Horn to Snail

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

where:

• α 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
• β is a parameter with the fixed value 1 for this plot
• γ 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 2π, scanning the angle of each circular section of the surface
• v ranges from 0 to 2π, defining the diameter of the "cylinder" within which the surface lies.

References:

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