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
