Interactive Surface Modeling and Analysis
Abstract
Computer Aided Geometric Design (CAGD) is concerned with efficiently modeling physical objects with a surface or a collection of surfaces and has applications in CAD/CAM, computer graphics, robotics, and computer vision. This thesis introduces multi-sided A-patches and investigates their properties. A-patches are smooth and single-sheeted implicit algebraic surface patches in Bernstein-Bézier (BB) form. A new technique is described for filling an n-sided hole smoothly using a single implicit surface patch or a network of implicit patches with a geometrically intuitive compact representation. The free parameters of the A-patches are used to achieve fair surfaces with desirable properties. The main impediments to the widespread use of implicit surfaces for geometric modeling are multiple sheets, self-intersections and several other undesirable singularities. Our A-patch technique provides simple ways to guarantee that the constructed implicit surface is single-sheeted and free of undesirable singularities. The technique uses the zero contouring surfaces of trivariate Bernstein-Bézier polynomials to construct a piecewise smooth surface. We call such iso-splines A-patches, where "A" stands for algebraic. We have designed four different algorithms to construct smooth A-patch surfaces that interpolate or approximate scattered 3D point data or simple polyhedra of arbitrary topology. In the first algorithm we first construct a Gk curvilinear wire frame and then create implicit surface patches that interpolate the curves. A single patch is used for each triangle, quadrilateral, and pentagon in the input. In the second algorithm we triangulate the quadrilaterals and pentagons. The constructed surface passes through the vertices of the discretization and has the given normals at the vertices. This solution uses piecewise functions defined on a hull that consists of tetrahedra. The third algorithm uses piecewise rational functions defined on a hull that consists of tetrahedra and pyramids. And the fourth algorithm uses piecewise rational functions defined on a hull that consists of prisms.
Degree
Ph.D.
Advisors
Bajaj, Purdue University.
Subject Area
Computer science
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