| 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 methods for 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. We show that growth of derivative magnitudes with order is a general property of constructions with locally supported basis functions, 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 functional is 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 polyharmonic function on a mesh, such as the fourth-order biharmonic or the sixth-order triharmonic. These equations can be thought of as linearizations 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. We show that this method, compared to more ad-hoc discretizations, has higher degree of mesh independence and yields surfaces of better quality. |
