Enhancing spatial deformation for virtual sculpting

The task of computer-based free-form shape design is fraught with practical and conceptual difficulties. Incorporating elements of traditional clay sculpting has long been recognised as a means of shielding a user from the complexities inherent in this form of modelling. The premise is to deform a mathematically-defined solid in a fashion that loosely simulates the physical moulding of an inelastic substance, such as modelling clay or silicone putty. Virtual sculpting combines this emulation of clay sculpting with interactive feedback. Spatial deformations are a class of powerful modelling techniques well suited to virtual sculpting. They indirectly reshape an object by warping the surrounding space. This is analogous to embedding a flexible shape within a lump of jelly and then causing distortions by flexing the jelly. The user controls spatial deformations by manipulating points, curves or a volumetric hyperpatch. Directly Manipulated Free-Form Deformation (DMFFD), in particular, merges the hyperpatchand point-based approaches and allows the user to pick and drag object points directly. This thesis embodies four enhancements to the versatility and validity of spatial deformation: 1. We enable users to specify deformations by manipulating the normal vector and tangent plane at a point. A first derivative frame can be tilted, twisted and scaled to cause a corresponding distortion in both the ambient space and inset object. This enhanced control is accomplished by extending previous work on bivariate surfaces to trivariate hyperpatches. 2. We extend DMFFD to enable curve manipulation by exploiting functional composition and degree reduction. Although the resulting curve-composed DMFFD introduces some modest and bounded approximation, it is superior to previous curve-based schemes in other respects. Our technique combines all three forms of spatial deformation (hyperpatch, point and curve), can maintain any desired degree of derivative continuity, is amenable to the automatic detection and prevention of self-intersection, and achieves interactive update rates over the entire deformation cycle. 3. The approximation quality of a polygon-mesh object frequently degrades under spatial deformation to become either oversaturated or undersaturated with polygons. We have devised an efficient adaptive mesh refinement and decimation scheme. Our novel contributions include: incorporating fully symmetrical decimation, reducing the computation cost of the refinement/decimation trigger, catering for boundary and crease edges, and dealing with sampling problems. 4. The potential self-intersection of an object is a serious weakness in spatial deformation. We have developed a variant of DMFFD which guards against self-intersection by subdividing manipulations into injective (one-to-one) mappings. This depends on three novel contributions: analytic conditions for identifying self-intersection, and two injectivity tests (one exact but computationally costly and the other approximate but efficient). i