Geometric Modelling for Computer Integrated Road Construction

Total Page:16

File Type:pdf, Size:1020Kb

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. In beiden Projekten war es notwendig, das Objekt Straße mit seinen Einbauschichten (hier im wesentlichen die Asphaltschichten) so zu modellieren, daß die gesetzten Ziele der Datenkommunikation und der Einbaukontrolle erreicht werden können. Es gibt durchaus praktikable und moderne Instrumente, das Objekt Straße in Form von Längs- und Querprofilen und auch räumlichen CAD-Methoden zu beschreiben und die entsprechenden Daten für die Ausführung zur Verfügung zu stellen. Für CIRC und OSYRIS sind diese jedoch nicht ausreichend. Geht es hier doch darum, die Vorgabedaten direkt auf die Maschine zu übertragen, sie dort in Kleincomputern als Steuerbefehle abzubilden und entweder automatisch oder über MMI (Man-Machine Interfaces) zu nutzen. Hierfür ist eine Modellierung erforderlich, die über die Möglichkeiten gängiger Modellierungen hinausgeht. Das geforderte Modell muß Prozeßdaten verarbeiten können und die Kommunikation verschiedener Maschinen untereinander ermöglichen. Dieser Aufgabe stellte sich Herr Jaroslaw Jurasz und präsentiert die Ergebnisse in seiner Dissertation. Im Hauptkapitel 4. wird ein digitales Umgebungsmodell präsentiert, das die Erfordernisse einer Computer Integrated Road Construction erfüllt. Rein geometrisch vorgegebene Ausgangsdaten werden dabei mit Maschinenparametern und der Prozeßkontrolle verknüpft. Die Vor- und Nachteile akzeptierter und verwendeter Geländemodelle werden dabei untersucht. Die Notwendigkeit einer neuen Modellierung wird herausgearbeitet. Dabei wird immer im Auge behalten, daß die Praxis des Straßenbaues genauso zu beachten ist, wie die Wirtschaftlichkeit der Datenverarbeitung, der Speicherplätze und der Rechenzeit der verwendeten Computer an Bord und im steuernden Büro. Als Lösung wird die modifizierte Ribbon-Technologie vorgeschlagen, implementiert und in Praxistests auf der Baustelle verifiziert. Die besondere Leistung der vorgelegten Dissertation besteht darin, diese Technologie angepaßt zu haben und dabei die Vorgaben "so komplex wie nötig" (ohne Prozeßdaten zu verlieren) und "so einfach wie möglich" (um Rechnerkapazität und Zeit zu sparen) konsequent beachtet zu haben. Dabei werden immer die praktischen Erfordernisse beachtet, die darin bestehen, zu hinterfragen, was ist die Straße, was kann dort vorkommen, wie können diese Vorgaben in Programme umgesetzt werden, wie geht das Modell damit um, aus dem GPS resultierende Abweichungen von der Soll-Position zu verarbeiten? Eine besondere Leistung in der vorgelegten Dissertation besteht auch darin, daß in dem vorgelegten Modell Vereinfachungen, die sich aus der Straßengeometrie ergeben, konsequent genutzt werden, um Rechenkapazität und Rechenzeit zu verringern. Das präsentierte Modell wird auch daraufhin ausgerichtet, daß es eine Kommunikation zwischen den einzelnen Einbaustellen (Maschinen) ermöglicht (Site Networking). Die Arbeit beschränkt sich nicht nur darauf, die neuen Modellierungskonzepte zu präsentieren, sondern weist in ausgeführten Praxistests nach, daß die Modellierung und zugehörigen Datenübertragungstechniken funktionieren. Die MMI werden dargestellt und in ihrer Funktionsweise beschrieben. Fritz Gehbauer VORWORT DES VERFASSERS Die vorliegende Arbeit ist ein Ergebnis meiner fünfeinhalbjährigen Forschungs- und Entwicklungstätigkeit am Institut für Technologie und Management im Baubetrieb (TMB, bis 2001 Institut für Maschinenwesen im Baubetrieb) der Universität Karlsruhe (TH). An diesem Institut arbeitet unter der Führung von Herrn Professor Fritz Gehbauer ein interdisziplinäres Team zusammen, welches in der Lage ist, Lösungen für die vielfältigen Probleme moderner Baustellen zu erarbeiten. Zu