# A Beautiful Ray/Mesh Intersection Algorithm

In my last post, I talked about a beautiful method for computing ray/triangle intersections. In this post, I will extend it to computing intersections with triangle fans. Since meshes are often stored in a corner table, which is simply an array of triangle fans, this gives an efficient algorithm for ray tracing triangle meshes.

The aforementioned algorithm computes ray/triangle intersections with 1 division, 20 multiplications, and up to 18 additions. It also required storing an affine transformation matrix, which takes up 4/3 as much space as just storing the vertices. But if we have many triangles which all share a common vertex, we can exploit that structure to save time and memory.

Say our triangle fan is composed of triangles $$ABC$$, $$ACD$$, $$ADE$$, etc. As before, we compute

$$P_{ABC} = \begin{bmatrix} \overrightarrow{AB} & \overrightarrow{AC} & \overrightarrow{AB} \times \overrightarrow{AC} & \overrightarrow{A} \\ 0 & 0 & 0 & 1 \end{bmatrix}^{-1}.$$

Computing the change of basis from here to the next triangle is even easier. We want to find new coordinates $$\langle u', v', w' \rangle$$ such that

$$\begin{eqnarray*} \vec{p} &=& u\hphantom{'}\,\overrightarrow{AB} &+& v\hphantom{'}\,\overrightarrow{AC} &+& w\hphantom{'}\,\overrightarrow{AB}\times\overrightarrow{AC} &+& \overrightarrow{A} \\ {} &=& u'\,\overrightarrow{AC} &+& v'\,\overrightarrow{AD} &+& w'\,\overrightarrow{AC}\times\overrightarrow{AD} &+& \overrightarrow{A}. \end{eqnarray*}$$

Two things are apparent: the first is that there is no further translation to perform because both coordinate systems have their origin at $$\overrightarrow{A}$$. The second is that only $$u'$$ depends on $$v$$. This means the matrix $$P^{ABC}_{ACD}$$ that takes us to the new basis has the special form

$$P^{ABC}_{ACD} = P_{ACD}\,P^{-1}_{ABC} = \begin{bmatrix} a & 1 & d & 0 \\ b & 0 & e & 0 \\ c & 0 & f & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix},$$

so we only need to store two of its columns, and transforming the ray into the new space can be done much faster than with a full matrix multiplication. The following transformation is applied to both the ray origin and direction:

\begin{align*} u' &= a\,u + v + d\,w \\ v' &= b\,u + e\,w \\ w' &= c\,u + f\,w. \end{align*}

In the end, for a triangle fan with $$n$$ vertices, the ray intersection can be computed with

\begin{align*} n & \;\mathrm{divisions,} \\ 6 + 14\,n & \;\mathrm{multiplications, and} \\ 7 + 11\,n & \;\mathrm{additions}. \end{align*}

The multiplications and additions are also easily vectorisable. The storage requirement is $$6\,(n + 1)$$ floating-point values, which is equivalent to storing all the vertices and precomputed normals.

The implementation of this algorithm in my ray tracer Dimension is available here.

## 3 thoughts on “A Beautiful Ray/Mesh Intersection Algorithm”

1. Mo says:

Pretty cool, I wonder if this can be extended to triangle strips as well. That one might be a lot more challenging though.

 Also I wonder if these dollar signs will break your page with a MathJax injection. I imagine it's more intelligent than that.

1. Mo says:

Nope. :)

2. tavianator says:

Haha yeah the MathJax is all done with a WordPress plugin, I write $m^at_h$ and it prettifies it. I'm hoping it doesn't scan the page and replace $m^at_h$ with MathJax.

And you're right, it looks hard to do this well with triangle strips because they don't all share a common vertex. The above still works except the translation comes back which wastes time and space.