Energy Duality Methods for Piecewise Bézier Curve Construction

Piecewise Bézier Curves are constructed using a minimization principle. C^k and GC^k continuity is imposed by linear constraints. The corresponding quadratic programming with linear constraints problem is introduced and solved by duality methods. Bordering matrices methods are implemented to deal with local refinement (subdivision). The result is a versatile tool for defining/editing contours made of piecewise Bézier curves.     

Boundary evaluation for interval Bézier curve

The objective of this paper is to provide an efficient and reliable algorithm for representing and evaluating the boundary of the interval Bézier curve in 2- and 3-D. The boundary of the planar Bézier curve is represented by a sequence of Bézier curve segments with same degree and line segments in the order they are encountered when marching counter-clockwise along its boundary. The boundary can also be represented as a single B-spline curve having the same degree with the interval Bézier curve. The boundary of the 3-D interval Bézier curve is made up of trimmed Bézier surface patches and rectangular patches. Some examples illustrate our algorithms.     

A shape controled fitting method for Bézier curves

The problem of controling a shape when fitting a curve to a set of digitized data points by proceeding to a least squares approximation is considered. A nonlinear method of solving this problem, dedicated to the obtention of planar curves with a smooth and monotonous variation of curvature is introduced. This method uses particular Bézier curves, called typical curves, whose control polygon is partially constrained in order to provide the desired curve shape. The curve fitting principle is based on variations of the tangent direction at the ends of the curve. These variations are controled by the displacement of a given curve point. An automatic procedure using this method to get a curve close to a set of data points has been implemented. An application to car body shape design and a comparison with the least squares approximation method is presented and discussed.     

Progressive iterative approximation for triangular Bézier surfaces

Recently, for the sake of fitting scattered data points, an important method based on the PIA (progressive iterative approximation) property of the univariate NTP (normalized totally positive) bases has been effectively adopted. We extend this property to the bivariate Bernstein basis over a triangle domain for constructing triangular Bézier surfaces, and prove that this good property is satisfied with the triangular Bernstein basis in the case of uniform parameters. Due to the particular advantages of triangular Bézier surfaces or rational triangular Bézier surfaces in CAD (computer aided design), it has wide application prospects in reverse engineering.     

A Bernstein-Bézier Based Approach to Soft Tissue Simulation

This paper discusses a Finite Element approach for volumetric soft tissue modeling in the context of facial surgery simulation. We elaborate on the underlying physics and address some computational aspects of the finite element discretization.
In contrast to existing approaches speed is not our first concern, but we strive for the highest possible accuracy of simulation. We therefore propose an extension of linear elasticity towards incompressibility and nonlinear material behavior, in order to describe the complex properties of human soft tissue more accurately. Furthermore, we incorporate higher order interpolation functions using a Bernstein-Bézier formulation, which has various advantageous properties, such as its integral polynomial form of arbitrary degree, efficient subdivision schemes, and suitability for geometric modeling and rendering. In addition, the use of tetrahedral Finite Elements does not put any restriction on the geometry of the simulated volumes.
Experimental results obtained from a synthetic block of soft tissue and from the Visible Human Data Set illustrate the performance of the envisioned model.     

Multi-degree reduction of Bézier curves using reparameterization

L₂-norms are often used in the multi-degree reduction problem of Bézier curves or surfaces. Conventional methods on curve cases are to minimize , where and are the given curve and the approximation curve, respectively. A much better solution is to minimize , where is the closest point to point , that produces a similar effect as that of the Hausdorff distance. This paper uses a piecewise linear function L(t) instead of t to approximate the function φ(t) for a constrained multi-degree reduction of Bézier curves. Numerical examples show that this new reparameterization-based method has a much better approximation effect under Hausdorff distance than those of previous methods.     

A comparison of local parametric C Bézier interpolants for triangular meshes

Parametric curved shape surface schemes interpolating vertices and normals of a given triangular mesh with arbitrary topology are widely used in computer graphics for gaming and real-time rendering due to their ability to effectively represent any surface of arbitrary genus. In this context, continuous curved shape surface schemes using only the information related to the triangle corresponding to the patch under construction, emerged as attractive solutions responding to the requirements of resource-limited hardware environments. In this paper we provide a unifying comparison of the local parametric C⁰ curved shape schemes we are aware of, based on a reformulation of their original constructions in terms of polynomial Bézier triangles. With this reformulation we find a geometric interpretation of all the schemes that allows us to analyse their strengths and shortcomings from a geometrical point of view. Further, we compare the four schemes with respect to their computational costs, their reproduction capabilities of analytic surfaces and their response to different surface interrogation methods on arbitrary triangle meshes with a low triangle count that actually occur in their real-world use.     

一种分层海洋中求取本征声线的新方法

提出了一种在恒定深度的分层海洋中计算本征声线的方法。该方法在跨度模型的基础上，将本征声线表示为三段声线跨度的组合，建立跨度方程组以确定本征声线。采用计算本征声线的方法，可以计算若干条本征声线的出射角、到达角、到达时间和传播损失。如果给定声源的辐射频率以及海底信息，还可以计算声线携带声波的强度和相位，这样在经典射线理论的基础上可以简明的描述声场分布，给水声通信时变特性分析和通信系统的设计奠定一定基础。