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?
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$\begingroup$ How do you change the location of the triangles then? $\endgroup$– lightxbulbCommented Jun 29, 2020 at 18:12
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$\begingroup$ @lightxbulb , I use model matrix to change the location of the triangle. The algorithm is using the vertices of this triangle. When I change the location of triangle, the input vertices of the algorithm should also change according to new location of the triangle. I am asking for this change, I am not asking how to change the location of the triangle on the screen. Sorry if it was ambiguous. $\endgroup$– cemklknCommented Jun 29, 2020 at 18:18
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1$\begingroup$ To change the location of the triangle you multiply the vertices with the model matrix. That's it really - the new location is what you get from the matrix-vector product. $\endgroup$– lightxbulbCommented Jun 29, 2020 at 19:03
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1$\begingroup$ A matrix-vector product yields a vector. If a vertex has position $(x,y,z)$ and your model matrix is M, then make a 4d vector $v=(x,y,z,1)$ and then compute $v' = Mv$. The first 3 components of $v'$ are the coordinates of $v$ transformed with $M$. If you do that for the 3 vertices you can compute the new locations. $\endgroup$– lightxbulbCommented Jun 30, 2020 at 5:08
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1$\begingroup$ Thank you very much! That was exactly answer of my question. Can you please send it as an answer so I select it ? It might help people with similar question. $\endgroup$– cemklknCommented Jun 30, 2020 at 15:26
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1 Answer
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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 $v_1, v_2, v_3$.
answered Jun 30, 2020 at 19:26