A Simpler form for Equation (3)
 Mathematical Details (and comments) for an article by Mandelbrot and Frame: Self-Contacting Fractal Trees. by Don West, Department of Mathematics, SUNY Plattsburgh

Mandelbrot and Frame give an equation, similar to equation (2), as their condition for tip contact in case the angle theta is in the range 90 to 135 degrees. Then they solve for r and call the result equation (3):

This expression can be simplified to

Their form comes right out of the quadratic formula, but it is undefined when the angle theta = 135°. The formula given here is simpler than equation (3) and has the advantage that it, and its derivative, are defined on the closed interval [90, 135]. This helps if one is trying to analyze the behavior of r as a function of theta. For example, it would be nice to prove that the (r, theta) graph (their Figure 9, p. 23) is concave-up on each segment where it is smooth. I have finished only part of this task.

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