point(-3.8, 0.5))
for pnt in POINTS:
a, b, c = TRI.vertices
isornot = "is" if iswithin(pnt, a, b, c) else "is not"
print("Point", pnt, isornot, "within the triangle", TRI)
 
Output:
Point Point2D(0, 0) is within the triangle Triangle(Point2D(3/2, 12/5), Point2D(51/10, -31/10), Point2D(-19/5, 1/2))

Racket[edit]

Racket has exact numbers in its numerical tower... so I don't see much motivation to accomodate rounding errors. This is why the implementation _fails_ the second imprecise test, whereas other implementations pass it. That point is very close to the edge of the triange. If your edge is fat enough (epsilon), it will fall inside. If it is infinitessimal (i.e. exact), it is on the outside.

I would probably use the dot-product version, if only because it requires less (no) division.

#lang racket/base
 
(define-syntax-rule (all-between-0..1? x ...)
(and (<= 0 x 1) ...))
 
(define (point-in-triangle?/barycentric x1 y1 x2 y2 x3 y3)
(let* ((y2-y3 (- y2 y3))
(x1-x3 (- x1 x3))
(x3-x2 (- x3 x2))
(y1-y3 (- y1 y3))
(d (+ (* y2-y3 x1-x3) (* x3-x2 y1-y3))))
(λ (x y)
(define a (/ (+ (* x3-x2 (- y y3)) (* y2-y3 (- x x3))) d))
(define b (/ (- (* x1-x3 (- y y3)) (* y1-y3 (- x x3))) d))
(define c (- 1 a b))
(all-between-0..1? a b c))))
 
(define (point-in-triangle?/parametric x1 y1 x2 y2 x3 y3)
(let ((dp (+ (* x1 (- y2 y3)) (* y1 (- x3 x2)) (* x2 y3) (- (* y2 x3)))))
(λ (x y)
(define t1 (/ (+ (* x (- y3 y1)) (* y (- x1 x3)) (- (* x1 y3)) (* y1 x3)) dp))
(define t2 (/ (+ (* x (- y2 y1)) (* y (- x1 x2)) (- (* x1 y2)) (* y1 x2)) (- dp)))
(all-between-0..1? t1 t2 (+ t1 t2)))))
 
(define (point-in-triangle?/dot-product X1 Y1 X2 Y2 X3 Y3)
(λ (x y)
(define (check-side x1 y1 x2 y2)
(>= (+ (* (- y2 y1) (- x x1)) (* (- x1 x2) (- y y1))) 0))
(and
(check-side X1 Y1 X2 Y2)
(check-side X2 Y2 X3 Y3)
(check-side X3 Y3 X1 Y1))))
 
(module+ main
(require rackunit)
 
(define (run-tests point-in-triangle?)
(define pit?-1 (point-in-triangle? #e1.5 #e2.4 #e5.1 #e-3.1 #e-3.8 #e1.2))
(check-true (pit?-1 0 0))
(check-true (pit?-1 0 1))
(check-false (pit?-1 3 1))
(check-true ((point-in-triangle? 1/10 1/9 25/2 100/3 25 10/9) #e5.414285714285714 #e14.349206349206348))
 ; exactly speaking, point is _not_ in the triangle
(check-false ((point-in-triangle? 1/10 1/9 25/2 100/3 -25/2 50/3) #e5.414285714285714 #e14.349206349206348)))
 
(run-tests point-in-triangle?/barycentric)
(run-tests point-in-triangle?/parametric)
(run-tests point-in-triangle?/dot-product))
Output:

no output means all tests passed


Raku[edit]

Reusing code from the Convex hull task and some logic from the Determine if two triangles overlap task.

class Point {
has Real $.x is rw;
has Real $.y is rw;
method gist { [~] '(', self.x,', ', self.y, ')' };
}
 
sub sign (Point $a, Point $b, Point $c) {
($b.x - $a.x)*($c.y - $a.y) - ($b.y - $a.y)*($c.x - $a.x);
}
 
sub triangle (*@points where *.elems == 6) {
@points.batch(2).map: { Point.new(:x(.[0]),:y(.[1])) };
}
 
sub is-within ($point, @triangle is copy) {
my @signs = sign($point, |(@triangle.=rotate)[0,1]) xx 3;
so (all(@signs) >= 0) or so(all(@signs) <= 0);
}
 
my @triangle = triangle((1.5, 2.4), (5.1, -3.1), (-3.8, 0.5));
 
for Point.new(:x(0),:y(0)),
Point.new(:x(0),:y(1)),
Point.new(:x(3),:y(1))
-> $point {
say "Point {$point.gist} is within triangle {join ', ', @triangle».gist}: ",
$point.&is-within: @triangle
}
Output:
Point (0, 0) is within triangle (1.5, 2.4), (5.1, -3.1), (-3.8, 0.5): True
Point (0, 1) is within triangle (1.5, 2.4), (5.1, -3.1), (-3.8, 0.5): True
Point (3, 1) is within triangle (1.5, 2.4), (5.1, -3.1), (-3.8, 0.5): False

