Affine-Invariant Distances, Envelopes and Symmetry Sets

Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams.     

Approximation by Conic Splines

We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature to within Hausdorff distance ɛ is c ₁ɛ^−1/4 + O(1), if the spline consists of parabolic arcs, and c ₂ɛ^−1/5 + O(1), if it is composed of general conic arcs of varying type. The constants c ₁ and c ₂ are expressed in the Euclidean and affine curvature of the curve. We also show that the Hausdorff distance between a curve and an optimal conic arc tangent at its endpoints is increasing with its arc length, provided the affine curvature along the arc is monotone. This property yields a simple bisection algorithm for the computation of an optimal parabolic or conic spline. The research of SG and GV was partially supported by grant 6413 of the European Commission to the IST-2002 FET-Open project Algorithms for Complex Shapes in the Sixth Framework Program.     

Unitals of PG(2,q2) Containing Conics

A unital in PG(2, q²) is a set of points such that each line meets in 1 or points. The well‐known example is the classical unital consisting of all absolute points of a unitary polarity of PG(2, q²). Unitals other than the classical one also exist in PG(2, q²) for every . Actually, all known unitals are of Buekenhout–Metz type [see F. Buekenhout, Geom Dedicata 5 (1976), 189–194, R. Metz, Geom Dedicata 8 (1979), 125–126.], and they can be obtained by a construction due to F. Buekenhout, (Geom Dedicata 5 (1976), 189–194).. The unitals constructed by R. D. Baker and G. L. Ebert (J Combin Theory Ser A 60 (1992), 67–84), and independently by J. W. P. Hirschfeld and T. Sz?nyi (Discrete Math 97 (1991), 229–242), are the union of q conics. Our Theorem  1.1 shows that this geometric property characterizes the Baker–Ebert–Hirschfeld–Sz?nyi unitals. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 101–111, 2013     

Generalization of a Circular Property

We generalize a property of a circle concerning the power of a point.     

Pencils of Hyperconics in Projective Planes of Characteristic Two

Let F be any field, finite or infinite, of characteristic 2. Put =PG(2,F). Let H ₁,H ₂ be hyperconics in . In this note we study the intersectionH ₁ H ₂. In particular we obtain canonical forms for H ₁,H ₂ in the cases where |H ₁ H ₂|=4,5,6. One interesting consequence is that the case |H ₁ H ₂|=6 can only occur if F contains a subfield of order 4. Related results concerning pencils of hyperconics are presented in Theorems 6 through 9. This work also leads to an extension to general fields of characteristic 2 of the well-known even intersection property for hyperovals in PG(2,4) which is pursued elsewhere ([2]).     

带脉冲的三维引力辅助变轨研究

在引力辅助过程中施加脉冲可以有效地改善变轨效果.目前只能对施加小脉冲的情况进行近似计算,当脉冲大于近拱点速度的1%时无法进行分析.针对这一问题,提出了一种解析分析方法,可以计算施加任意大小和方向脉冲的三维引力辅助变轨.基于二体问题,建立了带任意脉冲的三维引力辅助模型,采用8个相互独立的参数对模型进行描述,其中5个参数表征三维引力辅助、一个参数表征脉冲的大小、两个参数表征脉冲的方向;建立了一组坐标系,可以方便地对轨道进行描述;以施加脉冲为界,将轨道划分为前后两段,分别进行公式推导;应用双曲线轨道动力学与坐标变换等技术方法,可以将飞行器的位置矢量和速度矢量表示为上述8个参数的解析公式,进而可以求出变轨导致的速度、能量和轨道倾角的变化量.通过与基于圆型限制性三体问题的数值仿真结果进行对比,验证公式的有效性.应用导出的解析公式分析了施加脉冲的大小和方向对飞行器能量和轨道倾角的影响,并给出了相应规律.结果表明:以最大能量改变为优化目标,施加脉冲的最优方向往往并不是该点速度方向;轨道倾角受到脉冲方向的影响显著.      

Incorporating Angular Information into Parametric Models

The problem is considered of fitting a parametrically defined model in two or three dimensions to observed data, when angular information about the measured data points is available. Gauss-Newton methods based on correct separation of variables are developed. Some numerical results are included.     

Rational cubic/quartic Said-Ball conics

In CAGD, the Said-Ball representation for a polynomial curve has two advantages over the Bézier representation, since the degrees of Said-Ball basis are distributed in a step type. One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomial curve runs twice as fast as the de Casteljau algorithm of Bézier curve. Another is that the operations of degree elevation and reduction for a polynomial curve in Said-Ball form are simpler and faster than in Bézier form. However, Said-Ball curve can not exactly represent conics which are usually used in aircraft and machine element design. To further extend the utilization of Said-Ball curve, this paper deduces the representation theory of rational cubic and quartic Said-Ball conics, according to the necessary and sufficient conditions for conic representation in rational low degree Bézier form and the transformation formula from Bernstein basis to Said-Ball basis. The results include the judging method for whether a rational quartic Said-Ball curve is a conic section and design method for presenting a given conic section in rational quartic Said-Ball form. Many experimental curves are given for confirming that our approaches are correct and effective.