David Gay: Visualizing Families of Functions On Surfaces

The output of this iVRG will be a collection of computer-generated animations related to certain interesting families of functions on surfaces. For example, imagine the surface of a donut in 3-dimensional space. Given any unit vector in R^3, we can orthogonally project the surface onto the line spanned by that vector, and get a function from the surface to the reals. We have one such function for each direction in R^3, i.e. one such function for each point on the unit sphere. This is a family of functions parametrized by points on the sphere. We can study qualitative features of these functions, such as critical points and gradient vector fields, as we move around on the sphere, and figure out how to present our insights through computation and computer graphics. The only prerequisites are multivariable calculus, linear algebra and some kind of programming experience.