  A common representation of a three dimensional object in computer
applications, such as graphics and design, is in the form of a triangular mesh.
In many instances, individual or groups of triangles in such representation
need to satisfy spatial constraints that are imposed either by observation from
the real world, or by concrete design specifications of the object. As these
problems tend to be of large scale, choosing a mathematical optimization
approach can be particularly challenging. In this paper, we model various
geometric constraints as convex sets in Euclidean spaces, and find the
corresponding projections in closed forms. We also present an interesting idea
to successfully maneuver around some important nonconvex constraints while
still preserving the intrinsic nature of the original design problem. We then
use these constructions in modern first-order splitting methods to find optimal
solutions.
