First off, apologies if I cannot properly articulate my question in the most formal way. However, I believe my question should be simple enough to grasp anyhow.

In $\mathbb{R}^3$, I would like to determine the time of contact, if any, between:

- An unmoving unit sphere, whose center is at the origin
- A triangle, each of whose vertices follow independent, linear motion from $t = 0$ to $t = 1$. In other words, each triangle vertex has a start and end position, which are linearly interpolated by $t$.

The sphere may touch the triangle on a *vertex*, an *edge*, or on the *triangle's surface*. Vertex testing is simple enough as it's analogous to static line segment-sphere intersection.

For edge testing, parameterizing a line-sphere intersection test by $t$ appears to lead to solving a degree 4-polynomial, which isn't ideal. I believe that doing the same for triangle surface testing (parameterizing a sphere-plane intersection with $t$) would lead to solving a 6-degree polynomial.

Would there be any applicable non-analytical methods to approximate the intersection (other than directly approximating the polynomial roots)? Or maybe there is an analytical method that I'm not thinking of. In addition, would further constraining the motion of the triangle potentially simplify a solution?