Lawrence - A Catalog of Special Plane Curves



Dover

Classical Geometry

"This report is an illustrated study of plane algebraic and transcendental curves, emphasizing analytic equations and parameter studies."

= Chapter 1: Properties of Curves =

Coordinate Systems

 * Table 2: Pedal Equations : It's confusing as to what this even means at first. Eventually, I realized that a pedal equation essentially gives a differential equation for a curve.  Specifically, if the curve C is given parametrically by x(t) and y(t), then the tangent line L to C at P = (x(t), y(t)) has slope y'(t)/x'(t), and consequently has the equation $$Y - y(t) = \frac{y'(t)}{x'(t)} ( X - x(t) ),$$ and the perpendicular bisector to L through the origin has the equation $$Y = - \frac{x'(t)}{y'(t)} X$$. Consequently, "pedal coordinates" are given by $$r(t)^2 = x(t)^2 + y(t)^2$$ and $$p(t)^2 = x(t) - \frac{x'(t)}{y'(t)} y(t).$$  To see this in action, look at the equation $$p^2 = a r$$ for a parabola given in the table.  Converting to Cartesian coordinates to get a single differential equation, we have $$x - \frac{y(x)}{y'(x)} = a \sqrt{x^2 + y(x)^2}.$$.  Solving this differential equation gives a family of parabolas.

Geometry
= Chapter 2: Types of Derived Curves =

= Chapter 3: Conics and Polynomials =

= Chapter 4: Cubic Curves =

= Chapter 5: Quartic Curves =

= Chapter 6: Algebraic Curves of High Degree =

= Chapter 7: Transcendental Curves =