REXX[edit]

Translation of: Python


Extra certification code was added to verify that the   X,Y   coördinates for the points are not missing and are numeric.

/*REXX program determines if a   specified point   is within a   specified triangle.    */
parse arg p a b c . /*obtain optional arguments from the CL*/
if p=='' | p=="," then p= '(0,0)' /*Not specified? Then use the default.*/
if a=='' | a=="," then a= '(1.5,2.4)' /* " " " " " " */
if b=='' | b=="," then b= '(5.1,-3.1)' /* " " " " " " */
if c=='' | c=="," then c= '(-3.8,0.5)' /* " " " " " " */
if  ?(p, a, b, c) then @= ' is ' /*Is the point inside the triangle ? */
else @= " isn't " /* " " " outside " " */
say 'point' p @ " within the triangle " a ',' b "," c
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
cert: parse arg z,W; if datatype(z,'N') then return z; call serr z /*return coördinate.*/
serr: say W 'data point ' z " isn't numeric or missing."; exit 13 /*tell error message*/
x: procedure; parse arg "(" x ','  ; return cert(x,"X") /*return the X coördinate.*/
y: procedure; parse arg ',' y ")"; return cert(y,"Y") /* " " Y " */
$: parse arg aa,bb,cc; return (x(aa)-x(cc)) *(y(bb)-y(cc)) - (x(bb)-x(cc)) *(y(aa)-y(cc))
?: #1=$(p,a,b); #2=$(p,b,c); #3=$(p,c,a); return (#1>=0&#2>=0&#3>=0) | (#1<=0&#2<=0&#3<=0)
output   when using the default triangle and the point at:   (0,0)
point (0,0)   is   within the triangle  (1.5,2.4) , (5.1,-3.1) , (-3.8,0.5)
output   when using the default triangle and the point at:   (0,1)
point (0,1)   is   within the triangle  (1.5,2.4) , (5.1,-3.1) , (-3.8,0.5)
output   when using the default triangle and the point at:   (3,1)
point (3,1)   isn't   within the triangle  (1.5,2.4) , (5.1,-3.1) , (-3.8,0.5)

Ruby[edit]

Translation of: Go
EPS = 0.001
EPS_SQUARE = EPS * EPS
 
def side(x1, y1, x2, y2, x, y)
return (y2 - y1) * (x - x1) + (-x2 + x1) * (y - y1)
end
 
def naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
checkSide1 = side(x1, y1, x2, y2, x, y) >= 0
checkSide2 = side(x2, y2, x3, y3, x, y) >= 0
checkSide3 = side(x3, y3, x1, y1, x, y) >= 0
return checkSide1 && checkSide2 && checkSide3
end
 
def pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y)
xMin = [x1, x2, x3].min - EPS
xMax = [x1, x2, x3].max + EPS
yMin = [y1, y2, y3].min - EPS
yMax = [y1, y2, y3].max + EPS
return !(x < xMin || xMax < x || y < yMin || yMax < y)
end
 
def distanceSquarePointToSegment(x1, y1, x2, y2, x, y)
p1_p2_squareLength = (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1)
dotProduct = ((x - x1) * (x2 - x1) + (y - y1) * (y2 - y1)) / p1_p2_squareLength
if dotProduct < 0 then
return (x - x1) * (x - x1) + (y - y1) * (y - y1)
end
if dotProduct <= 1 then
p_p1_squareLength = (x1 - x) * (x1 - x) + (y1 - y) * (y1 - y)
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength
end
return (x - x2) * (x - x2) + (y - y2) * (y - y2)
end
 
def accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
if !pointInTriangleBoundingBox(x1, y1, x2, y2, x3, y3, x, y) then
return false
end
if naivePointInTriangle(x1, y1, x2, y2, x3, y3, x, y) then
return true
end
if distanceSquarePointToSegment(x1, y1, x2, y2, x, y) <= EPS_SQUARE then
return true
end
if distanceSquarePointToSegment(x2, y2, x3, y3, x, y) <= EPS_SQUARE then
return true
end
if distanceSquarePointToSegment(x3, y3, x1, y1, x, y) <= EPS_SQUARE then
return true
end
return false
end
 
def main
pts = [[0, 0], [0, 1], [3, 1]]
tri = [[1.5, 2.4], [5.1, -3.1], [-3.8, 1.2]]
print "Triangle is ", tri, "\n"
x1, y1 = tri[0][0], tri[0][1]
x2, y2 = tri[1][0], tri[1][1]
x3, y3 = tri[2][0], tri[2][1]
for pt in pts
x, y = pt[0], pt[1]
within = accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
print "Point ", pt, " is within triangle? ", within, "\n"
end
print "\n"
 
tri = [[0.1, 1.0 / 9.0], [12.5, 100.0 / 3.0], [25.0, 100.0 / 9.0]]
print "Triangle is ", tri, "\n"
x1, y1 = tri[0][0], tri[0][1]
x2, y2 = tri[1][0], tri[1][1]
x3, y3 = tri[2][0], tri[2][1]
x = x1 + (3.0 / 7.0) * (x2 - x1)
y = y1 + (3.0 / 7.0) * (y2 - y1)
pt = [x, y]
within = accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
print "Point ", pt, " is within triangle? ", within, "\n"
print "\n"
 
tri = [[0.1, 1.0 / 9.0], [12.5, 100.0 / 3.0], [-12.5, 100.0 / 6.0]]
print "Triangle is ", tri, "\n"
x3, y3 = tri[2][0], tri[2][1]
within = accuratePointInTriangle(x1, y1, x2, y2, x3, y3, x, y)
print "Point ", pt, " is within triangle? ", within, "\n"
end
 
main()
Output:
Triangle is [[1.5, 2.4], [5.1, -3.1], [-3.8, 1.2]]
Point [0, 0] is within triangle? true
Point [0, 1] is within triangle? true
Point [3, 1] is within triangle? false

Triangle is [[0.1, 0.1111111111111111], [12.5, 33.333333333333336], [25.0, 11.11111111111111]]
Point [5.414285714285714, 14.349206349206348] is within triangle? true

Triangle is [[0.1, 0.1111111111111111], [12.5, 33.333333333333336], [-12.5, 16.666666666666668]]
Point [5.414285714285714, 14.349206349206348] is within triangle? true

Wren[edit]

Library: Wren-math

This is a translation of the ActionScript code for the 'accurate' method in the first referenced article above.

import "/math" for Math
 
var EPS = 0.001
var EPS_SQUARE = EPS * EPS
 
var side = Fn.new { |x1, y1, x2, y2, x, y|
return (y2 - y1)*(x - x1) + (-x2 + x1)*(y - y1)
}
 
var naivePointInTriangle = Fn.new { |x1, y1, x2, y2, x3, y3, x, y|
var checkSide1 = side.call(x1, y1, x2, y2, x, y) >= 0
var checkSide2 = side.call(x2, y2, x3, y3, x, y) >= 0
var checkSide3 = side.call(x3, y3, x1, y1, x, y) >= 0
return checkSide1 && checkSide2 && checkSide3
}
 
var pointInTriangleBoundingBox = Fn.new { |x1, y1, x2, y2, x3, y3, x, y|
var xMin = Math.min(x1, Math.min(x2, x3)) - EPS
var xMax = Math.max(x1, Math.max(x2, x3)) + EPS
var yMin = Math.min(y1, Math.min(y2, y3)) - EPS
var yMax = Math.max(y1, Math.max(y2, y3)) + EPS
return !(x < xMin || xMax < x || y < yMin || yMax < y)
}
 
var distanceSquarePointToSegment = Fn.new { |x1, y1, x2, y2, x, y|
var p1_p2_squareLength = (x2 - x1)*(x2 - x1) + (y2 - y1)*(y2 - y1)
var dotProduct = ((x - x1)*(x2 - x1) + (y - y1)*(y2 - y1)) / p1_p2_squareLength
if (dotProduct < 0) {
return (x - x1)*(x - x1) + (y - y1)*(y - y1)
} else if (dotProduct <= 1) {
var p_p1_squareLength = (x1 - x)*(x1 - x) + (y1 - y)*(y1 - y)
return p_p1_squareLength - dotProduct * dotProduct * p1_p2_squareLength
} else {
return (x - x2)*(x - x2) + (y - y2)*(y - y2)
}
}
 
