Intersection of a polyhedron with a line

In computational geometry, the intersection of a polyhedron with a line is the problem of computing the intersection of a convex polyhedron and a ray in Euclidean space. This problem has important applications in computer graphics, optimization, and even in some Monte Carlo methods.

Statement of the problem

In general, a convex polyhedron is defined as the intersection of a finite number of halfspaces. That is, a convex polyhedron is the set of solutions of a system of inequations of the form

$Ax \le b.$

The formal statement of our problem is to find the intersection of the set $\{x\in\mathbb{R}^n|Ax \le b\}$ with the line defined by $c%2Btv$ , where $t\in\mathbb{R}$ and $c, v\in\mathbb{R}^n$ .

General solution

To this end, we would like to find $t$ such that $A(c%2Btv)\le b$ , which is equivalent to finding a $t$ such that

$[Av]_it \leq [b-Ac]_i$

for $i=1,\ldots,n$ .

Thus, we can bound $t$ as follows:

$t \leq \frac{[b-Ac]_i}{[Av]_i} \;\;\;{\rm if}\;[Av]_i > 0$

$t \geq \frac{[b-Ac]_i}{[Av]_i} \;\;\;{\rm if}\;[Av]_i < 0$

$t {\rm\; unbounded} \;\;\;{\rm if}\;[Av]_i = 0, [b-Ac]_i > 0$

$t {\rm\; infeasible} \;\;\;{\rm if}\;[Av]_i = 0, [b-Ac]_i < 0$

The last two lines follow from the cases when the direction vector $v$ is parallel to the halfplane defined by the $i^{th}$ row of $A$ : $a_i^Tx \leq b_i$ . In the second to last case, the point $c$ is on the inside of the halfspace; in the last case, the point $c$ is on the outside of the halfspace, and so $t$ will always be infeasible.

As such, we can find $t$ as all points in the region (so long as we do not have the fourth case from above)

$\left[ \max_{i:[Av]_i\leq 0}\frac{[b-Ac]_i}{[Av]_i}, \min_{i:[Av]_i\geq 0}\frac{[b-Ac]_i}{[Av]_i}\right],$

which will be empty if there is no intersection.

External links

Intersection of convex hull with a line with pseudo code