Geometric Modeling with High Order Derivatives

Candidate: Elif Tosun

Advisor: Denis Zorin

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.