An Improved Curvature Circle Algorithm for Orthogonal Projection onto a Planar Algebraic Curve

Point orthogonal projection onto planar algebraic curve plays an important role in computer graphics, computer aided design, computer aided geometric design and other fields. For the case where the test point p is very far from the planar algebraic curve, we propose an improved curvature circle algorithm to find the footpoint. Concretely, the first step is to repeatedly iterate algorithm (the Newton’s steepest gradient descent method) until the iterated point could fall on the planar algebraic curve. Then seek footpoint by using the algorithm (computing footpoint q ) where the core technology is the curvature circle method. And the next step is to orthogonally project the footpoint q onto the planar algebraic curve by using the algorithm (the hybrid tangent vertical foot algorithm). Repeatedly run the algorithm (computing footpoint q ) and the algorithm (the hybrid tangent vertical foot algorithm) until the distance between the current footpoint and the previous footpoint is near 0. Furthermore, we propose Second Remedial Algorithm based on Comprehensive Algorithm B. In particular, its robustness is greatly improved than that of Comprehensive Algorithm B and it achieves our expected result. Numerical examples demonstrate that Second Remedial Algorithm could converge accurately and efficiently no matter how far the test point is from the plane algebraic curve and where the initial iteration point is.

The $2$-Hessian and sextactic points on plane algebraic curves

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated $2$-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the $2$-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the $2$-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.

The complex moment problem: determinacy and extendibility

Complex moment sequences are exactly those which admit positive definite extensions on the integer lattice points of the upper diagonal half-plane. Here we prove that the aforesaid extension is unique provided the complex moment sequence is determinate and its only representing measure has no atom at $0$. The question of converting the relation is posed as an open problem. A partial solution to this problem is established when at least one of representing measures is supported in a plane algebraic curve whose intersection with every straight line passing through $0$ is at most one point set. Further study concerns representing measures whose supports are Zariski dense in $\mathbb{C} $ as well as complex moment sequences which are constant on a family of parallel “Diophantine lines”. All this is supported by a bunch of illustrative examples.

Design of Mechanisms to Trace Plane Curves

This paper describes a mechanism design methodology that assembles standard components to trace plane curves that have a Fourier series parameterization. This approach can be used to approximate complex plane curves to interpolate image boundaries constructed from points. We describe three ways to construct a mechanism that generates a curve from a Fourier series parameterization. One uses Scotch yoke linkages for each term of Fourier series which are added using a belt drive. The second approach uses a coupled serial chain for each coordinate Fourier parameterization. The third method uses one constrained coupled serial chain to trace a specified plane curve. This work can be viewed as a version of the Kempe Universality Theorem that states that a linkage exists that can trace any plane algebraic curve. In our case, we include belts and pulleys, and obtain linkages that trace curves that have Fourier parameterizations.

Distinct Distances on Algebraic Curves in the Plane

LetSbe a set ofnpoints in${\mathbb R}^{2}$contained in an algebraic curveCof degreed. We prove that the number of distinct distances determined bySis at leastcdn4/3, unlessCcontains a line or a circle.We also prove the lower boundcd′ min{m2/3n2/3,m2,n2} for the number of distinct distances betweenmpoints on one irreducible plane algebraic curve andnpoints on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer and Solymosi in [19].

About the Tasks of Descriptive Geometry With Imaginary Solutions

The article is devoted to the discussion of the scientific methodological problems of presentation tasks of descriptive geometry along with having real and imaginary solutions. Examples of such problems are given, graphics solutions who give the wrong answers. As a consequence they resulted in some the textbooks on descriptive geometry to the emergence false claims type “ the curve degenerates to a point”, “a torus is a surface of the second order”, “conical and cylindrical surfaces are a special cases of the torsoboy surface in the case of degeneration of the ribs return torsoboy the surface at the point, etc.” In the article gives a correct mathematical interpretation of imaginary solutions the tasks by considering of examples an the determine the order and class of plane algebraic curve, the isolated point touch, of the line of intersection of surfaces of the second order with a common plane of symmetry. To obtain a mathematically valid answers the conclusion about the need for a combination of graphical and analytical solutions. This approach meets the requirements of the GEF on ensure as intrasubject discussed in this publication, and so interdisciplinary competencies. The latter have a broad outlet of descriptive geometry in complex space in the theory of algebraic curves and surfaces, kremenovic transformations, field theory, etc.

A bound for the Milnor number of plane curve singularities

Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].