| There has been a growing interest in the use of implicit representations for curves and surfaces. Applications arise in numerous areas of physical sciences, such as engineering, robotics, vision, graphics, geometric optics, and in many areas of pure mathematics, such as differential geometry, complex analysis, number theory and differential equations. Implicit algebraic models have proved to be very useful representations for 2D curves and 3D surfaces in several model-based applications including vision, graphics, computational geometry and CAD. Their interpolation properties can compensate for certain amounts of missing-data and/or occlusion. Moreover, for more accurate representations of object data, higher degree curve or surface models can be used. Furthermore, geometric and/or algebraic invariants can be defined for implicit algebraic models, and subsequently used for identifying objects in arbitrary configurations.; The primary purpose of this thesis is to present some new insight into implicit algebraic models. We do this by presenting several ramifications of a fundamental new result, called the unique decomposition theorem for algebraic curves. This theorem allows us to express higher degree algebraic curves as a unique sum of real conic-line products. We use this compact representation to establish several new results, such as canonical representations for identifying and comparing free-form objects, constructing a complete set of functionally independent geometric invariants for curves of arbitrary degree, developing elliptical-circular representations for quartic curves, which imply a particularly transparent set of geometric invariants, modifying free-form shapes through conic primitive alterations, and generating sweep surfaces using conic interpolations. These results represent the main contribution of this thesis. |
