Surface interpolation is a very important tool for complex surface modeling and widely applied in computer aided design, computer graphics and visualization. Generally, the objective surface, optimal surface, is required not only to interpolate all given points and curves, but also to meet some other preferred properties of industry or application-related senses such as minimum plate energy and maximum developability. Constructing an optimal surface (variational surface design) is challenging, mainly because interpolation conditions are generally arbitrary, and industry preferred properties may be highly nonlinear. Meeting these conditions simultaneously will make the construction of objective surfaces extremely complicated or even impossible....[ Read more ]

Surface interpolation is a very important tool for complex surface modeling and widely applied in computer aided design, computer graphics and visualization. Generally, the objective surface, optimal surface, is required not only to interpolate all given points and curves, but also to meet some other preferred properties of industry or application-related senses such as minimum plate energy and maximum developability. Constructing an optimal surface (variational surface design) is challenging, mainly because interpolation conditions are generally arbitrary, and industry preferred properties may be highly nonlinear. Meeting these conditions simultaneously will make the construction of objective surfaces extremely complicated or even impossible.

In this thesis, we focus on two particular optimal surface interpolation problems: optimal skinned surface and developable mesh surface interpolation. For optimal skinned surface, the currently existing skinning methods fall short of constructing surface of an optimal topology, which can improve the final shape quality by alleviating or removing surface anomalies such as ripples, bumps or even self-intersection. To solve this problem, we propose a novel algorithm to construct a “fair” skinned surface; The proposed algorithm is a discrete method: all curves are first sampled with a certain number of points, a curve network is constructed to interpolate all sample points and then patches fill the holes of the curve network to obtain a composite surface. However, in our algorithm the curve network is not specified in a heuristic method, instead, it is dynamically computed in an optimal sense, which can remove anomalies on the resultant surface. In a nutshell, the proposed algorithm combines the local construction of triangular patches and that of globally optimal curve network to obtain the best composite surface minimizing variation of principle curvatures. All patches are self-defined as Hermite Triangular Patches (HT) and a G¹ continuity is obtained among all neighboring HTs.

For the developable mesh surface interpolation problem, in which arbitrary positions are required to be interpolated, a new algorithm is proposed to deform a given mesh model in Gauss-Newton iteration and maximize its developability and at the same time respecting all interpolation conditions. If the initial model is flat or developable, the resultant mesh surface will be developable also and keep an isometric mapping to its initial one. We also use this algorithm to solve the length-stretch and collision problems in the field of cloth deformation simulation, where final deformation shapes are always noticeably stretched by currently existing methods.