What's the most efficient way to detect triangle-triangle intersections?

Question

How can I tell whether two triangles intersect in 2D Euclidean space? (i.e. classic 2D geometry) given the (X,Y) coordinates of each vertex in each triangle.

Answer 1

Python implementation for line intersection and point in triangle test, with a little modification.

def line_intersect2(v1,v2,v3,v4):
    '''
    judge if line (v1,v2) intersects with line(v3,v4)
    '''
    d = (v4[1]-v3[1])*(v2[0]-v1[0])-(v4[0]-v3[0])*(v2[1]-v1[1])
    u = (v4[0]-v3[0])*(v1[1]-v3[1])-(v4[1]-v3[1])*(v1[0]-v3[0])
    v = (v2[0]-v1[0])*(v1[1]-v3[1])-(v2[1]-v1[1])*(v1[0]-v3[0])
    if d<0:
        u,v,d= -u,-v,-d
    return (0<=u<=d) and (0<=v<=d)

def point_in_triangle2(A,B,C,P):
    v0 = [C[0]-A[0], C[1]-A[1]]
    v1 = [B[0]-A[0], B[1]-A[1]]
    v2 = [P[0]-A[0], P[1]-A[1]]
    cross = lambda u,v: u[0]*v[1]-u[1]*v[0]
    u = cross(v2,v0)
    v = cross(v1,v2)
    d = cross(v1,v0)
    if d<0:
        u,v,d = -u,-v,-d
    return u>=0 and v>=0 and (u+v) <= d

def tri_intersect2(t1, t2):
    '''
    judge if two triangles in a plane intersect 
    '''
    if line_intersect2(t1[0],t1[1],t2[0],t2[1]): return True
    if line_intersect2(t1[0],t1[1],t2[0],t2[2]): return True
    if line_intersect2(t1[0],t1[1],t2[1],t2[2]): return True
    if line_intersect2(t1[0],t1[2],t2[0],t2[1]): return True
    if line_intersect2(t1[0],t1[2],t2[0],t2[2]): return True
    if line_intersect2(t1[0],t1[2],t2[1],t2[2]): return True
    if line_intersect2(t1[1],t1[2],t2[0],t2[1]): return True
    if line_intersect2(t1[1],t1[2],t2[0],t2[2]): return True
    if line_intersect2(t1[1],t1[2],t2[1],t2[2]): return True
    inTri = True 
    inTri = inTri and point_in_triangle2(t1[0],t1[1],t1[2], t2[0])
    inTri = inTri and point_in_triangle2(t1[0],t1[1],t1[2], t2[1])
    inTri = inTri and point_in_triangle2(t1[0],t1[1],t1[2], t2[2])
    if inTri == True: return True
    inTri = True
    inTri = inTri and point_in_triangle2(t2[0],t2[1],t2[2], t1[0])
    inTri = inTri and point_in_triangle2(t2[0],t2[1],t2[2], t1[1])
    inTri = inTri and point_in_triangle2(t2[0],t2[1],t2[2], t1[2])
    if inTri == True: return True
    return False

Answer 2

Sort the vertices of both triangle by decreasing ordinate. That takes at most three comparisons per triangle. Then merge the two sequences. I guess that this takes at most five comparisons.

Now for every ordinate, consider an horizontal line. It intersects both triangles in at most one line segment, and it is an easy matter to check it the segments do overlap. If they do, or if they change order between two lines, then the triangles intersect.

---

Update:

There is an affine transformation that can normalize one of the triangles to (0, 0)-(1, 0)-(0, 1). Apply it to the other and many computations will simplify.

Answer 3

Here is my attempt at the triangle-triangle collision problem (implemented in python):

#2D Triangle-Triangle collisions in python
#Release by Tim Sheerman-Chase 2016 under CC0

import numpy as np

def CheckTriWinding(tri, allowReversed):
    trisq = np.ones((3,3))
    trisq[:,0:2] = np.array(tri)
    detTri = np.linalg.det(trisq)
    if detTri < 0.0:
        if allowReversed:
            a = trisq[2,:].copy()
            trisq[2,:] = trisq[1,:]
            trisq[1,:] = a
        else: raise ValueError("triangle has wrong winding direction")
    return trisq

def TriTri2D(t1, t2, eps = 0.0, allowReversed = False, onBoundary = True):
    #Trangles must be expressed anti-clockwise
    t1s = CheckTriWinding(t1, allowReversed)
    t2s = CheckTriWinding(t2, allowReversed)

    if onBoundary:
        #Points on the boundary are considered as colliding
        chkEdge = lambda x: np.linalg.det(x) < eps
    else:
        #Points on the boundary are not considered as colliding
        chkEdge = lambda x: np.linalg.det(x) <= eps

    #For edge E of trangle 1,
    for i in range(3):
        edge = np.roll(t1s, i, axis=0)[:2,:]

        #Check all points of trangle 2 lay on the external side of the edge E. If
        #they do, the triangles do not collide.
        if (chkEdge(np.vstack((edge, t2s[0]))) and
            chkEdge(np.vstack((edge, t2s[1]))) and  
            chkEdge(np.vstack((edge, t2s[2])))):
            return False

    #For edge E of trangle 2,
    for i in range(3):
        edge = np.roll(t2s, i, axis=0)[:2,:]

        #Check all points of trangle 1 lay on the external side of the edge E. If
        #they do, the triangles do not collide.
        if (chkEdge(np.vstack((edge, t1s[0]))) and
            chkEdge(np.vstack((edge, t1s[1]))) and  
            chkEdge(np.vstack((edge, t1s[2])))):
            return False

    #The triangles collide
    return True

if __name__=="__main__":
    t1 = [[0,0],[5,0],[0,5]]
    t2 = [[0,0],[5,0],[0,6]]
    print (TriTri2D(t1, t2), True)

    t1 = [[0,0],[0,5],[5,0]]
    t2 = [[0,0],[0,6],[5,0]]
    print (TriTri2D(t1, t2, allowReversed = True), True)

    t1 = [[0,0],[5,0],[0,5]]
    t2 = [[-10,0],[-5,0],[-1,6]]
    print (TriTri2D(t1, t2), False)

    t1 = [[0,0],[5,0],[2.5,5]]
    t2 = [[0,4],[2.5,-1],[5,4]]
    print (TriTri2D(t1, t2), True)

    t1 = [[0,0],[1,1],[0,2]]
    t2 = [[2,1],[3,0],[3,2]]
    print (TriTri2D(t1, t2), False)

    t1 = [[0,0],[1,1],[0,2]]
    t2 = [[2,1],[3,-2],[3,4]]
    print (TriTri2D(t1, t2), False)

    #Barely touching
    t1 = [[0,0],[1,0],[0,1]]
    t2 = [[1,0],[2,0],[1,1]]
    print (TriTri2D(t1, t2, onBoundary = True), True)

    #Barely touching
    t1 = [[0,0],[1,0],[0,1]]
    t2 = [[1,0],[2,0],[1,1]]
    print (TriTri2D(t1, t2, onBoundary = False), False)

It works based based on the fact that the triangles do not overlap if all the points of triangle 1 are on the external side of at least one of the edges of triangle 2 (or vice versa is true). Of course, triangles are never concave.

I don't know if this approach is more or less efficient than the others.

Bonus: I ported it to C++ https://gist.github.com/TimSC/5ba18ae21c4459275f90

Answer 4

I realize that this is a very old question, but here is Rosetta code for this task in many languages: https://rosettacode.org/wiki/Determine_if_two_triangles_overlap