Smooth Hypocycloidal Paths with Collision-Free and Decoupled Multi-Robot Path Planning
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On Ramified Covers of the Projective Plane I: Segre's Theory And
ON RAMIFIED COVERS OF THE PROJECTIVE PLANE I: SEGRE’S THEORY AND CLASSIFICATION IN SMALL DEGREES (WITH AN APPENDIX BY EUGENII SHUSTIN) MICHAEL FRIEDMAN AND MAXIM LEYENSON1 Abstract. We study ramified covers of the projective plane P2. Given a smooth surface S in Pn and a generic enough projection Pn → P2, we get a cover π : S → P2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. Several questions arise: First, What is the geography of branch curves among all cuspidal-nodal curves? And second, what is the geometry of branch curves; i.e., how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e., form a special 0-cycle on the plane. We start with reviewing what is known about the answers to these questions, both simple and some non-trivial results. Secondly, the classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in P3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. We also review examples of small degree. In addition, the Appendix written by E. Shustin shows the existence of new Zariski pairs. -
Interactive $ G^ 1$ and $ G^ 2$ Hermite Interpolation Using
Interactive G1 and G2 Hermite Interpolation Using Extended Log-aesthetic Curves Ferenc Nagy1,2, Norimasa Yoshida3, and Mikl´osHoffmann1,4 1University of Debrecen, Faculty of Informatics 2University of Debrecen, Doctoral School of Informatics 3Nihon University, College of Industrial Technology 4Eszterh´azyK´arolyUniversity, Institute of Mathematics and Computer Science March 2021 Abstract In the field of aesthetic design, log-aesthetic curves have a significant role to meet the high industrial requirements. In this paper, we propose a new interactive G1 Hermite interpolation method based on the algorithm of Yoshida et al. [13] with a minor boundary condition. In this novel approach, we compute an extended log-aesthetic curve segment that may include inflection point (S-shaped curve) or cusp. The curve segment is defined by its endpoints, a tangent vector at the first point, and a tangent direction at the second point. The algorithm also determines the shape parameter of the log-aesthetic curve based on the length of the first tangent that provides control over the curvature of the first point and makes the method capable of joining log-aesthetic curve segments with G2 continuity. 1 Introduction Aesthetic curves are primarily used in computer-aided design to meet arXiv:2105.09762v1 [math.NA] 20 Apr 2021 the high aesthetic requirements of the industry. Levien et al. stated [5] that the log-aesthetic curve is the most promising curve for aesthetic design and a large number of research papers are published since their introduction. The log-aesthetic curve is originated from Harada et al. [3, 4]. They analyzed the characteristics of aesthetic curves, and insisted that natu- ral aesthetic curves have such a property that their logarithmic distribu- tion diagram of curvature (LDDC) can be approximated by straight lines meanwhile there is a strong correlation between the slopes of the lines and the impressions of the curves. -
Combination of Cubic and Quartic Plane Curve
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis. -
Horizontal Curves.Pdf
Geospatial Science RMIT ENGINEERING SURVEYING 1 HORIZONTAL CURVES CIRCULAR CURVES, COMPOUND CIRCULAR CURVES, REVERSE CIRCULAR CURVES TRANSITION CURVES AND COMPOUND CURVES R.E.Deakin, August 2005 1. TYPES OF HORIZONTAL CURVES The types of horizontal curves usually encountered in engineering surveying application may be broadly categorised as (i) Circular curves: curves of constant radius joining two intersecting straights. C θ nt ge rc tan a A B chord A' B' s iu d a R r θ O Figure 1.1 In Figure 1.1, a circular curve of constant radius R, centred at O, joins two straights A'A and BB' which intersect at C. A and B are tangent points to the circular arc. OA and OB are radials, which meet the straights at right angles, and the angle at O is equal to the intersection angle at C. Horizontal Curves.doc 1 Geospatial Science RMIT (ii) Compound circular curves: two or more consecutive circular curves of different radii. C θ = θ 12 + θ D A B R A' 1 2 R B' θ1 O 1 R 2 11 RR − 2 R θ2 O2 Figure 1.2 In Figure 1.2, a compound circular curve ADB joins two straights A'A and BB' which intersect at C. A and B are tangent points to circular arcs of radii R1 and R2 respectively. D is a common tangent point. Horizontal Curves.doc 2 Geospatial Science RMIT (iii) Reverse circular curves: two or more consecutive circular curves, of the same or different radii whose centres lie on different sides of a common tangent point. -
What Is Form?
