Horn Twisting

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 with the fixed value 0.2 for this plot (as α approaches 1, the horn approaches a snail)
• 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
• n is a parameter ranging from 0 (straight horn) to 3 (three twists) and back, with a step of 0.1, giving the 60 frames shown here
• 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.

See Also:

Horn-to-Snail

References:

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