# How to get intersection points between a plane and a triangle (3D)?

This is an illustration of the scenario I have:

I have the plane equation of a plane P $(A_px + B_py + C_pz + D_p = 0)$ and the coordinates of points A $(X_a, Y_a, Z_a)$, B $(X_b, Y_b, Z_b)$ and C $(X_c, Y_c, Z_c)$. The plane P will intersect the triangle ABC which will result in creating intersection points D and E. How can I calculate the coordinates for these two new points with the values I have?

• Do you know that you have an intersection, or do you also need to test for that? – amd Oct 20 '17 at 19:48
• @amd I will always have an intersection, so no need for testing. – delux Oct 20 '17 at 20:34

I like using homogeneous coordinates and the Plücker matrix representation of a line for this since that lets you crank out the answer directly. The Plücker matrix of the line through two points with homogeneous coordinate vectors $\mathbf p$ and $\mathbf q$ is $\mathcal L = \mathbf p \mathbf q^T-\mathbf q \mathbf p^T$. If the homogeneous vector that represents the plane is $\mathbf\pi$, then the intersection of the line and plane is simply $$\mathcal L \mathbf\pi = (\mathbf p \mathbf q^T-\mathbf q \mathbf p^T)\mathbf\pi = (\mathbf q^T \mathbf\pi) \mathbf p - (\mathbf p^T \mathbf\pi) \mathbf q.$$ In other words, multiply each point by the dot product of the plane and the other point, and subtract. To convert back to Cartesian coordinates, you’ll of course have to divide through by the last coordinate of the result. (If the last coordinate is $0$, there’s no intersection.) For this problem, we have in homogeneous coordinates the plane $\mathbf P=[A_p:B_p:C_p:D_p]$, the point $\mathbf A=[X_a:Y_a:Z_a:1]$ and so forth.