A technique is presented for generating implicit sweep objects that support direct specification and manipulation of the surface with no topological limitations on the 2D sweep template. The novelty of this method is that the underlying scalar field is bounded and C continuous, apart from surface creases. Bounded scalar fields guarantee local influence when modeling with implicit surfaces, an important usbility requirement for interactive modeling. A discrete approximation is also described that supports fast evaluation for bounded scalar fields. The new sweep objects are implemented in an interactive BlobTree modeling tool, providing an intuitive and expressive free-form implicit modeling component. This sweep representation permits conversion of parametric sweep surfaces to implicit volumes. An application to volume reconstruction from parallel contours is also explored.

Interactive tools for shape modeling with hierarchical implicit surfaces have been limited both by the high computational cost of visualization, and the lack of techniques for direct surface manipulation. To address the visualization issue, Hierarchical Spatial Caching is developed. This novel technique combines caching and spatial approximation to accelerate queries of the functional model tree. By reducing the cost of evaluating cached branches from O(N) to O(1), an order-of-magnitude improvement in visualization speed is realized. A new implicit sweep surface formulation which supports direct manipulation of the sweep profile is also developed, providing a powerful and flexible free-form implicit primitive. These new techniques form the core of a proofof-concept interactive modeling environment, called ShapeShop. ShapeShop provides a level of interactive control over hierarchical implicit models which has not been previously available. A survey of current techniques for shape modeling with implicit surfaces is also provided. iii Acknowledgements Where does one begin? Clearly, there is Ailidh. Thank you for your patience (particularly at

Nowadays, there is a lack of existing practical modeling tools suitable for specification of free-form implicit shapes. In this paper, we propose a new interpolation technique supplemented by interactive tools aimed at shape modeling by feature preserving metamorphosis of two-dimensional implicit shapes. The metamorphosis is controlled by global parameters and a set of correspondence vectors that are drawn directly to the scene. The underlying environment is the XISL package for definition and manipulation of implicit objects, which offers additional modeling and rendering possibilities. CR Categories: I.3.5 [COMPUTER GRAPHICS]: Computational Geometry and Object Modeling—Curve, surface, solid and object

We define b-compatibility for planar curves and propose three ball morphing techniques (b-morphs) between pairs of b-compatible curves. B-morphs use the automatic ball-map correspondence, proposed by Chazel et al. [11], from which they derive vertex trajectories (Linear, Circular, Parabolic). All are symmetric, meeting both curves with the same angle, which is a right angle for the Circular and Parabolic. We provide simple constructions for these b-morphs using the maximal disks in the finite region bounded by the two curves. We compare the b-morphs to each other and to other simple morphs (Linear Interpolation (LI), Closest Projection (CP), Curvature Interpolation (CI), Laplace Blending (LB), Heat Propagation (HP)) using seven measures of quality deficiency (travel distance, distortion, stretch, local acceleration, surface area, average curvature, maximal curvature). We conclude that the ratios of these measures depends heavily on the test case, especially for LI, CI, and LB which compute correspondence from a uniform geodesic parameterization. Nevertheless, we found that the Linear b-morph has consistently the shortest travel distance and that the Circular b-morph has the least amount of distortion. 36 37 Figure 1: A morph between an apple and pear along Circular b-morph trajectories (top left). morphing solutions and (2) How do the b-morphs introduced here compare to other approaches.

Interactive tools for shape modeling with hierarchical implicit surfaces have been limited both by the high computational cost of visualization, and the lack of techniques for direct surface manipulation. To address the visualization issue, Hierarchical Spatial Caching is developed. This novel technique combines caching and spatial approximation to accelerate queries of the functional model tree. By reducing the cost of evaluating cached branches from O(N) to O(1), an order-of-magnitude improvement in visualization speed is realized. A new implicit sweep surface formulation which supports direct manipulation of the sweep profile is also developed, providing a powerful and flexible free-form implicit primitive. These new techniques form the core of a proofof-concept interactive modeling environment, called ShapeShop. ShapeShop provides a level of interactive control over hierarchical implicit models which has not been previously available. A survey of current techniques for shape modeling with implicit surfaces is also provided.