# Line segment-triangle intersection algorithm [closed]

currently in my project I'm using signed tetrahedron volume to check whether a line segment intersects a triangle. Initially I've found this approach in the great answer provided by professor O'Rourke: Stackoverflow link.

The method works perfectly fine, but my concern is if there are any scientific papers, which study and prove it.

Thank you in advance.

## 1 Answer

This problem can be reduced to solving the following linear equations in the variables $$\lambda,\mu_1,\mu_2,\mu_3$$: $$(1-\lambda)x_1+\lambda x_2=\mu_1 z_1+\mu_2 z_2+\mu_3 z_3,\ \mu_1+\mu_2+\mu_3=1,$$ where the $$x$$’s are the points determining the line and the $$z$$’s the vertices of the triangle. In the generic case you will have a unique solution and the line will pass through the interior of the triangle if and only if the $$\mu$$‘s are all positive. The other cases (line parallel to or lying in the plane of the triangle) can be dealt with easily.