diesen Problemen gehört unter anderem der rechnergestützte Straßenbau, der in der vorliegenden Arbeit unter dem Schlagwort „Computer Integrated Road Construction“ (CIRC) ausführlich behandelt wird. Zwei Generationen von CIRC- Systemen wurden am TMB im Rahmen der europäischen Verbundforschungsprojekte CIRC und OSYRIS (Open System for Road Information Support) erfolgreich implementiert. Als Entwickler des CIRC-Bordsystems für die Straßenwalze und Leiter des TMB-OSYRIS-Entwicklungsteams habe ich hierbei Lösungen für den rechnergestützten Straßenbau erarbeitet und erfolgreich auf Baustellen getestet. Insbesondere haben mich dabei die mathematischen und informationstechnischen Grundlagen solcher Systeme interessiert, die im folgenden vertieft und untersucht werden. Darüber hinaus wurden neue Ansätze und Algorithmen entwickelt, die sich dank Flexibilität und Leistungsfähigkeit besonders für die Vernetzung des Maschinenverbandes eignen. Ohne die Unterstützung vieler Menschen wäre die Dissertation in dieser Form nie zustande gekommen. Mein besonderer Dank gilt an erster Stelle Herrn Professor Fritz Gehbauer, für die Möglichkeit, auf diesem faszinierendem Gebiet zu arbeiten und für die mir gewährte Freiheit bei der Ausführung dieser Arbeit. Sehr herzlich möchte ich mich bei meiner Korreferentin, Frau Professor Maria Hennes und Herrn Dr. Norbert Rösch bedanken, für die fruchtbaren Diskussionen, guten Fragen und wertvollen Anregungen, insbesondere zur 3D-Genauigkeitsanalyse sowie für die kritische Durchsicht des Manuskripts. Die CIRC-Implementation ist eine Teamleistung. Ich bin allen Teammitgliedern am TMB und Projektpartnern sehr dankbar für ihre Beteiligung und Feedback. Besondere Unterstützung erhielt ich von meiner Ehefrau und Arbeitspartnerin Dr. Agata Ligier und meinem Freund und Kollegen Johannes Fliedner, der mich in diese Thematik einführte. Ich schätzte sehr die anspruchsvollen Fragen von François Peyret vom französischem Laboratoire Central des Ponts et Chaussées, der das CIRC Projekt initiiert hatte. Herrn Alfons Horn von der Firma Moba bin ich für den inspirierenden Gedankenaustausch sehr dankbar. Da die kleine CIRC-Gesellschaft sehr international geprägt ist, habe ich mich entschieden, die Arbeit in englischer Sprache zu verfassen. Jarek Jurasz Karlsruhe, im Februar 2003 Ausführliche Zusammenfassung GEOMETRISCHE MODELLIERUNG FÜR DEN RECHNERINTEGRIERTEN STRAßENBAU - AUSFÜHRLICHE ZUSAMMENFASSUNG Schlagworte: Bänder, Baumaschinensteuerung, Bordrechner, Digitales Geländemodell, Digitales Umgebungsmodell, Annäherung des Straßenentwurfs, Entwurfsübertragung, Flächendeckende dynamische Verdichtungskontrolle, Hermitesches Spline, Nivellierung, Rechnergestützter Bau und Straßenbau, Rechnerintegrierter Bau und Straßenbau, Straßenmodell, Ribbons, Tensorprodukt-Fläche. KAPITEL 1. EINFÜHRUNG Heutzutage werden Baumaschinen immer häufiger mit EDV-Systemen ausgestattet, die das Bedienpersonal unterstützen und aufwendige Kontroll- und Dokumentationsaufgaben übernehmen. Ihre Anwendung ist dank der jüngsten Fortschritte der Positionierungstechnologie und mobiler EDV möglich. Vor allem der Straßen- und Erdbau bietet ein vielversprechendes Anwendungsfeld durch sich wiederholende und klar definierte Aufgaben. Gerade hier spielt die kontinuierliche Qualitätskontrolle eine große Rolle. Besonders in den neuen Vertragsmodellen, wie z. B. funktionelle Verträge oder Build-Own-Operate wünschen die Benutzer, sehr aktiv an der Qualitätskontrolle teilzunehmen. Hohe zusätzliche Kosten, sowie die beschränkte Funktionalität und niedrige Robustheit sind Faktoren, die den Fortschritt der Bord-EDV Technologie bremsen. Die erwähnte EDV-Technologie gehört zu dem allgemeinen Konzept des rechnerintegrierten Bauens (engl. Computer Integrated Construction, CIC), das als eine EDV-basierende Integration von Informationen aus Baumanagement, -Planung, -Entwurf und -Durchführung definiert werden kann. Im deutschsprachigem Raum
Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • On a Class of Polar Log-Aesthetic Curves
    Proceedings of the ASME 2021 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2021 August 17-20, 2021, Virtual, Online, 2021, Online, Virtual DETC2021/72184 ON A CLASS OF POLAR LOG-AESTHETIC CURVES Victor Parque System Design Laboratory Department of Modern Mechanical Engineering Waseda University Tokyo, 169-8555 Email: [email protected] ABSTRACT 1 INTRODUCTION Curves are essential concepts that enable compounded aes- Curves are ubiquitous entities in industrial design, enabling thetic curves, e.g., to assemble complex silhouettes, match a spe- the creation of functional and aesthetic machines and consumer cific curvature profile in industrial design, and construct smooth, electronics [1, 2], and the smooth and energy-efficient motion comfortable, and safe trajectories in vehicle-robot navigation planning in robots [3, 4]. The research on conceptual mecha- systems. New mechanisms able to encode, generate, evaluate, nisms allowing to construct aesthetic curves is relevant, not only and deform aesthetic curves are expected to improve the through- to render sketches to enable designers to extrapolate ideas to ac- put and the quality of industrial design. In recent years, the study curate symbolic expressions in computers but also to facilitate of (log) aesthetic curves have attracted the community’s atten- the characterization of the first principles of aesthetics in geo- tion due to its ubiquity in natural phenomena such as bird eggs, metric design [5, 6]. Thus, studying mechanisms to generate, butterfly wings, falcon flights, and manufactured products such evaluate, and deform aesthetic curves is expected to improve the as Japanese swords and automobiles. throughput and the quality of the industrial design process.
    [Show full text]
  • Discrete Differential Geometry MA5210 Reading Report 2 Ang Yan Sheng A0144836Y
    Discrete Differential Geometry MA5210 Reading Report 2 Ang Yan Sheng A0144836Y Thanks to the contributions of Gauss, Riemann, Grassmann, Poincare,´ Cartan, and many others, we now have a comprehensive classical theory of differential ge- ometry. The most famous use of this theory might be in Einstein’s theory of general relativity, but even today, differential geometry sees substantial applications in di- verse areas such as computational biology, computer graphics, industrial design, and architecture—in short, any field which requires some form of digital geometry processing. Of course, a computer does not directly work with smooth surfaces or differ- ential forms, but with certain finite, discrete approximations of such objects, such as meshes. However, discretising the smooth objects and concepts of classical dif- ferential geometry poses some nontrivial problems. Crane and Wardetzky (2017) describe a “game” which is often played in this context: 1. Start with several equivalent definitions in the smooth setting; 2. Apply each definition to an object in the discrete setting; 3. Analyse the trade-offs among these (usually inequivalent) discrete definitions. In this report, I will look at some instances of this phenomenon, also known as the no free lunch scenario, and highlight some of the results in the area of discrete differential geometry, which attempts to study the link between discrete geometric structures and their smooth counterparts. Curvature As a first example, we consider the curvature of a smooth plane curve γ : [0; L] ! R2. Assuming that γ is parametrised by arc length, its unit tangent vector 0 -1 is given by T(t) := γ0(s), and its unit normal by N(s) := T(s), ie.
