 # 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
• 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
• 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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