From Osher Doctorow Ph.D.
From r = a(1 - cos(u)), we get alternatively to the last posting:
1) dr/du = asin(u) Since sin^2(u) = 1 - cos^(u) and from the first equation cos(u) = 1 - r/a, we get: 2) dr/du = asin(u) = sqrt(1 - (1 - r/a)^2) when we're in the first and third quadrants, and the minus sign goes in front of sqrt in the second and fourth quadrants. Since discontinuities occur for simple real-valued functions mainly of types y = 1/(polynomial function of x), y = log(x), and y = sqrt(x) at x = 0 or x = root of polynomial (x = 0 for polynomial = x, etc.), we see that the discontinuity in the Mandelbroit Cardioids of the previous posting are not simply "removable singularities" or irrelevant as they might be considered in complex analysis, quite important. The Mandelbroit Cardioids are characterized in terms of Existence among the various types of Existence discussed by infinities or discontinuities - not at all points, but at some points (often only one). Osher Doctorow Ph.D.