Ghosh, Sunayana

English

Dissertations of the University of Groningen

   University of GroningenGeometric approximation of curves and singularities of secant mapsGhosh, SunayanaIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2010Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Ghosh, S. (2010). Geometric approximation of curves and singularities of secant maps: a differentialgeometric approach. s.n.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-10-2022SummaryIn this thesis we study: (i) geometric approximation of curves in the planeand in space, and (ii) singularities of secant maps of immersed surfaces froma geometric perspective.Geometric Curve Approximation.Geometric modeling is a branch of applied mathematics and computationalgeometry that studies methods and algorithms for the mathematical descrip-tion of shapes. The shapes studied are mostly two or three dimensional,although many of the tools and principles can be applied to sets in any fi-nite dimension. Today geometric modeling is done with computers and forcomputer-based applications.Curve modeling is a part of geometric modeling and in the current thesisour focus is on problems related to approximation of parametric curves in theplane with conic arcs and biarcs, and in space with bihelical arcs. Biarcs arecurves formed by joining two circular arcs in a tangent continuous fashion.Similarly, a bihelical arc is formed by joining two circular helices in a tangentcontinuous manner. Two curves are said to join in a tangent continuousfashion at a point if they meet at a point and the derivative of both curves atthat point are parallel to each other.Approximating a parametric curve with some spline curve is always basedon some metric with which we measure how close the original curve andits approximation are to each other. A spline curve in general is definedpiecewise by some special class of functions. In this thesis we consider tangentcontinuous splines, e.g., a conic spline is made up of piecewise conic arcs,where two consecutive conic arcs join in a tangent continuous manner. In ourstudy we primarily consider the Hausdorff distance and, in case of parabolicand conic arcs, we also consider approximation with respect to the symmetric176 Summarydifference distance. The Hausdorff distance in case of plane curves is defined asthe maximum of the distance function between two curves: For every point ona given parametric curve we consider the Frenet-Serret frame and consider thedistance between the given point and the point of intersection of the normalline at that point with the approximating curve. The symmetric differencedistance between two curves sharing the endpoints is the area enclosed by thecurves.Example of a biarc One-parameter family of biarcsFigure 6.1: Example of a biarc formed by two circular arcs and a one-parameterfamily of bitangent biarcs.Complexity. Our goal is to compute the complexity (minimum number ofelements) of approximation of a sufficiently smooth curve, with non-vanishingcurvature, in the plane with biarc, parabolic and conic splines. We also de-termine the complexity of approximation of space curves with bihelix splines.Circles are the only curves in plane with non-zero constant curvature, con-ics are the only curves in plane with constant affine curvature, where affinecurvature is a differential invariant arising in affine differential geometry andis a notion similar to curvature. Furthermore, circular helices are curves inspace with constant curvature and constant torsion. We exploit these proper-ties of the curves in our computation of the Hausdorff distance and show thatthe approximation error improves by one order of magnitude in going fromSummary 177biarc splines (third order) to parabolic splines (fourth order) to conic splines(fifth order). We also compute the leading term in the approximation error,and, hence, in the complexity of the approximating spline. Consequently thecomplexity can be expressed in terms of differential geometric quantities likecurvature and affine curvature and for:• biarc splines depends on the derivative of the curvature,• parabolic splines depends on the affine curvature, and for conic splinesdepends on the derivative of the affine curvature, and for• bihelical splines depends on the derivative of the curvature.One-parameter family of conics Family of displacement functionsFigure 6.2: One-parameter family of bitangent conics and their corresponding dis-placement functions.Geometry of Approximation. We consider the approximation of spiralarcs with bitangent biarcs. Spirals are curves in the plane with monotonicallyincreasing/decreasing curvature. A bitangent biarc to a spiral arc