Geometric Modeling with High Order Derivatives
Candidate: Elif Tosun
Advisor: Denis Zorin

Abstract

Modeling of high quality surfaces is the core of geometric modeling. Such models are used in many computer-aided design and computer graphics applications. Irregular behavior of higher-order differential parameters of the surface (e.g. curvature variation) may lead to aesthetic or physical imperfections. In this work, we consider approaches to constructing surfaces with high degree of smoothness.

One direction is based on a manifold-based surface definition which ensures well-defined high-order derivatives that can be explicitly computed at any point. We extend previously proposed manifold-based construction to surfaces with piecewise-smooth boundary and explore trade-offs in some elements of the construction. We show that growth of derivative magnitudes with order is a general property of constructions with locally supported basis functions and derive a lower bound for derivative growth and numerically study flexibility of resulting surfaces at arbitrary points.

An alternative direction to using high-order surfaces is to define an approximation to high-order quantities for meshes, with high-order surface implicit. These approximations do not necessarily converge point-wise, but can nevertheless be successfully used to solve surface optimization problems. Even though fourth-order problems are commonly solved to obtain high quality surfaces, in many cases, these formulations may lead to reflection-line and curvature discontinuities. We consider two approaches to further increasing control over surface properties.

The first approach is to consider data-dependent functionals leading to fourth-order problems but with explicit control over desired surface properties. Our fourth-order functionals are based on reflection line behavior. Reflection lines are commonly used for surface interrogation and high-quality reflection line patterns are well-correlated with high-quality surface appearance. We demonstrate how these can be discretized and optimized accurately and efficiently on general meshes.

A more direct approach is to consider a poly-harmonic function on a mesh, such as the fourth-order biharmonic or the sixth-order triharmonic. The biharmonic and the triharmonic equations can be thought of as a linearization of curvature and curvature variation Euler-Lagrange equations respectively. We present a novel discretization for both problems based on the mixed finite element framework and a regularization technique for solving the resulting, highly ill-conditioned systems of equations. We show that this method, compared to more ad-hoc discretizations, has higher degree of mesh independence and yields surfaces of better quality.