GEOMETRY OF CURVES CHAPMAN & HALL/CRC. Boca Raton London New York Washington, D.C.

Transcription

1 GEOMETRY OF CURVES JOHN W. RUTTER CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C.

2 Contents Introduction 0.1 Cartesian coordinates 0.2 Polar coordinates 0.3 The Argand diagram 0.4 Polar equations 0.5 Angles 0.6 Orthogonal and parallel vectors 0.7 Trigonometry Lines, circles, and conics Lines The circle Conics The ellipse in canonical position The hyperbola in canonical position The parabola in canonical position Classical geometric constructions of conics Polar equation of a conic with a focus as pole History and applications of conics 26 2 Conics: general position Geometrical method for diagonalisation Algebra Algebraic method for diagonalisation Translating to canonical form Central conics referred to their centre Practical procedures for dealing with the general conic Rational parametrisations of conics 55

3 viii Contents 3 Some higher curves The semicubical parabola: a cuspidal cubic A crunodal cubic An acnodal cubic A cubic with two parts History and applications of algebraic curves A tachnodal quartic curve Limagons Equi-angular (logarithmic) spiral Archimedean spiral Application - Watt's curves 73 4 Parameters, tangents, normal Parametric curves Tangents and normals at regulär points Non-singular points of algebraic curves Parametrisation of algebraic curves Tangents and normals at non-singular points Arc-length parametrisation Some results in analysis 96 5 Contact, inflexions, undulations Contact Invariance of point-contact order Inflexions and undulations Geometrical Interpretation of n-point contact An analytical Interpretation of contact Cusps, non-regular points Cusps Tangents at cusps Contact between a line and a curve at a cusp Higher singularities Curvature Cartesian coordinates Curves given by polar equation Curves in the Argand diagram An alternative formula Curvature: applications Inflexions of parametric curves at regulär points Vertices and undulations at regulär points Curvature of algebraic curves 157

4 Contents 8.4 Limiting curvature of algebraic curves at cusps Circle of curvature Centre of curvature and circle of curvature Contact between curves and circles 173 ix 10 Limagons 10.1 The equation 10.2 Curvature 10.3 Non-regular points 10.4 Inflexions 10.5 Vertices 10.6 Undulations 10.7 The five classes of limacons 10.8 An alternative equation Evolutes Definition and special points A matrix method for calculating evolutes Evolutes of the cycloid and the cardioid Parallels, involutes Parallels of a curve Involutes Roulettes General roulettes Parametrisation of circles Cycloids: rolling a circle on a line Trochoids: rolling a circle on or in a circle Rigid motions Non-regular points and inflexions of roulettes Envelopes Evolutes as a model Singular-set envelopes Discriminant envelopes Different definitions and singularities of envelopes Limiting-position envelopes Orthotomics and caustics The relation between orthotomics and caustics Orthotomics of a circle Caustics of a circle 268

5 x Contents 15 Singular points of algebraic curves Intersection multiplicity with a given line Homogeneous polynomials Multiplicity of a point Singular lines at the origin Isolated singular points Tangents and branches at non-isolated singular points Branches for non-repeated linear factors Branches for repeated linear factors Cubic curves Curvature at singular points Projective curves The projective line The projective plane Projective curves The projective curve determined by a plane curve Affine views of a projective curve Plane curves as views of a projective curve Tangent lines to projective curves Boundedness of the associated affine curve Summary of the analytic viewpoint Asymptotes Singular points and inflexions of projective curves Equivalence of curves Examples of asymptotic behaviour Worked example Practical work Drawn curves Personalising MATLAB for metric printing Ellipse 1 and Ellipse Ellipse Parabola Parabola Parabola Hyperbola Semicubical parabola Polar graph paper Further reading 353 Index 355