Horn to Snail


Definition of surface, in parametric equations:

x(u,v) = (1-v/(2)) cos(n v) (1 + cos(u)) + cos(n v)
y(u,v) = (1-v/(2)) sin(n v) (1 + cos(u)) + sin(n v)
z(u,v) = (1-v/(2)) sin(u) + v/(2)

where:

  • is a parameter ranging from 0.15 (horn) to 1 (snail) and back, with a step of 0.05, giving the 35 frames shown here
  • is a parameter with the fixed value 1 for this plot
  • is a parameter with the fixed value 0.1 for this plot, determining the thickness of the horn-snail
  • n is a parameter with the fixed value 2 for this plot, determining the number of twists around the z-axis
  • u ranges from 0 to 2, scanning the angle of each circular section of the surface
  • v ranges from 0 to 2, defining the diameter of the "cylinder" within which the surface lies.

References:

Tore Nordstrand, Mathematical Surfaces. (Note: Nordstrand calls this a "seashell")


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