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Inversion of figure 8 knot

  • Figure 8 knot - Black curve
  • Inversion - Red curve

GeoGebra script

sigma = Slider(0.01, 3, 0.01, 1, 200) translateX = Slider(-3, 3, 0.01, 1, 200) epsilon = Slider(0.01, 0.5, 0.01, 1, 200) h = Slider(0, 2, 0.01, 1, 200) lambda = Slider(0, 0.5, 0.01, 1, 200) # Figure 8 knot A(x) = epsilon sin(4x) H(x) = (1+A(x)) / (1-A(x)) fx(x) = sigma * H(x) (lambda * sin(x) - (1-lambda) sin(3x)) + translateX fy(x) = sigma * H(x) (lambda * cos(x) + (1-lambda) cos(3x)) fz(x) = sigma * H(x) (h sin(2x)) figureEightKnot = Curve(fx(t), fy(t), fz(t), t, 0, 2pi) # 2-sphere sx(x, y) = cos(x) sin(y) sy(x, y) = sin(x) sin(y) sz(x, y) = cos(y) twoSphere = Surface(sx(u, v), sy(u, v), sz(u, v), u, 0, 2pi, v, 0, pi) # Inversion norm2(x) = (fx(x))^2 + (fy(x))^2 + (fz(x))^2 figureEightKnotInverted = Curve(fx(t) / norm2(t), fy(t) / norm2(t) , fz(t) / norm2(t), t, 0, 2pi)