问题
I have a triangle defined by 3 points in 3d space. I also have a line segment defined by 2 points in 3d space. I want to know if they intersect. I don't really need to know the point of intersection.
I don't know any calculus but I know some trig. I know some about matrices but I understand vectors well (3d vectors specifically). Please keep it simple.
Can you walk me through the example problem:
triangle:
a: -4, 3, 0
b: 4, 3, 0
c: -3, -5, 4
line segment:
d: 1, -2, 0
e: -2, 6, 2
EDIT:
I am going to use this in a c++ physics engine.
One answer involved tetrahedron volume calculation from 4 vertices. Please provide formula or show it in code.
回答1:
Here is one way to solve your problem. Compute the volume of the tetrahedron Td = (a,b,c,d) and Te = (a,b,c,e). If either volume of Td or Te is zero, then one endpoint of the segment de lies on the plane containing triangle (a,b,c). If the volumes of Td and Te have the same sign, then de lies strictly to one side, and there is no intersection. If Td and Te have opposite signs, then de crosses the plane containing (a,b,c).
From there there are several strategies. One is to compute the point p where de crosses that plane. Then project down to 2D, and solve the point-in-triangle problem in 2D.
Another route is to compute the volumes of the tetrahedra (a,b,d,e), (b,c,d,e), and (c,a,d,e). Then only if all three have the same sign, does de intersect the triangle (a,b,c).
How to compute the volume of a tetrahedron from its corner coordinates is all over the web, and also in Computational Geometry in C.
回答2:
I implemented the great answer that Joseph gave in python and thought I would share. The function takes a set of line segments and triangles and computes for each line segment if it intersects any of the given triangles.
The first input to the function is a 2xSx3 array of line segments where the first index specifies the start or end point of the segment, the second index refers to the s^th line segment, and the third index points to the x, y,z coordinate of the line segment point.
The second input is a 3XTX3 array of triangle vertices, where the first index specifies one of the three vertices (which don't have to be in any particular order), the second index refers to the t^th triangle, and the third index points to the x,y,z coordinates the the triangle vertex.
The output of this function is a binary array of size S which tells you whether the s^th line segment intersects any of the triangles given. If you want to know which triangles the segments intersect, then just remove the summation of the last line of the function.
def signedVolume(a, b, c, d):
"""Computes the signed volume of a series of tetrahedrons defined by the vertices in
a, b c and d. The ouput is an SxT array which gives the signed volume of the tetrahedron defined
by the line segment 's' and two vertices of the triangle 't'."""
return np.sum((a-d)*np.cross(b-d, c-d), axis=2)
def segmentsIntersectTriangles(s, t):
"""For each line segment in 's', this function computes whether it intersects any of the triangles
given in 't'."""
# compute the normals to each triangle
normals = np.cross(t[2]-t[0], t[2]-t[1])
normals /= np.linalg.norm(normals, axis=1)[:, np.newaxis]
# get sign of each segment endpoint, if the sign changes then we know this segment crosses the
# plane which contains a triangle. If the value is zero the endpoint of the segment lies on the
# plane.
# s[i][:, np.newaxis] - t[j] -> S x T x 3 array
sign1 = np.sign(np.sum(normals*(s[0][:, np.newaxis] - t[2]), axis=2)) # S x T
sign2 = np.sign(np.sum(normals*(s[1][:, np.newaxis] - t[2]), axis=2)) # S x T
# determine segments which cross the plane of a triangle. 1 if the sign of the end points of s is
# different AND one of end points of s is not a vertex of t
cross = (sign1 != sign2)*(sign1 != 0)*(sign2 != 0) # S x T
# get signed volumes
v1 = np.sign(signedVolume(t[0], t[1], s[0][:, np.newaxis], s[1][:, np.newaxis])) # S x T
v2 = np.sign(signedVolume(t[1], t[2], s[0][:, np.newaxis], s[1][:, np.newaxis])) # S x T
v3 = np.sign(signedVolume(t[2], t[0], s[0][:, np.newaxis], s[1][:, np.newaxis])) # S x T
same_volume = np.logical_and((v1 == v2), (v2 == v3)) # 1 if s and t have same sign in v1, v2 and v3
return (np.sum(cross*same_volume, axis=1) > 0)
来源：https://stackoverflow.com/questions/53962225/how-to-know-if-a-line-segment-intersects-a-triangle-in-3d-space