    [Show full text]
  • Evolution of Plane Curves Driven by a Nonlinear Function of Curvature and Anisotropy∗
    SIAM J. APPL. MATH. c 2001 Society for Industrial and Applied Mathematics Vol. 61, No. 5, pp. 1473–1501 EVOLUTION OF PLANE CURVES DRIVEN BY A NONLINEAR FUNCTION OF CURVATURE AND ANISOTROPY∗ KAROL MIKULA† AND DANIEL SEVˇ COVIˇ Cˇ ‡ Abstract. In this paper we study evolution ofplane curves satisfyinga geometric equation v = β(k, ν), where v is the normal velocity and k and ν are the curvature and tangential angle ofa plane curve Γ. We follow the direct approach and we analyze the so-called intrinsic heat equation governing the motion ofplane curves obeying such a geometric equation. The intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional α. We show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities. From an analytical point ofview we present some new results on short time existence ofa regular familyofevolving curves in the degenerate case when β(k, ν)=γ(ν)km,0<m≤ 2, and the governing system ofequations includes a nontrivial tangential velocity functional. Key words. nonlinear curve evolution, intrinsic heat equation, degenerate parabolic equations, tangential velocity, numerical solution AMS subject classifications. 35K65, 65N40, 53C80 PII. S0036139999359288 1. Introduction. The goal of this paper is to study curvature-driven evolution of a family of closed smooth plane curves. We consider the case when the normal velocity v of an evolving family of plane curves Γt : S1 → R2,t∈ (0,T), is a function of the curvature k and the tangential angle ν: (1.1) v = β(k, ν). In past years, geometric equations of the form (1.1) have attracted a lot of attention from both the theoretical and the practical point of view.
    [Show full text]
  • Isoptics of Log-Aesthetic Curves
    Isoptics of log-aesthetic curves Ferenc Nagy1 1University of Debrecen, Faculty of Informatics, 26 Kassai St., 4028 Debrecen, Hungary 1University of Debrecen, Doctoral School of Informatics, 26 Kassai St., 4028 Debrecen, Hungary January 2021 Abstract The log-aesthetic curve has a significant factor in the field of aesthetic design to meet the high industrial requirements. It has much exceptional property and a large number of research papers are published, since its introduction. It can be parametrized by tangential angle, and it is au- toevolute, meaning that its evolute is also a log-aesthetic curve. In this paper, we will examine the isoptics of the log-aesthetic curve and deter- mine if it is autoisoptic as well. We will show that the log-aesthetic curve only coincides its isoptic in some specific cases. We will also provide a new formula to calculate the isoptic curve of any tangential angle parametrized curves. 1 Introduction To design highly aesthetic objects, the use of high-quality curves is very important. The main property of such curves is the curvature, which should be monotonically varying. Among them, the log-aesthetic stated as the most promising for aesthetic design [10] because its curvature and torsion graphs are arXiv:2104.11327v1 [math.GM] 20 Apr 2021 both linear [19]. The log-aesthetic curve is originated from Harada et al. [7, 6]. They revealed that natural aesthetic curves have such a property that their logarithmic dis- tribution diagram of curvature (LDDC) can be approximated by straight lines. Based on their work Miura et al. [12, 11] have defined the logarithmic curvature graph (LCG), an analytical version of the LDDC as the following: when a curve (given with arc length s and radius of curvature ρ) is subdivided into infinites- imal segments such that ∆ρ/ρ is constant, the LCG represents the relationship between ρ and ∆s in a double logarithmic graph.