What is form? Draft December 19, 2013 Abstract What is the form of a set? Though there are many vague descrip- tions of the form of a set, there remains one inherent property: the form does not change by rotations and translations. So there are on one side the objects and on the other side the transactions which can be carried out on an object without changing its form. It is of significance, that the transactions themselves may be con- sidered as objects. The leading thought for the following is to create a kind of basis of the transaction objects in order to describe all other objects. We will focus on plane, closed, rectifiable curves building the border of a simply connected domain. It turns out, that there exists a simple uniform relationship of four fundamental entities up to normalization for orthogonal trajectories: 1. The sum of the changes of geodesic curvature is zero. 2. The difference of the changes of geodesic curvature is the real part of the Schwarzian derivative. 3. The difference of the changes of geodesic acceleration is the imaginary part of the Schwarzian derivative. 4. The sum of the changes of geodesic acceleration is the Gaussian curvature. 1 Plane rotations and translations The plane rotations are a commutative group, any representation is equiva- lent to a representation consisting of representations of first order, Smirnov [8]. The complex circular functions are the striking examples if the respec- tive angle is an integer multiple of the circumference. By that way we get a representation for any integer and thus an infinite number of representa- tions. -
Catenaries in Viscous Fluid Arxiv:1509.01282V2 [Physics.Flu
Catenaries in viscous fluid Brato Chakrabarti∗ Department of Biomedical Engineering and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. J A Hannay Department of Biomedical Engineering and Mechanics, Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. January 5, 2016 Abstract This work explores a simple model of a slender, flexible structure in a uniform flow, providing analytical solutions for the translating, axially flowing equilibria of strings sub- jected to a uniform body force and drag forces linear in the velocities. The classical catenaries are extended to a five-parameter family of curves. A sixth parameter affects the tension in the curves. Generic configurations are planar, represented by a single first order equation for the tangential angle. The effects of varying parameters on represen- tative shapes, orbits in angle-curvature space, and stress distributions are shown. As limiting cases, the solutions include configurations corresponding to \lariat chains" and the towing, reeling, and sedimentation of flexible cables in a highly viscous fluid. Regions of parameter space corresponding to infinitely long, semi-infinite, and finite length curves are delineated. Almost all curves subtend an angle less than π radians, but curious spe- cial cases with doubled or infinite range occur on the borders between regions. Separate transitions in the tension behavior, and counterintuitive results regarding finite towing tensions for infinitely long cables, are presented. Several physically inspired boundary value problems are solved and discussed. arXiv:1509.01282v2 [physics.flu-dyn] 4 Jan 2016 1 Introduction The catenary, or hanging chain, is one of the oldest problems in analytical mechanics. -
Geometric Modelling for Computer Integrated Road Construction
Geometric Modelling for Computer Integrated Road Construction Zur Erlangung des akademischen Grades eines DOKTOR-INGENIEURS von der Fakultät für Bauingenieur-, Geo- und Umweltwissenschaften der Universität Fridericiana zu Karlsruhe (TH) genehmigte DISSERTATION von Mgr Inż. Jarosław Jurasz aus Łódź (Lodz), Polen Tag der mündlichen Prüfung: 10. Februar 2003 Hauptreferent: o. Prof. Dr.-Ing. Fritz Gehbauer, M.S. Korreferentin: o. Prof. Dr.