is tangentto a spiral arc at its endpoints. We prove that there exists a one-parameterfamily of biarcs bitangent to a spiral arc. Furthermore, we show that in thisone-parameter family of biarcs there exists a unique biarc which is closest tothe curve with respect to the Hausdorff distance. Similar to the notion ofspirals, there is the notion of affine spirals in the plane. Affine spirals are178 Summarycurves with monotonically increasing/decreasing affine curvature. We showthat there exists a one-parameter family of bitangent conic arcs to a givenaffine spiral arc. Moreover, the distance function of this one-parameter familyis bimodal as shown in Figure 6.2.We prove that there exists a unique bitangent conic which is closest tothe given affine spiral with respect to the Hausdorff distance. In this casethe bimodal displacement function is equioscillatory in nature as shown bythe red curve in Figure 6.2. In case of biarcs as well as conics we also provethat the displacement function and hence the Hausdorff distance function ismonotonic, i.e., if we reduce the length of the given spiral or affine spiralarc the Hausdorff distance of its best approximating biarc or conic arc keepsdecreasing respectively. Due to these properties of the approximating biarcand conic spline we are able to design an algorithm for computing a bestapproximating biarc or conic spline of a given spiral or affine spiral curve. Incase of approximation of regular space curves, we prove a general result inChapter 4 regarding the Hausdorff distance between a regular space curve andan approximating G1-spline consisting of two smooth arcs which are tangentto each other at one of their endpoints (junction point) and the spline formedby these two smooth arcs is bitangent to the given space curve. Imposingthe condition that the derivative of curvature of this kind of spline curve iszero at the endpoints shared with the curve, we obtain a lower bound on theHausdorff distance between a curve and such a bitangent spline. Furthermore,we obtain an asymptotic characterization of the junction point and tangentat the junction point (junction tangent) for a bihelix spline where the lowerbound of the Hausdorff distance is obtained. We present an algorithm usingthese characterizations of the junction point and the junction tangent andconclude that the experimentally measured complexity of the bihelix splinematches its theoretical complexity almost exactly. Nonetheless, in this casethe problem of finding a bitangent bihelix which is closest to the curve iscomplex and still remains open.Summary 179Singularities of Secant MapsSingularity theory is a far-reaching generalization of the study of functions atits critical points. It provide some useful tools for the study of the geometryof curves and surfaces locally i.e., properties around a small neighborhoodof a point lying on a curve or surface. Given a function f : I → R, x0 iscalled its critical point if f ′(x0) = 0. A critical point or a singular pointgives crucial information about the shape of the graph f(x). The maximal orminimal points of a function are its nondegenerate critical points. Moreoverx0 is a degenerate critical point if f′(x0) = f′′(x0) = 0. In Whitney’s the-ory functions are replaced by mappings, i.e., collections of several functionsof several variables. Generically all critical points are non-degenerate, butdegenerate critical points may occur in generic families of functions. Theproblem considered in this thesis is an extension of Bruce’s work where hestudies the singularities of secant maps of curves in space. Given a spacecurve γ, which is a regular, smooth embedding from R to R3, the project-ivized secant map Ŝ : R × R → P2 maps a pair of distinct points t1 and t2in R to an unoriented line onto a real projective space P2. Furthermore, itmaps two non-distinct points onto the corresponding unoriented tangent line.The goal is then to study the local behavior of the secant map for points onthe diagonal ∆ = {(t, t) | t ∈ R}. In this thesis we study the secant mapsand projectivized secant maps of a surface immersed in Rn, where n ≥ 3.A secant map acting on two points p, q on a surface, maps them to the linep− q. We discuss in detail the singularities of such maps and its projectivizedcounterpart in this thesis along the points on the diagonal.

Geometric approximation of curves and singularities of secant maps:a differential geometric approach

https://pure.rug.nl/ws/files/14477961/09summ.pdf

Geometric approximation of curves and singularities of secant maps:a differential geometric approach

Abstract

Similar works

Full text

Available Versions

Dissertations of the University of Groningen