    [Show full text]
  • The Cissoid of Diocles
    Playing With Dynamic Geometry by Donald A. Cole Copyright © 2010 by Donald A. Cole All rights reserved. Cover Design: A three-dimensional image of the curve known as the Lemniscate of Bernoulli and its graph (see Chapter 15). TABLE OF CONTENTS Preface.................................................................................................................. xix Chapter 1 – Background ............................................................ 1-1 1.1 Introduction ............................................................................................................ 1-1 1.2 Equations and Graph .............................................................................................. 1-1 1.3 Analytical and Physical Properties ........................................................................ 1-4 1.3.1 Derivatives of the Curve ................................................................................. 1-4 1.3.2 Metric Properties of the Curve ........................................................................ 1-4 1.3.3 Curvature......................................................................................................... 1-6 1.3.4 Angles ............................................................................................................. 1-6 1.4 Geometric Properties ............................................................................................. 1-7 1.5 Types of Derived Curves ....................................................................................... 1-7 1.5.1 Evolute
    [Show full text]
  • MF-$0.75 HC Not Available from EDRS. PLUS POSTAGE Geometry
    DOCUMENT RESUME ED 100 648 SE 018 119 AUTHOR Yates, Robert C. TITLE Curves and Their Properties. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 74 NOTE 259p.; Classics in Mathematics Education, Volume 4 AVAILABLE FROM National Council of Teachers of Mathematics, Inc., 1906 Association Drive, Reston, Virginia 22091 ($6.40) EDRS PRICE MF-$0.75 HC Not Available from EDRS. PLUS POSTAGE DESCRIPTORS Analytic Geometry; *College Mathematics; Geometric Concepts; *Geometry; *Graphs; Instruction; Mathematical Enrichment; Mathematics Education;Plane Geometry; *Secondary School Mathematics IDENTIFIERS *Curves ABSTRACT This volume, a reprinting of a classic first published in 1952, presents detailed discussions of 26 curves or families of curves, and 17 analytic systems of curves. For each curve the author provides a historical note, a sketch orsketches, a description of the curve, a a icussion of pertinent facts,and a bibliography. Depending upon the curve, the discussion may cover defining equations, relationships with other curves(identities, derivatives, integrals), series representations, metricalproperties, properties of tangents and normals, applicationsof the curve in physical or statistical sciences, and other relevantinformation. The curves described range from thefamiliar conic sections and trigonometric functions through tit's less well knownDeltoid, Kieroid and Witch of Agnesi. Curve related - systemsdescribed include envelopes, evolutes and pedal curves. A section on curvesketching in the coordinate plane is included. (SD) U S DEPARTMENT OFHEALTH. EDUCATION II WELFARE NATIONAL INSTITUTE OF EDUCATION THIS DOCuME N1 ITASOLE.* REPRO MAE° EXACTLY ASRECEIVED F ROM THE PERSON ORORGANI/AlICIN ORIGIN ATING 11 POINTS OF VIEWOH OPINIONS STATED DO NOT NECESSARILYREPRE INSTITUTE OF SENT OFFICIAL NATIONAL EDUCATION POSITION ORPOLICY $1 loor oiltyi.4410,0 kom niAttintitd.: t .111/11.061 .
    [Show full text]
  • Smooth Hypocycloidal Paths with Collision-Free and Decoupled Multi-Robot Path Planning
    International Journal of Advanced Robotic Systems ARTICLE SHP: Smooth Hypocycloidal Paths with Collision-free and Decoupled Multi-robot Path Planning Regular Paper Abhijeet Ravankar1*, Ankit A. Ravankar1, Yukinori Kobayashi1 and Takanori Emaru1 1 Laboratory of Robotics and Dynamics, Division of Human Mechanical Systems and Design, Hokkaido University, Sapporo, Japan *Corresponding author(s) E-mail: [email protected] Received 16 September 2015; Accepted 04 April 2016 DOI: 10.5772/63458 © 2016 Author(s). Licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A novel ‵multi-robot map update‵ in case of dynamic obstacles in the map is proposed, such that robots update Generating smooth and continuous paths for robots with other robots about the positions of dynamic obstacles in the collision avoidance, which avoid sharp turns, is an impor‐ map. A timestamp feature ensures that all the robots have tant problem in the context of autonomous robot naviga‐ the most updated map. Comparison between SHP and other tion. This paper presents novel smooth hypocycloidal paths path smoothing techniques and experimental results in real (SHP) for robot motion. It is integrated with collision-free environments confirm that SHP can generate smooth paths and decoupled multi-robot path planning. An SHP diffuses for robots and avoid collision with other robots through local (i.e., moves points along segments) the points of sharp turns communication. in the global path of the map into nodes, which are used to generate smooth hypocycloidal curves that maintain a safe Keywords Path Smoothing, Robot Path Planning, Multi- clearance in relation to the obstacles.
    [Show full text]