How can I find the point B(t) along a cubic Bezier curve that is closest to an arbitrary point P in the plane?
After lots of searching I found a paper that discusses a method for finding the closest point on a Bezier curve to a given point:
Improved Algebraic Algorithm On Point Projection For Bezier Curves, by Xiao-Diao Chen, Yin Zhou, Zhenyu Shu, Hua Su, and Jean-Claude Paul.
Furthermore, I found Wikipedia and MathWorld's descriptions of Sturm sequences useful in understanding the first part of the algoritm, as the paper itself isn't very clear in its own description.