Guo-Jin Wang

2-D shape blending: an intrinsic solution to the vertex path problem

This paper presents an algorithmfor determiningthe paths along which corresponding vertices travel in a 2–D shape blending. Rather than considering the vertex paths explicitly, the algorithm defines the intermediate shapes by interpolating the intrinsic definitions of the initial and final shapes. The algorithm produces shape blends which generally are more satisfactory than those produced using linear or cubic curve paths. Particularly, the algorithm can avoid the shrinkage that normally occurs when rotating rigid bodies are linearly blended, and avoids kinks in the blend when there were none in the key polygons.

Easy Mesh Cutting

We present Easy Mesh Cutting, an intuitive and easy‐to‐use mesh cutout tool. Users can cut meaningful components from meshes by simply drawing freehand sketches on the mesh. Our system provides instant visual feedback to obtain the cutting results based on an improved region growing algorithm using a feature sensitive metric. The cutting boundary can be automatically optimized or easily edited by users. Extensive experimentation shows that our approach produces good cutting results while requiring little skill or effort from the user and provides a good user experience. Based on the easy mesh cutting framework, we introduce two applications including sketch‐based mesh editing and mesh merging for geometry processing.

Comparison of interval methods for plotting algebraic curves

This paper compares the performance and efficiency of different function range interval methods for plotting f(x, y)=0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD. The methods considered are interval arithmetic methods (using the power basis, Bernstein basis, Homer form and centred form), an affine arithmetic method, a Bernstein coefficient method, Taubins method, Rivlins method, Gopalsamys method, and related methods which also take into account derivative information. Our experimental results show that the affine arithmetic method, interval arithmetic using the centred form, the Bernstein coefficient method, Taubins method, Rivlins method, and their related derivative methods have similar performance, and generally they are more accurate and efficient than Gopalsamys method and interval arithmetic using the power basis, the Bernstein basis, and Horner form methods.

Parametric representation of a surface pencil with a common spatial geodesic

In this paper, we study the problem of constructing a family of surfaces from a given spatial geodesic curve. We derive a parametric representation for a surface pencil whose members share the same geodesic curve as an isoparametric curve. By utilizing the Frenet trihedron frame along the given geodesic, we express the surface pencil as a linear combination of the components of this local coordinate frame, and derive the necessary and sufficient conditions for the coefficients to satisfy both the geodesic and the isoparametric requirements. We illustrate and verify the method by finding exact surface pencil formulations for some simple surfaces, such as surfaces of revolution and ruled surfaces. Finally, we demonstrate the use of this method in a garment design application. q 2003 Elsevier Ltd. All rights reserved.

A mesh reconstruction algorithm driven by an intrinsic property of a point cloud

This paper presents an algorithm for reconstructing a triangle mesh surface from a given point cloud. Starting with a seed triangle, the algorithm grows a partially reconstructed triangle mesh by selecting a new point based on an intrinsic property of the point cloud, namely, the sampling uniformity degree. The reconstructed mesh is essentially an approximate minimum-weight triangulation to the point cloud constrained to be on a two-dimensional manifold. Thus, the reconstructed surface has only small topological difference from the surface of the sampled object. Topological correct reconstruction can be guaranteed by adding a post-processing step. q 2003 Elsevier Ltd. All rights reserved.

Optimal multi-degree reduction of Bézier curves with constraints of endpoints continuity

Given a Bezier curve of degree n, the problem of optimal multi-degree reduction (degree reduction of more than one degree) by a Bezier curve of degree m (m > n - 1) with constraints of endpoints continuity is investigated. With respect to L2 norm, this paper presents one approximate method (MDR by L2) that gives an explicit solution to deal with it. The method has good properties of endpoints interpolation: continuity of any r, s (r, s ≥ 0) orders can be preserved at two endpoints respectively. The method in the paper performs multi-degree reduction at one time and does not need the stepwise computing. When applied to the multi-degree reduction with endpoints continuity of any orders, the MDR by L2 obtains the best least squares approximation. Comparison with another method of multi-degree reduction (MDR by L∞), which achieves the nearly best uniform approximation with respect to L∞ norm, is also given. The approximate effect of the MDR by L2 is better than that of the MDR by L∞. Explicit approximate error analysis of the multi-degree reduction methods is presented.

Non-iterative approach for global mesh optimization

This paper presents a global optimization operator for arbitrary meshes. The global optimization operator is composed of two main terms, one part is the global Laplacian operator of the mesh which keeps the fairness and another is the constraint condition which reserves the fidelity to the mesh. The global optimization operator is formulated as a quadratic optimization problem, which is easily solved by solving a sparse linear system. Our global mesh optimization approach can be effectively used in at least three applications: smoothing the noisy mesh, improving the simplified mesh, and geometric modeling with subdivision-connectivity. Many experimental results are presented to show the applicability and flexibility of the approach.

Hodographs and normals of rational curves and surfaces

Abstract Derivatives and normals of rational Bezier curves and surface patches are discussed. A non-uniformly scaled hodograph of a degree m × n tensor-product rational surface, which provides correct derivative direction but not magnitude, can be written as a degree (2 m − 2) × 2 n or 2 m × (2 n − 2) vector function in polynomial Bezier form. Likewise, the scaled normal direction is degree (3 m − 2) × (3 n − 2). Efficient methods are developed for bounding these directions and the derivative magnitude.

Partial derivatives of rational Be´zier surfaces

Abstract A rational surface is the locus of a rational curve that is moving through space and thereby changing its shape by changing its control points and weights. This intuitive definition can be used to derive hodographs of rational Bezier surfaces and their bounds of magnitude.

The rational cubic Be´zier representation of conics

Abstract The rational cubic Bezier curve is a very useful tool in CAGD. It incorporates both conic sections and parametric cubic curves as special cases, so its advantage is that one can deal with curves of these two kinds in one computer procedure. In this paper, the necessary and sufficient conditions for representing conics by the rational cubic Bezier form in proper parametrization are investigated; these conditions can be divided into two parts: one for weights and the other for Bezier vertices.