Mandelbrot set: Difference between revisions

Another approach to reduce the glitches is the so-called ''rebasing''. See [https://gbillotey.github.io/Fractalshades-doc/math.html#avoiding-loss-of-precision Avoiding loss of precision] (Fractalshades) and [https://fractalforums.org/fractal-mathematics-and-new-theories/28/another-solution-to-perturbation-glitches/4360 Another solution to perturbation glitches] (Fractalforums) for details. Remarkably, the condition for the rebasing abs2(z) < abs2(epsilon) or abs(z) < abs(epsilon) does not define a circular area but a half-plane. If you want a circular area, you can insert a factor. For 0 < gamma < 1 the condition abs(z) < gamma * abs(epsilon) defines a circular area containing the origin. The size of this circle depends on the distance of the reference point S from the origin and on the gamma factor. The smaller gamma, the smaller the circle and the closer its center is to the origin. For gamma = 1/2 and below, the circle lies entirely within the rescue circle around the origin O(0, 0) with escape radius r > 2, even if the associated reference point S is on the edge of the rescue circle. For example, if the reference point is at S(6, 0), the equation abs(z) = abs(epsilon) defines the set of all points that have the same distance from the origin O and the reference point S. This is exactly the perpendicular bisector x = 3. If,Considering on the other hand, you look atinstead the equation abs(z) = 1/2 * abs(epsilon), you lookone forfinds all points that are half as far from the origin as from the reference point. The set ofAll these points lieslie on a circle with center M(-2, 0) and radius R = 4. This circle lies completely in a rescue circle around the origin with escape radius r = 6 around the origin. If you choose a factor gamma > 1, you will get corresponding circles aroundcontaining the reference point. However, the condition abs(z) < abs(epsilon) or abs2(z) < abs2(epsilon) is optimal because Sit keeps the differences epsilon to the reference sequence as small as possible. Rebasing occurs if and only if the distance abs(epsilon) to the reference sequence becomes smaller as a result.

if abs2_z < abs2_epsilon # rebaserebasing if zthis is closer toreduces the origin thandistance to the reference sequence