-Ing. Maria Hennes Karlsruhe 2003 VORWORT DES HERAUSGEBERS Das Institut für Technologie und Management im Baubetrieb (früher: Maschinenwesen im Baubetrieb) beschäftigt sich seit 5 Jahren mit der Computer Integrated Road Construction (CIRC). Die Forschungs- und Entwicklungsarbeiten wurden in zwei europäischen Verbundprojekten durchgeführt. Das erste unter dem Titel "Computer Integrated Road Construction" (CIRC), das zweite unter dem Titel "OSYRIS ("Open System for Road Information Support"). Im ersten Projekt stand die GPS-gestützte Überwachung der Einbau- und Verdichtungsgeräte zur Qualitätssicherung im Mittelpunkt. Das zweite Projekt hat die Aufgabenstellung erweitert mit dem Ziel, die Planungsdaten direkt auf den Maschinen zu deren Steuerung - entweder automatisch oder über Informationen für den Bediener - zur Verfügung zu stellen und zu nutzen. Dafür wurde ein Straßenplanungs-Programm dergestalt modifiziert, daß daraus Vorgabedaten für die Einstellung der Maschinenparameter abgeleitet werden können. An Bord der Maschinen installierte Computer wurden entwickelt, die in der Lage sind, diese Informationen in Steuerungselemente umzuwandeln. Zur Datenübertragung wird sowohl GPS als auch eine bodengestützte "Robotic Total Station" mit der jeweiligen drahtlosen Datenfernübertragung verwendet. Das gleiche System wird auch verwendet, um die "As Built" Daten zu erfassen und im zentralen Computer niederzulegen. Diese Daten können sowohl kurzfristig zur Einbausteuerung als auch langfristig zur Planung der Unterhaltung verwendet werden. -
Deltoid* the Deltoid Curve Was Conceived by Euler in 1745 in Con
Deltoid* The Deltoid curve was conceived by Euler in 1745 in con- nection with his study of caustics. Formulas in 3D-XplorMath: x = 2 cos(t) + cos(2t); y = 2 sin(t) − sin(2t); 0 < t ≤ 2π; and its implicit equation is: (x2 + y2)2 − 8x(x2 − 3y2) + 18(x2 + y2) − 27 = 0: The Deltoid or Tricuspid The Deltoid is also known as the Tricuspid, and can be defined as the trace of a point on one circle that rolls inside * This file is from the 3D-XplorMath project. Please see: http://3D-XplorMath.org/ 1 2 another circle of 3 or 3=2 times as large a radius. The latter is called double generation. The figure below shows both of these methods. O is the center of the fixed circle of radius a, C the center of the rolling circle of radius a=3, and P the tracing point. OHCJ, JPT and TAOGE are colinear, where G and A are distant a=3 from O, and A is the center of the rolling circle with radius 2a=3. PHG is colinear and gives the tangent at P. Triangles TEJ, TGP, and JHP are all similar and T P=JP = 2 . Angle JCP = 3∗Angle BOJ. Let the point Q (not shown) be the intersection of JE and the circle centered on C. Points Q, P are symmetric with respect to point C. The intersection of OQ, PJ forms the center of osculating circle at P. 3 The Deltoid has numerous interesting properties. Properties Tangent Let A be the center of the curve, B be one of the cusp points,and P be any point on the curve. -
Slipping on an Arbitrary Surface with Friction
Slipping on an Arbitrary Surface with Friction Felipe Gonz´alez-Cataldo∗ and Gonzalo Guti´errezy Departamento de F´ısica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile. Julio M. Y´a~nezz Departamento de F´ısica, Facultad de Ciencias, Universidad Cat´olica del Norte (Dated: August 22, 2018) The motion of a block slipping on a surface is a well studied problem for flat and circular surfaces, but the necessary conditions for the block to leave (or not) the surface deserve a detailed treatment. In this article, using basic differential geometry, we generalize this problem to an arbitrary surface, including the effects of friction, providing a general expression to determine under which conditions the particle leaves the surface. An explicit integral form for the velocity is given, which is analytically integrable for some cases, and we find general criteria to determine the critical velocity at which the particle immediately leaves the surface. Five curves, a circle, ellipse, parabola, catenary and cycloid, are analyzed in detail. Several interesting features appear, for instance, in the absense of friction, a particle moving on a catenary leaves the surface just before touching the floor, and in the case of the parabola, the particle never leaves the surface, regardless of the friction. A phase diagram that separates the conditions that lead to a particle stopping in the surface to those that lead to a particle departuring from the surface is also discussed. I. INTRODUCTION A natural question that arises from studying this situa- tion is how to incorporate friction on the circular surface. -
Steiner's Hat: a Constant-Area Deltoid Associated with the Ellipse
