Geometric model representation schemes have become more advanced in structure and mathematical complexity over the years. Boundary representations are widely accepted in geometric modeling due to their effectiveness in graphics rendering. Complex geometries are formed by polygonal mesh or patching analytical surfaces. Polygonal mesh representation is favoured due to its flexibility in shape modeling and ease in rendering while it suffers from large data size and failure to represent sharp features accurately. Patched analytical surface modelling has a much small data size thus favoring file transfer in mobile network, but complex shape modelling demanding higher order representation with stringent smoothness requirement remains a challenge. Individual extensions of polygonal m...[ Read more ]

Geometric model representation schemes have become more advanced in structure and mathematical complexity over the years. Boundary representations are widely accepted in geometric modeling due to their effectiveness in graphics rendering. Complex geometries are formed by polygonal mesh or patching analytical surfaces. Polygonal mesh representation is favoured due to its flexibility in shape modeling and ease in rendering while it suffers from large data size and failure to represent sharp features accurately. Patched analytical surface modelling has a much small data size thus favoring file transfer in mobile network, but complex shape modelling demanding higher order representation with stringent smoothness requirement remains a challenge. Individual extensions of polygonal mesh and patched analytical surfaces for complex object modeling are continuously being proposed while build complex shape model by integrating the two representation have not been fully exploited.

In this thesis, shapes are classified into features with two separate levels of shape complexity. For low complexity shape features, they are defined using parametric patched analytical surface with up to C² continuity requirement. On the other hand, complex shape features such as sculptures, streamlined objects, organic shape, are represented by polygonal mesh. Two separate methods are proposed respectively for two-dimensional and three-dimensional complex shape models.

For two dimensional shape models, a Non-Uniform Degree B-Spline Scheme has been proposed to represent objects with sharp features embedded in a parametric curve. By allowing non-uniform degree distribution along the polynomial basis, the proposed parametric spline can combine different continuity shapes into one single compact form. The scheme also works in higher dimensional space by taking the tensor product of lower dimensional domains. The proposed scheme generalizes the current basic-spline representation and provides a simple and efficient mathematical formulation in comparison with prior work. The work has been extended to three-dimensional shape modelling showing significant limitations which remain to be resolved.

For complex three-dimensional shape models, a Hybrid PN Mesh/Parametric Surface Scheme has been proposed to blend a low level shape feature represented by a patched analytical surface and a higher level shape feature represented by a triangular mesh using a transitional surface. The planar mesh elements in the triangular mesh shape model are transformed into PN triangles of cubic degree. This transformation facilitates the blending of the trimmed low level shape feature to the complex high level meshed shape feature with prescribed constraints. Energy minimization is also applied to the transition surface to ensure smooth blending between the features. The resulting shape model will be smooth, continuous and a 2-manifold model.

This work is the first step in an effort to construct complex shape model using a mixed representation scheme of minimum data size which is much needed in the modern world of wireless communication.