Jiaye Wang

An algebraic condition for the separation of two ellipsoids

Given two ellipsoids, we show that their characteristic equation has at least two negative roots and that the ellipsoids are separated by a plane if and only if their characteristic equation has two distinct positive roots. Furthermore, the ellipsoids touch each other externally if and only if the characteristic equation has a positive double root. An advantage of this characterization is that only the signs of the roots matter.  2001 Elsevier Science B.V. All rights reserved.

Medial axis tree-an internal supporting structure for 3D printing

Saving material and improving strength are two important but conflicting requirements in 3D printing. We propose a novel method for designing the internal supporting frame structures of 3D objects based on their medial axis such that the objects are fabricated with minimal amount of material but can still withstand specified external load. Our method is inspired by the observation that the medial axis, being the skeleton of an object, serves as a natural backbone structure of the object to improve its resistance to external loads. A hexagon-dominant framework beneath the boundary surface is constructed and a set of tree-like branching bars are designed to connect this framework to the medial axis. The internal supporting structure is further optimized to minimize the material cost subject to strength constraints. Models fabricated with our method are intended to withstand external loads from various directions, other than just from a particular direction as considered in some other existing methods. Experimental results show that our method is capable of processing various kinds of input models and producing stronger and lighter 3D printed objects than those produced with other existing methods.

Efficient collision detection for moving ellipsoids using separating planes

We present a simple, accurate and efficient algorithm for collision detection among moving ellipsoids. Its efficiency is attributed to two results: (i) a simple algebraic test for the separation of two ellipsoids, and (ii) an efficient method for constructing a separating plane between two disjoint ellipsoids. Inter-frame coherence is exploited by using the separating plane to reduce collision detection to simpler subproblems of testing for collision between the plane and each of the ellipsoids. Compared with previous algorithms (such as the GJK method) which employ polygonal approximation of ellipsoids, our algorithm demonstrates comparable computing speed and much higher accuracy.

Using signature sequences to classify intersection curves of two quadrics

We present a method that uses signature sequences to classify the intersection curve of two quadrics (QSIC) or, equivalently, quadric pencils in PR^3 (3D real projective space), in terms of the shape, topological properties, and algebraic properties of the QSIC. Specifically, for a QSIC we consider its singularity, reducibility, the number of its components, and the degree of each irreducible component, etc. There are in total 35 different types of non-degenerate quadric pencils. For each of the 35 types of QSICs given by these non-degenerate pencils, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of the quadric pencil. We show how to compute a signature sequence with rational arithmetic and use it to determine the type of the intersection curve of any two quadrics which form a non-degenerate pencil. As an example of application, we discuss how to apply our results to collision detection of cones in 3D affine space.

Classifying the nonsingular intersection curve of two quadric surfaces

We present new results on classifying the morphology of the nonsingular intersection curve of two quadrics by studying the roots of the characteristic equation, or the discriminant, of the pencil spanned by the two quadrics. The morphology of a nonsingular algebraic curve means the structural (or topological) information about the curve, such as the number of disjoint connected components of the curve in P/spl Ropf//sup 3/ (the 3D real projective space), and whether a particular component is a compact set in any affine realization of P/spl Ropf//sup 3/. For example, we show that two quadrics intersect along a nonsingular space quartic curve in P/spl Ropf//sup 3/ with one connected component if and only if their characteristic equation has two distinct real roots and a pair of complex conjugate roots. Since the number of the real roots of the characteristic equation can be counted robustly with exact arithmetic, our results can be used to obtain structural information reliably before computing the parameterization of the intersection curve; thus errors in the subsequent computation that is most likely done using floating point arithmetic will not lead to erroneous topological classification of the intersection curve. The key technique used to prove our results is to reduce two quadrics into simple forms using a projective transformation, a technique equivalent to the simultaneous block diagonalization of two real symmetric matrices, a topic that has been studied in matrix algebra.

Local computation of curve interpolation knots with quadratic precision

There are several prevailing methods for selecting knots for curve interpolation. A desirable criterion for knot selection is whether the knots can assist an interpolation scheme to achieve the reproduction of polynomial curves of certain degree if the data points to be interpolated are taken from such a curve. For example, if the data points are sampled from an underlying quadratic polynomial curve, one would wish to have the knots selected such that the resulting interpolation curve reproduces the underlying quadratic curve; in this case, the knot selection scheme is said to have quadratic precision. In this paper, we propose a local method for determining knots with quadratic precision. This method improves on our previous method that entails the solution of a global equation to produce a knot sequence with quadratic precision. We show that this new knot selection scheme results in better interpolation error than other existing methods, including the chord-length method, the centripetal method and Foleys method, which do not possess quadratic precision.

Fast Updating of Delaunay Triangulation of Moving Points by Bi-cell Filtering

Updating a Delaunay triangulation when data points are slightly moved is the bottleneck of computation time in variational methods for mesh generation and remeshing. Utilizing the connectivity coherence between two consecutive Delaunay triangulations for computation speedup is the key to solving this problem. Our contribution is an effective filtering technique that confirms most bi‐cells whose Delaunay connectivities remain unchanged after the points are perturbed. Based on bi‐cell flipping, we present an efficient algorithm for updating two‐dimensional and three‐dimensional Delaunay triangulations of dynamic point sets. Experimental results show that our algorithm outperforms previous methods.

Parallel B-Spline Surface Interpolation on a Mesh-Connected Processor Array

A parallel implementation of Chebyshev method is presented for the B-spline surface interpolation problem. The algorithm finds the control points of a uniform bicubic B-spline surface that interpolates m × n data points on an m × n mesh-connected processor array in constant time. Hence it is optimal. Due to its numerical stability, the algorithm can successfully be used in finite precision floating-point arithmetic.

A maximum power point tracking algorithm for buoy-rope-drum wave energy converters

The maximum power point tracking control is the key link to improve the energy conversion efficiency of wave energy converters (WEC). This paper presents a novel variable step size Perturb and Observe maximum power point tracking algorithm with a power classification standard for control of a buoy-rope-drum WEC. The algorithm and simulation model of the buoy-rope-drum WEC are presented in details, as well as simulation experiment results. The results show that the algorithm tracks the maximum power point of the WEC fast and accurately.

Minimizing the Number of Separating Circles for Two Sets of Points in the Plane

Given two sets of points