KoG•24–2020 R. Garcia, D. Reznik, H. Stachel, M. Helman: Steiner’s Hat: a Constant-Area Deltoid ... https://doi.org/10.31896/k.24.2 RONALDO GARCIA, DAN REZNIK Original scientific paper HELLMUTH STACHEL, MARK HELMAN Accepted 5. 10. 2020. Steiner's Hat: a Constant-Area Deltoid Associated with the Ellipse Steiner's Hat: a Constant-Area Deltoid Associ- Steinerova krivulja: deltoide konstantne povrˇsine ated with the Ellipse pridruˇzene elipsi ABSTRACT SAZETAKˇ The Negative Pedal Curve (NPC) of the Ellipse with re- Negativno noˇziˇsnakrivulja elipse s obzirom na neku nje- spect to a boundary point M is a 3-cusp closed-curve which zinu toˇcku M je zatvorena krivulja s tri ˇsiljka koja je afina is the affine image of the Steiner Deltoid. Over all M the slika Steinerove deltoide. Za sve toˇcke M na elipsi krivulje family has invariant area and displays an array of interesting dobivene familije imaju istu povrˇsinui niz zanimljivih svoj- properties. stava. Key words: curve, envelope, ellipse, pedal, evolute, deltoid, Poncelet, osculating, orthologic Kljuˇcnerijeˇci: krivulja, envelopa, elipsa, noˇziˇsnakrivulja, evoluta, deltoida, Poncelet, oskulacija, ortologija MSC2010: 51M04 51N20 65D18 1 Introduction Main Results: 0 0 Given an ellipse E with non-zero semi-axes a;b centered • The triangle T defined by the 3 cusps Pi has invariant at O, let M be a point in the plane. The Negative Pedal area over M, Figure 7. Curve (NPC) of E with respect to M is the envelope of • The triangle T defined by the pre-images Pi of the 3 lines passing through points P(t) on the boundary of E cusps has invariant area over M, Figure 7. -
Steiner's Hat: a Constant-Area Deltoid Associated with the Ellipse
STEINER’S HAT: A CONSTANT-AREA DELTOID ASSOCIATED WITH THE ELLIPSE RONALDO GARCIA, DAN REZNIK, HELLMUTH STACHEL, AND MARK HELMAN Abstract. The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties. Keywords curve, envelope, ellipse, pedal, evolute, deltoid, Poncelet, osculat- ing, orthologic. MSC 51M04 and 51N20 and 65D18 1. Introduction Given an ellipse with non-zero semi-axes a,b centered at O, let M be a point in the plane. The NegativeE Pedal Curve (NPC) of with respect to M is the envelope of lines passing through points P (t) on the boundaryE of and perpendicular to [P (t) M] [4, pp. 349]. Well-studied cases [7, 14] includeE placing M on (i) the major− axis: the NPC is a two-cusp “fish curve” (or an asymmetric ovoid for low eccentricity of ); (ii) at O: this yielding a four-cusp NPC known as Talbot’s Curve (or a squashedE ellipse for low eccentricity), Figure 1. arXiv:2006.13166v5 [math.DS] 5 Oct 2020 Figure 1. The Negative Pedal Curve (NPC) of an ellipse with respect to a point M on the plane is the envelope of lines passing through P (t) on the boundary,E and perpendicular to P (t) M. Left: When M lies on the major axis of , the NPC is a two-cusp “fish” curve. Right: When− M is at the center of , the NPC is 4-cuspE curve with 2-self intersections known as Talbot’s Curve [12]. -
The Unexpected Hanging and Other Mathematical Diversions
AND OTHER MATHEMATICAL nIVERSIONS OLLECTION OF GI ES FRON A ERIC" The Unexpected Hanging A Partial List of Books by Martin Gardner: The Annotated "Casey at the Bat" (ed.) The Annotated Innocence of Father Brown (ed.) Hexaflexagons and Other Mathematical Diversions Logic Machines and Diagrams The Magic Numbers of Dr. Matrix Martin Gardner's Whys and Wherefores New Ambidextrous Universe The No-sided Professor Order and Surprise Puzzles from Other Worlds The Sacred Beetle and Other Essays in Science (ed.) The Second Scientific American Book of Mathematical Puzzles and Diversions Wheels, Life, and Other Mathematical Entertainments The Whys of a Philosopher Scrivener MARTIN GARDNER The Unexpected Hanging And Other Mathematical Diversions With a new Afterword and expanded Bibliography The University of Chicago Press Chicago and London Material previously published in Scientific American is copyright 0 1961, 1962, 1963 by Scientific American, Inc. All rights reserved. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London O 1969, 1991 by Martin Gardner All rights reserved. Originally published 1969. University of Chicago Press edition 1991 Printed in the United States of America Library of Congress Cataloging-in-Publication Data Gardner, Martin, 1914- The unexpected hanging and other mathematical diversions : with a new afterword and expanded bibliography 1 Martin Gardner. - University of Chicago Press ed. p. cm. Includes bibliographical references. ISBN 0-226-28256-2 (pbk.) 1. Mathematical recreations. I. Title. QA95.G33 1991 793.7'4-dc20 91-17723 CIP @ The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences-Permanence of Paper for Printed Library Materials, ANSI 239.48-1984.