"This report is an illustrated study of plane algebraic and transcendental curves, emphasizing analytic equations and parameter studies."
Chapter 1: Properties of Curves Edit
Coordinate Systems Edit
- 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.