Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials

In a recent article, Ge et al. (1997) identify a special class of rational curves (Harmonic Rational Bézier (HRB) curves) that can be reparameterized in sinusoidal form. Here we show how this family of curves strongly relates to the class of p-Bézier curves, curves easily expressible as single-valued in polar coordinates. Although both subsets do not coincide, the reparameterization needed in both cases is exactly the same, and the weights of a HRB curve are those corresponding to the representation of a circular arc as a p-Bézier curve. We also prove that a HRB curve can be written as a combination of its control points and certain Bernstein-like trigonometric basis functions. These functions form a normalized totally positive B-basis (that is, the basis with optimal shape preserving properties) of the space of trigonometric polynomials {1, sint, cost, …. sinmt, cosmt} defined on an interval of length < π.     

基于四点分段的一类三角多项式曲线

提出了一类m(m=1,2,3)次分段三角多项式曲线,通过引入形状参数,给出了加权三角多项式曲线,与三次B样条曲线类似。每段三角多项式曲线由4个相继的控制点生成,对于等距节点的情形,所提出的三角多项式曲线是C^2m-1连续;给出了三角开曲线和闭曲线的构造方法。论述了椭圆的表示方法,给出了三角多项式曲线与三次B样条曲线的对比,通过改变次数m或调整形状参数,可以得到不同程度地接近于控制多边形的曲线,因此,所给曲线的生成方法是一种结构简单和使用方便的曲线生成方法。     

Corner cutting with trapezoidal augmentation for area-preserving smoothing of polygons and polylines

Recently, a new subdivision method was introduced by the author for smoothing polygons and polylines while preserving the enclosed area [Gordon D. Corner cutting and augmentation: an area-preserving method for smoothing polygons and polylines. Computer Aided Geometric Design 2010; 27(7):551–62]. The new technique, called “corner cutting and augmentation” (CCA), operates by cutting corners with line segments and adding the cut area of each corner to two augmenting structures constructed on the two incident edges; this operation can be iterated as needed. Area is preserved in a local sense, meaning that when a corner is cut, the cut area is added to the other side of the line in immediate proximity to the cut corner. Thus, CCA is also applicable to self-intersecting polygons and polylines, and it enables local control. CCA was originally developed with triangular augmentation, which was called CCA1. This work presents CCA2, in which the augmenting structures are trapezoids. A theoretical result from previous work is used to show that certain implementation restrictions guarantee the existence and the G¹-continuity of the limit curve of CCA2, and also the preservation of convexity. The main difference between CCA1 and CCA2 is that the limit curve of CCA1 does not contain straight line segments, while CCA2 can contain such segments. CCA2 allows the user to determine how closely each iteration follows its previous polygon. Potential applications include computer aided geometric design, an alternative to spline approximation, an aid to artistic design, and a possible alternative to multiresolution curves.     

A new algorithm for dominant points detection and polygonization of digital curves

A new algorithm for detecting dominant points and polygonal approximation of digitized closed curves is presented. It uses an optimal criterion for determining the region-of-support of each boundary point, and a new mechanism for selecting the dominant points. The algorithm does not require an input parameter, and can handle shapes that contain features of multiple sizes efficiently. In addition, the approximating polygon preserves the symmetry of the shape.     

Geometric construction of spline curves with tension properties

In this paper a class of C²∩FC³ spline curves possessing tension properties is described. These curves can be constructed using a simple modification of the well-known geometric construction of C⁴ quintic splines; therefore their shape can be easily controlled using the control net. Their applications in approximation and interpolation of spatial data will be discussed.     

Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves

Given a real valued function f(X,Y), a box region B₀⊆R² and ε>0, we want to compute an ε-isotopic polygonal approximation to the restriction of the curve S=f⁻¹(0)={p∈R²:f(p)=0} to B₀. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga & Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities.     

基于Chebyshev多项式的椭圆曲线密码系统算法

通过结合Chebyshev多项式与椭圆曲线, 构造基于Chebyshev多项式的椭圆曲线密码系统算法。利用有限域上Chebyshev良好的半群特性和椭圆曲线上的性质, 实现了在椭圆曲线上的加密算法。该算法具有混沌密码和椭圆曲线密码算法的优点。通过对该算法的分析, 认为算法简单、安全性高、方案可行。     

Poisson Manifold Reconstruction — Beyond Co-dimension One

Screened Poisson Surface Reconstruction creates 2D surfaces from sets of oriented points in 3D (and can be extended to co-dimension one surfaces in arbitrary dimensions). In this work we generalize the technique to manifolds of co-dimension larger than one. The reconstruction problem consists of finding a vector-valued function whose zero set approximates the input points. We argue that the right extension of screened Poisson Surface Reconstruction is based on exterior products: the orientation of the point samples is encoded as the exterior product of the local normal frame. The goal is to find a set of scalar functions such that the exterior product of their gradients matches the exterior products prescribed by the input points. We show that this setup reduces to the standard formulation for co-dimension 1, and leads to more challenging multi-quadratic optimization problems in higher co-dimension. We explicitly treat the case of co-dimension 2, i.e., curves in 3D and 2D surfaces in 4D. We show that the resulting bi-quadratic problem can be relaxed to a set of quadratic problems in two variables and that the solution can be made effective and efficient by leveraging a hierarchical approach.