Note: This blog post is still a work in progress. Right now it's mostly ramblings of a madman, I'm still trying to organizing my thoughts about this.

All uppercase bolded variables are vectors in $R^3$.

Drawing

Let $\mathbf{R}(t) = \mathbf{Q} + t\mathbf{D}$ be a ray originating at $\mathbf{Q}$, in the direction of $\mathbf{D}$. We can describe a cube of radius $r$ centered at $\mathbf{C}$ with the equation

$$||\mathbf{C} - \mathbf{P}||_\infty = r$$

With $\mathbf{P}$ a vector in $R^3$. To be more specific: $\mathbf{P}$ is a point on the square if it satisfies the above equation for the given $\mathbf{C}$ and $r$.

If we want to know if (and where) a ray $\mathbf{R}(t)$ intersects a given cube. We can substitute $\mathbf{P}$ in for $\mathbf{R}(t)$

$$\begin{align*} ||\mathbf{C} - \mathbf{R}(t)||_\infty &= r \\ ||\mathbf{C} - \mathbf{Q} + t\mathbf{D}||_\infty &= r \\ \end{align*}$$

But now by definition of the infinity norm, we can write

$$\begin{align*} ||\mathbf{C} - \mathbf{Q} + t\mathbf{D}||_\infty = \max\{&|C_x - Q_x + tD_x|,\\ &|C_y - Q_y + tD_y|,\\ &|C_z - Q_z + tD_z|\} = r \end{align*}$$

Here, since we're looking for a $t$ that will satisfy this equation, we can solve for it element-wise. That is, we can look at the expression for $x$

$$\begin{align*} |C_x - Q_x + tD_x| &= r \\ C_x - Q_x + tD_x &= \pm r \\ t &= (Q_x - C_x \pm r) / D_x \end{align*}$$

This gives us two values for $t$ which we can then plug back into the three expressions above. If either value $t^*$ is a maximum, then $\mathbf{R}(t^*)$ a point on the square. We do this same process for the $y$ and $z$ expressions, finding six separate values for $t$, and expecting zero, one, or two solutions.

See also

• Ray Tracing in One Weekend which inspired me to ray trace my voxel engine.