How to construct a 3D polygonal region with a given domain in 3-space

I have a point (x,y,z), and I want to construct a spherical region around that point with a given number of vertices. When I add to the domain, I want the spherical polygon to selectively stretch/expand to enclose the modified domain. Most importantly, I have to check whether a given point (a,b,c) is in that polygonal region.

Does anyone know how to go about this? I'm working on a personal project and ran into this roadblock. Any help is appreciated.