We present a technique for visualizing complicated mathematical surfaces that is inspired by hand-designed topological illustrations. Our approach generates exploded views that expose the internal structure of such a surface by partitioning it into parallel slices, which are separated from each other along a single linear explosion axis. Our contributions include a set of simple, prescriptive design rules for choosing an explosion axis and placing cutting planes, as well as automatic algorithms for applying these rules. First we analyze the input shape to select the explosion axis based on the detected rotational and reflective symmetries of the input model. We then partition the shape into slices that are designed to help viewers better understand how the shape of the surface and its cross-sections vary along the explosion axis. Our algorithms work directly on triangle meshes, and do not depend on any specific parameterization of the surface. We generate exploded views for a variety of mathematical surfaces using our system.

We introduce SmoothSketch---a system for inferring plausible 3D free-form shapes from visible-contour sketches. In our system, a user's sketch need not be a simple closed curve as in Igarashi's Teddy [1999], but may have cusps and T-junctions, i.e., endpoints of hidden parts of the contour. We follow a process suggested by Williams [1994] for inferring a smooth solid shape from its visible contours: completion of hidden contours, topological shape reconstruction, and smoothly embedding the shape via relaxation. Our main contribution is a practical method to go from a contour drawing to a fairly smooth surface with that drawing as its visible contour. In doing so, we make several technical contributions: • extending Williams' and Mumford's work [Mumford 1994] on figural completion of hidden contours containing T-junctions to contours containing cusps as well, • characterizing a class of visible-contour drawings for which inflation can be proved possible, • finding a topological embedding of the combinatorial surface that Williams creates from the figural completion, and • creating a fairly smooth solid shape by smoothing the topological embedding using a mass-spring system.We handle many kinds of drawings (including objects with holes), and the generated shapes are plausible interpretations of the sketches. The method can be incorporated into any sketch-based free-form modeling interface like Teddy.

With the advent of sketch-based methods for shape construction, there's a new degree of power available in the rapid creation of approximate shapes. Sketch [Zeleznik, 1996] showed how a gesture-based modeler could be used to simplify conventional CSG-like shape creation. Teddy [Igarashi, 1999] extended this to more free-form models, getting much of its power from its ``inflation'' operation (which converted a simple closed curve in the plane into a 3D shape whose silhouette, from the current point of view, was that curve on the view plane) and from an elegant collection of gestures for attaching additional parts to a shape, cutting a shape, and deforming it. But despite the powerful collection of tools in Teddy, the underlying polygonal representation of shapes intrudes on the results in many places. In this paper, we discuss our preliminary efforts at using variational implicit surfaces [Turk, 2000] as a representation in a free-form modeler. We also discuss the implementation of several operations within this context, and a collection of user-interaction elements that work well together to make modeling interesting hierarchies simple. These include ``stroke inflation'' via implicit functions, blob-merging, automatic hierarchy construction, and local surface modification via silhouette oversketching. We demonstrate our results by creating several models. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Modeling packages I.3.6 [Computer Graphics]: Interaction techniques