No abstract

Diffusion curves [OBW * 08] provide a flexible tool to create smooth-shaded images from curves defined with colors. The resulting image is typically computed by solving a Poisson equation that diffuses the curve colors to the interior of the image. In this paper we present a new method for solving diffusion curves by using ray tracing. Our approach is analogous to final gathering in global illumination, where the curves define source radiance whose visible contribution will be integrated at a shading pixel to produce a color using stochastic ray tracing. Compared to previous work, the main benefit of our method is that it provides artists with extended flexibility in achieving desired image effects. Specifically, we introduce generalized curve colors called shaders that allow for the seamless integration of diffusion curves with classic 2D graphics including vector graphics (e.g. gradient fills) and raster graphics (e.g. patterns and textures). We also introduce several extended curve attributes to customize the contribution of each curve. In addition, our method allows any pixel in the image to be independently evaluated, without having to solve the entire image globally (as required by a Poisson-based approach). Finally, we present a GPU-based implementation that generates solution images at interactive rates, enabling dynamic curve editing. Results show that our method can easily produce a variety of desirable image effects.

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean ¿-space E ¿ to the context of circle polyhedra in the ¾-sphere S ¾ . We prove that any two convex and proper non-unitary c-polyhedra with Möbiuscongruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere as well as that of certain hyperideal hyperbolic polyhedra in H ¿ .

We construct a 1-parameter family of geodesic shape metrics on a space of closed parametric curves in Euclidean space of any dimension. The curves are modeled on homogeneous elastic strings whose elasticity properties are described in terms of their tension and rigidity coefficients. As we change the elasticity properties, we obtain the various elastic models. The metrics are invariant under reparametrizations of the curves and induce metrics on shape space. Analysis of the geometry of the space of elastic strings and path spaces of elastic curves enables us to develop a computational model and algorithms for the estimation of geodesics and geodesic distances based on energy minimization. We also investigate a curve registration procedure that is employed in the estimation of shape distances and can be used as a general method for matching the geometric features of a family of curves. Several examples of geodesics are given and experiments are carried out to demonstrate the discriminative quality of the elastic metrics.

We verify the infinitesimal inversive rigidity of almost all triangulated circle polyhedra in the Euclidean plane E ¾ , as well as the infinitesimal inversive rigidity of tangency circle packings on the ¾-sphere S ¾ . From this the rigidity of almost all triangulated circle polyhedra follows. The proof adapts Gluck's proof in [7] of the rigidity of almost all Euclidean polyhedra to the setting of circle polyhedra, where inversive distances replace Euclidean distances and Möbius transformations replace rigid Euclidean motions.