Superquadrics and Angle-Preserving Transformations

Over the past 20 years, a great deal of interest has developed in the use of computer graphics and numerical methods for three-dimensional design. Significant progress in geometric modeling is being made, predominantly for objects best represented by lists of edges, faces, and vertices. One long-term goal of this work is a unified mathematical formalism, to form the basis of an interactive and intuitive design environment in which designers can simulate three-dimensional scenes with shading and texture, produce usable design images, verify numerical machining-control commands, and set up finite-element meshwork for structural and dynamic analysis. A new collection of smooth parametric objects and a new set of three-dimensional parametric modifiers show potential for helping to achieve this goal. The superquadric primitives and angle-preserving transformations extend the traditional geometric primitives-quadric surfaces and parametric patches-used in existing design packages, producing a new spectrum of flexible forms. Their chief advantage is that they allow complex solids and surfaces to be constructed and altered easily from a few interactive parameters

Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

Robust Feature Detection and Local Classification for Surfaces Based on Moment Analysis

The stable local classification of discrete surfaces with respect to features such as edges and corners or concave and convex regions, respectively, is as quite difficult as well as indispensable for many surface processing applications. Usually, the feature detection is done via a local curvature analysis. If concerned with large triangular and irregular grids, e.g., generated via a marching cube algorithm, the detectors are tedious to treat and a robust classification is hard to achieve. Here, a local classification method on surfaces is presented which avoids the evaluation of discretized curvature quantities. Moreover, it provides an indicator for smoothness of a given discrete surface and comes together with a built-in multiscale. The proposed classification tool is based on local zero and first moments on the discrete surface. The corresponding integral quantities are stable to compute and they give less noisy results compared to discrete curvature quantities. The stencil width for the integration of the moments turns out to be the scale parameter. Prospective surface processing applications are the segmentation on surfaces, surface comparison, and matching and surface modeling. Here, a method for feature preserving fairing of surfaces is discussed to underline the applicability of the presented approach.

Fourth-order flows in surface modelling

This short article is a brief account of the usage of fourth-order curvature flow in surface modelling

Computational Modeling, Visualization, and Control of 2-D and 3-D Grasping under Rolling Contacts

This chapter presents a computational methodology for modeling 2-dimensional grasping of a 2-D object by a pair of multi-joint robot fingers under rolling contact constraints. Rolling contact constraints are expressed in a geometric interpretation of motion expressed with the aid of arclength parameters of the fingertips and object contours with an arbitrary geometry. Motions of grasping and object manipulation are expressed by orbits that are a solution to the Euler-Lagrange equation of motion of the fingers/object system together with a set of first-order differential equations that update arclength parameters. This methodology is then extended to mathematical modeling of 3-dimensional grasping of an object with an arbitrary shape. Based upon the mathematical model of 2-D grasping, a computational scheme for construction of numerical simulators of motion under rolling contacts with an arbitrary geometry is presented, together with preliminary simulation results. The chapter is composed of the following three parts. Part 1 Modeling and Control of 2-D Grasping under Rolling Contacts between Arbitrary Smooth Contours Authors: S. Arimoto and M. Yoshida Part 2 Simulation of 2-D Grasping under Physical Interaction of Rolling between Arbitrary Smooth Contour Curves Authors: M. Yoshida and S. Arimoto Part 3 Modeling of 3-D Grasping under Rolling Contacts between Arbitrary Smooth Surfaces Authors: S. Arimoto, M. Sekimoto, and M. Yoshid