Dynamic Ray-Triangle Intersection

Computer Graphics Asked by cemklkn on August 27, 2021

I am working on a small simulation software which has a 3D Renderer that is using OpenGL. I render objects to the screen as triangle meshes. To be able to select the triangles on the screen, I have implemented the Möller–Trumbore ray-triangle intersection algorithm. The algorithm works fine. However, I have another problem. My triangles are not static in the world. When I change the location of the triangle on the screen, I have to change the input vertices of the triangle in the algorithm but I don’t know how to do it mathematically. Is there a way to use this algorithm for dynamic objects?

There are 2 ways to go about intersecting the triangle. Let the vertices of the triangle have positions $$v_1, v_2, v_3$$. Let the ray have origin $$o$$ and direction $$d$$. Let the model (4x4) matrix be $$M$$.

To find the new vertex coordinates one extends the positions with a 1 (to allow for translations) and multiplies by the model matrix. Let $$u_i = (v_{i,x}, v_{i,y}, v_{i,z}, 1)$$ then $$w_i = Mu_i$$. The resulting vertex positions are: $$v_i' = (w_{i,x}, w_{i,y}, w_{i,z})$$.

The other option is to transform the ray with the inverse matrix $$M^{-1}$$ and intersect with the non-transformed triangle. To achieve this extend $$o$$ with a 4th coord of 1 (to account for translation) and extend $$d$$ with a 4th coord of 0 (to ignore translation) then multiply both with $$M^{-1}$$: $$o' = M^{-1}(o_x, o_y, o_z, 1)$$ $$d' = M^{-1}(d_x, d_y, d_z, 0)$$ Drop the 4th coordinate of $$o'$$ and $$d'$$ then intersect with the triangle formed by $$v_1, v_2, v_3$$.

Correct answer by lightxbulb on August 27, 2021

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