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## Section: New Results

### Geometry Modeling and Computing

#### Computing the minimum distance between two Bézier curves

Keywords : Minimum distance, Bézier curve, Sweeping sphere clipping method.

Participants : Xiao-Diao Chen, Linqiang Chen, Yigang Wang, Gang Xu, Jun-Hai Yong, Jean-Claude Paul.

A sweeping sphere clipping method is presented for computing the minimum distance between two Bézier curves. The sweeping sphere is constructed by rolling a sphere with its center point along a curve. The initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve. The nearest point on a curve must be contained in the sweeping sphere along the other curve, and all of the parts outside the sweeping sphere can be eliminated. A simple sufficient condition when the nearest point is one of the two end points of a curve is provided, which turns the curve/curve case into a point/curve case and leads to higher efficiency. Examples are shown to illustrate efficiency and robustness of the new method [12] .

#### A torus patch approximation approach for point projection on surfaces

Keywords : Point projection; Torus patch approximation; Parametric surface.

Participants : Xiao-Ming Liu, Lei Yang, Jun-Hai Yong, He-Jin Gu, Jia-Guang Sun.

This paper proposes a second order geometric iteration algorithm for point projection and inversion on parametric surfaces. The iteration starts from an initial projection estimation. In each iteration, we construct a second order osculating torus patch to the parametric surface at the previous projection. Then we project the test point onto the torus patch to compute the next projection and its parameter. This iterative process is terminated when the parameter satisfies the required precision. Experiments demonstrate the convergence speed of our algorithm [19] .

#### Computing the minimum distance between a point and a clamped B-spline surface

Keywords : Minimum distance, B-spline surface, Newton' s method, Exclusion criterion.

Participants : Xiao-Diao Chen, Gang Xu, Jun-Hai Yong, Guozhao Wang, Jean-Claude Paul.

The computation of the minimum distance between a point and a surface is important for the applications such as CAD/CAM, NC verification, robotics and computer graphics. This paper presents a spherical clipping method to compute the minimum distance between a point and a clamped B-spline surface. The surface patches outside the clipping sphere which do not contain the nearest point are eliminated. Another exclusion criterion whether the nearest point is on the boundary curves of the surface is employed, which is proved to be superior to previous comparable criteria. Examples are also shown to illustrate efficiency and correctness of the new method [14] .

#### Computing lines of curvature for implicit surfaces

Keywords : Implicit surface, line of curvature, principal foliation, principal configuration, umbilical point, visualization.

Participants : Xiaopeng Zhang, Wujun Che, Jean-Claude Paul.

Lines of curvature are important intrinsic characteristics of a curved surface used in a wide variety of geometric analysis and processing. Although their differential attributes have been examined in detail, their global geometric distribution and topological pattern are very difficult to compute over the whole surface because of umbilical points and unstable numerical computation. No studies have yet been carried out on this problem, especially for an implicit surface. In this paper, we present a scheme for computing and visualizing the lines of curvature defined on an implicit surface. A key structure is introduced, conveying significant structure information about lines of curvature to facilitate their investigation, rather than computing their whole net. Our current framework is confined to a collection of manageable structures, consisting of algorithms to locate some seed umbilical points, to compute the lines of curvature through them, and finally to assemble this structure. The numerical implementations are provided in detail and a novel evaluation function measuring the violation of umbilical points in an implicit surface, i.e. indicating how much a point is to be umbilical, is also presented. This paper is the continuation of [Che, W.J., Paul, J.-C., Zhang, X.P., 2007. Lines of curvature and umbilical points for implicit surfaces. Computer Aided Geometric Design 24 (7), 395-409] [23] .

#### Constructing G1 quadratic Bézier curves with arbitrary endpoint tangent vectors

Keywords : Quadratic Bézier curve, geometric continuity, endpoint condition, smoothness.

Participants : He-Jin Gu, Jun-Hai Yong, Jean-Claude Paul, Fuhua Cheng [ Frank ] .

Quadratic Bézier curves are important geometric entities in many applications. However, it was often ignored by the literature the fact that a single segment of a quadratic Bézier curve may fail to fit arbitrary endpoint unit tangent vectors. The purpose of this paper is to provide a solution to this problem, i.e., constructing G1 quadratic Bézier curves satisfying given endpoint (positions and arbitrary unit tangent vectors) conditions. Examples are given to illustrate the new solution and to perform comparison between the G1 quadratic Bézier cures and other curve schemes such as the composite geometric Hermite curves and the biarcs [24] .

#### Approximate computation of curves on B-spline surfaces using quadratic reparameterization

Keywords : Approximation, B-spline, Curves on surfaces, Curve approximations.

Participants : Yi-Jun Yang, Jun-Hai Yong, Jean-Claude Paul.

Curves on surfaces play an important role in computer-aided geometric design. Because of the considerably high degree of exact curves on surfaces, approximation algorithms are preferred in CAD systems. To approximate the exact curve with a reasonably low degree curve which also lies completely on the B-spline surface, an algorithm is presented in this paper. The Hausdorff distance between the approximate curve and the exact curve is controlled under the user-specified distance tolerance. The approximate curve is TG1 continuous, where T is the user-specified angle tolerance. Examples are given to show the performance of our algorithm [22] .

#### B-spline Coons surface construction

Keywords : Coons surface, G 2 continuity, B-spline surface, geometric tolerance.

Participants : Kan-Le Shi, Jun-Hai Yong, Jia-Guang Sun, Jean-Claude Paul.

Coons surface is one of the most significant and widely used representations of shapes in computers. Its construction method is a recurring and essential operation in computer aided geometric design. This paper proposes an approach to construct a biquintic Coons surface having continuity with the specified boundary derivatives in the B-spline form, for arbitrary geometric tolerance vector . It presents the definition of this geometric invariant measure of angular and curvature tolerances. The methods of handling six types of the compatibility problems in constructing biquintic B-spline Coons surfaces are also proposed. Several examples are provided as well in this paper [20] .

#### Gn blending multiple surfaces in polar coordinates

Keywords : Multiple surface blending, Polar coordinate; NURBS surface; G n continuity; N-sided hole filling.

Participants : Kan-Le Shi, Jun-Hai Yong, Jia-Guang Sun, Jean-Claude Paul.

This paper proposes a method of Gn blending multiple parametric surfaces in polar coordinates. It models the geometric continuity conditions of parametric surfaces in polar coordinates and presents a mechanism of converting a Cartesian parametric surface into its polar coordinate form. The basic idea is first to reparameterize the parametric blendees into the form of polar coordinates. Then they are blended simultaneously by a basis function in the complex domain. To extend its compatibility, we also propose a method of converting polar coordinate blending surface into N NURBS patches. One application of this technique is to fill N -sided holes. Examples are presented to show its feasibility and practicability [21] .

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