The Advanced Geometry of Plane Curves and Their Applications BY C ZWIKKER

The Advanced Geometry of Plane Curves and Their Applications

The Advanced Geometry of Plane Curves and Their Applications

Contents of The Advanced Geometry of Plane Curves

  • CHAPTER I The plane
  • The plane considered as a collection of complex numbers z
  • Representation of curves by z= flu) First examples
  • Intersection of two curves Complex x- and y-values
  • The elements at infinity
  • Complex values of the parameter u; deficient curves
  • CHAPTER II The geometrical interpretation of analytic operations applied to complex numbers
  • Addition; subtraction; multiplication
  • Decomposition of complex numbers
  • Quotients, cross ratio
  • Conformal transforms; inversion
  • Non-conformal transforms; collineation
  • The first derivative, tangent
  • The second derivative, curvature
  • CHAPTER III The straight line
  • Collinearity of three points; concurrency of three lines
  • Theorems of Ceva, Menelaos and Desargues
  • Line coordinates
  • Dual transformation
  • Projective point assemblages and ray pencils
  • Projective geometry Involution
  • CHAPTER IV The triangle
  • Centre of gravity; orthocentre, circumcentre
  • Euler’s axis; nine points circle
  • Base points of perpendiculars, WALLACE’S theorem Triangle and deltoid
  • CHAPTER V The circle
  • Properties of constant angle and constant power
  • General circle formula
  • Circuit impedance and admittance
  • The circle transformation
  • Projective properties
  • CHAPTER VI Algebraic curves
  • Unicursal curves of n•th order
  • Synthetic construction of conics, cubics and quartics
  • Pole and polar with respect to conics
  • Pascal’s and Brianchon’s theorems
  • Cubics, Newton’s classification
  • Cubics, projective properties
  • CHAPTER VII Ellipse
  • Introduction
  • Conjugate diameters
  • The foci
  • Kepler orbits
  • Conic sections
  • The reflection law
  • The perimeter of the ellipse
  • CHAPTER VIII Hyperbola
  • Introduction
  • Medians, conjugate directions
  • The foci
  • Orthogonal hyperbola
  • Cartesian ovals
  • CHAPTER IX Parabola
  • Introduction
  • Right angles in the parabola
  • Medians, pole and polar
  • Concluding remark on conics
  • CHAPTER X Involutes, evolutes and anticaustics
  • Involute and evolute
  • Norwich spiral
  • Catenary and tractrix
  • Tractrices in general
  • The evolute of the parabola
  • Anticaustics
  • CHAPTER XI Pedals and other derived curve
  • Pedal and contrapedal
  • The pedal inversion theorem
  • Limalt n, conchoid
  • Pedals derived from the parabola
  • Cissoid and strophoid
  • Orthoptic curves
  • Palle
  • CHAPTER XII Areas and other integrals
  • Areas
  • Surfaces of revolution
  • Volumes of revolution
  • Centre of gravity of areas
  • Centre of gravity of contours
  • Inertia moments
  • CHAPTER XIII Envelopes
  • Definition of the envelope
  • Astroid
  • Caustics
  • Antipedals
  • Anticaustics as envelopes
  • CHAPTER XIV Orthogonal trajectories
  • General way of finding orthogonal trajectories
  • Laplace’s equation
  • The transformation u’ = u
  • Point sources
  • Two magnetic problems
  • Non-Laplacian orthogonal trajectories
  • CHAPTER XV Kinked curves
  • Discontinuous functions
  • Irrational functions
  • Schwarz’s theorem
  • Triangles and rectangles
  • Application to potential- and streamline problems
  • CHAPTER XVI Spirals
  • Archimedes’ spiral and hyperbolic spiral
  • Logarithmic spiral
  • Syntrepency
  • Klothoid
  • Miscellaneous spirals
  • CHAPTER XVII Lemniscate
  • Geometrical properties
  • Sinus spirals
  • Lemniscate and hyperbola
  • Wave impedance and wave admittance
  • Straight line guides
  • CHAPTER XVIII Cycloid
  • Geometrical properties
  • The projections of the screw, trochoids
  • Brachistochrone
  • Tautochrone
  • Path of electron in combined electric and magnetic fields, phygoid
  • The cycloid as envelope and as caustic
  • CHAPTER XIX Epi- and hypocycloids
  • Introduction
  • Natural equation, evolute
  • Cycloids as envelopes
  • Deltoid
  • Deltoid and orthogonal complete quadrangle
  • Projective properties of the deltoid
  • Roses
  • CHAPTER XX Cardioid and limaron
  • Various properties and applications of the cardioid
  • The cardioid as caustic
  • The cardioid as inversion of the parabola
  • Application in electric circuit theory
  • Pascal’s lima on
  • CHAPTER XXI Gear wheel tooth profiles
  • Coupling of epi and hypocycleids
  • The general problem
  • Evolvente wheel teeth
  • Pin wheels
  • Trochoids
  • Parallel curves
  • Slip

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