var accuratePointInTriangle = Fn.new { |x1, y1, x2, y2, x3, y3, x, y|
if (!pointInTriangleBoundingBox.call(x1, y1, x2, y2, x3, y3, x, y)) return false
if (naivePointInTriangle.call(x1, y1, x2, y2, x3, y3, x, y)) return true
if (distanceSquarePointToSegment.call(x1, y1, x2, y2, x, y) <= EPS_SQUARE) return true
if (distanceSquarePointToSegment.call(x2, y2, x3, y3, x, y) <= EPS_SQUARE) return true
if (distanceSquarePointToSegment.call(x3, y3, x1, y1, x, y) <= EPS_SQUARE) return true
return false
}
 
var pts = [ [0, 0], [0, 1], [3, 1]]
var tri = [ [3/2, 12/5], [51/10, -31/10], [-19/5, 1.2] ]
System.print("Triangle is %(tri)")
var x1 = tri[0][0]
var y1 = tri[0][1]
var x2 = tri[1][0]
var y2 = tri[1][1]
var x3 = tri[2][0]
var y3 = tri[2][1]
 
for (pt in pts) {
var x = pt[0]
var y = pt[1]
var within = accuratePointInTriangle.call(x1, y1, x2, y2, x3, y3, x, y)
System.print("Point %(pt) is within triangle ? %(within)")
}
System.print()
tri = [ [1/10, 1/9], [100/8, 100/3], [100/4, 100/9] ]
System.print("Triangle is %(tri)")
x1 = tri[0][0]
y1 = tri[0][1]
x2 = tri[1][0]
y2 = tri[1][1]
x3 = tri[2][0]
y3 = tri[2][1]
var x = x1 + (3/7)*(x2 - x1)
var y = y1 + (3/7)*(y2 - y1)
var pt = [x, y]
var within = accuratePointInTriangle.call(x1, y1, x2, y2, x3, y3, x, y)
System.print("Point %(pt) is within triangle ? %(within)")
System.print()
tri = [ [1/10, 1/9], [100/8, 100/3], [-100/8, 100/6] ]
System.print("Triangle is %(tri)")
x3 = tri[2][0]
y3 = tri[2][1]
within = accuratePointInTriangle.call(x1, y1, x2, y2, x3, y3, x, y)
System.print("Point %(pt) is within triangle ? %(within)")
Output:
Triangle is [[1.5, 2.4], [5.1, -3.1], [-3.8, 1.2]]
Point [0, 0] is within triangle ? true
Point [0, 1] is within triangle ? true
Point [3, 1] is within triangle ? false

Triangle is [[0.1, 0.11111111111111], [12.5, 33.333333333333], [25, 11.111111111111]]
Point [5.4142857142857, 14.349206349206] is within triangle ? true

Triangle is [[0.1, 0.11111111111111], [12.5, 33.333333333333], [-12.5, 16.666666666667]]
Point [5.4142857142857, 14.349206349206] is within triangle ? true

XPL0[edit]

func real Dot(W,X,Y,Z);         \Return the dot product of two 2D vectors
real W,X,Y,Z; \ (W-X) dot (Y-Z)
real WX(2), YZ(2);
[WX(0):= W(0)-X(0); WX(1):= W(1)-X(1);
YZ(0):= Y(0)-Z(0); YZ(1):= Y(1)-Z(1);
return WX(0)*YZ(0) + WX(1)*YZ(1);
];
 
real A,B,C; \triangle
 
func PointInTri(P); \Return 'true' if point P is inside triangle ABC
real P;
int S0,S1,S2; \signs
[S0:= Dot(P,A,B,A) >= 0.0;
S1:= Dot(P,B,C,B) >= 0.0;
S2:= Dot(P,C,A,C) >= 0.0;
return S0=S1 & S1=S2 & S2=S0;
];
 
[A:= [10.5, 6.3]; B:= [13.5, 3.6]; C:= [ 3.3, -1.6];
Text(0, if PointInTri([10.0, 3.0]) then "inside" else "outside"); CrLf(0);
Text(0, if PointInTri([-5.0,-2.2]) then "inside" else "outside"); CrLf(0);
Text(0, if PointInTri([10.5, 6.3]) then "inside" else "outside"); CrLf(0);
]
Output:
inside
outside
inside