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Implicit surface modeling is attracting attention, because a complex object can be constructed easily and intuitively from some simple primitive objects, defined by primitive defining functions, using successive compositions of blending operations. Blending operations play a major role in implicit surfaces, because they can join intersecting primitive objects (operands) smoothly with transitions generated automatically by blending operators. Hence, this dissertation proposes three new methods: (1) the scale method, (2) field functions with adjustable inner and outer radii, and (3) the translation method, for developing blending operations that have blending range controls. That is, the proposed blending operations provide blending range parameters to adjust the size and shape of the transition of the blending surface freely, without deforming the shapes of blended primitives totally. The first and the third methods offer blending range controls by developing new blending operators, whereas the second method does the similar things by developing new primitive defining functions. The scale method is a generalized method. It provides a framework to transform any existing blending operators or arc-shaped curves into the blending operator that has the following properties: (1) Provides blending range and curvature parameters to adjust the size and shape of the transition of the blending surface, without deforming the shapes of blended primitives totally. (2) Behaves like Max/Min(x1,¡K,xk) operators in non-blending regions in the entire domain. As a result, it gives a more intuitive shape control on modeling its subsequent blends. (3) Possesses C1 continuity in the entire domain except the origin. As a result, it can prevent from generating non-smooth surfaces on sequential blends with overlapped blending regions. (4) Works to blend both non-zero and zero implicit surfaces. (5) Can be a new primitive in other blends, especially in Soft blending. (6) Applies for bulge elimination. Field functions with adjustable inner and outer radii provide parameters to adjust the inner and the outer radii of influence, respectively. This dissertation proposes four different transforms to develop this kind of field functions. Thus, using the proposed field functions as the new primitive defining functions of soft object modeling, Soft blending, R-functions, Ricci¡¦s super-ellipsoid blends and Perlin¡¦s set operations: (1) Can retain their low computing complexity. (2) Can perform the blending range controls, by adjusting the inner and the outer radii of influence of the proposed field functions. The translation method is also a generalized method. It offers a framework to transform any existing blending operators or arc-shaped curves into controllable blending operators for blending zero implicit surfaces. A controllable blending operator has the following properties: (1) Offers blending range and curvature parameters to adjust the size and shape of transition of the blending surface, without deforming the shapes of blended primitives completely. (2) Provides parameters mi, i=1,2,¡K,k, to behave like Max/Min(x1/m1,¡K,xk/mk) operators on non-blending regions in the entire domain, and its zero level blending surface remains unchanged, whatever mi, i=1,2,¡K,k, are set. As a result, by adjusting mi, i=1,2,¡K,k, a controllable blending operator has the following abilities to control its primitives¡¦ subsequent blends:

Computing surface and/or object intersections is a cornerstone of many algorithms in Geometric Modeling and Computer Graphics, for example Set Operations between solids, or surface Ray Casting display. We present an object centered, information preserving, hierarchical representation for polyhedra called Prism Tree. We use the representation to decompose the intersection algorithms into two steps: the localization of intersections, and their processing. When dealing with polyhedra with many faces (typically more than one thousand), the first step is by far the most expensive. The Prism Tree structure is used to compute efficiently this localization step. A preliminary implementation of the Set Operations and Ray casting algorithms has been constructed.

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<p>Set-based methods have been leveraged in many engineering applications from robust control and global optimization, to probabilistic planning and estimation. While useful, these methods have most widely been applied to analysis over sets that are convex, due to their ease in both representation and calculation. The representation and analysis of nonconvex sets is inherently complex. When nonconvexity arises in design and control applications, the nonconvex set is often over-approximated by a convex set to provide conservative results. However, the level of conservatism may be large and difficult to quantify, often leading to trivial results and requiring repetitive analysis by the engineer. Nonconvexity is inherent and unavoidable in many applications, such as the analysis of hybrid systems and robust safety constraints. </p> <p>In this dissertation, I present a new nonconvex set representation named the hybrid zonotope. The hybrid zonotope builds upon a combination of recent advances in the compact representation of convex sets in the controls literature with methods leveraged in solving mixed-integer programming problems. It is shown that the hybrid zonotope is equivalent to the union of an exponential number of convex sets while using a linear number of continuous and binary variables in the set’s representation. I provide identities for, and derivations of, the set operations of hybrid zonotopes for linear mappings, Minkowski sums, generalized intersections, halfspace intersections, Cartesian products, unions, complements, point containment, set containment, support functions, and convex enclosures. I also provide methods for redundancy removal and order reduction to improve the compactness and computational efficiency of the represented sets. Therefore proving the hybrid zonotopes expressive power and applicability to many nonconvex set-theoretic methods. Beyond basic set operations, I specifically show how the exact forward and backward reachable sets of linear hybrid systems may be found using identities that are calculated algebraically and scale linearly. Numerical examples show the scalability of the proposed methods and how they may be used to verify the safety and performance of complex systems. These exact methods may also be used to evaluate the level of conservatism of the existing approximate methods provided in the literature.  </p>