I'm not exactly certain of the mathematical description of this surface (if I were I wouldn't have a question), but I basically want to make a "3D spiral" which is basically a sine wave "wrapped" around a logarithmic spiral in a continuous sense.
The only real-life example I can think of something like this is if you had a blanket that you grabbed a point on and twisted, but I can describe it more effectively with an example. Imagine that image as a logarithmic, rather than archimedian spiral. (The equation that generated that image is $z=sin(\theta+r)$ down to scaling factors, but I can think of absolutely no way to incorporate a power without it being discontinuous near $\theta=2\pi\cong0$. The best result I've achieved is demonstrated in this failed attempt, the formula for which I seem to have lost but it was a hack anyway.)
Because I'm writing this as part of a computer program which is rendering an image from the output, I need to cover every point on a grid, so I can't use a parametric approach which covers points in its own order.
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