# Warping Functions

Unfortunately, this goal doesn't appear possible. However, I was thinking about the following mathematical factoid:

y(x) = x − a⋅sin(x)

has the solution (for |a|<1, which is the well-posed case)

x(y) = y + ∑_{n=1}^{∞} (2/n)⋅J_{n}(a⋅n)⋅sin(ny)

(This type of series of Bessel functions is known as a Kapteyn series; also see Chapter 17 of GN Watson, *A Treatise on the Theory of Bessel Functions, 2nd Edition*.) Furthermore, J_{n}(a⋅n) decreases exponentially fast as n→∞. This means that for warps represented as low-order finite Fourier series, the inverse functions, although requiring an infinite number of terms to be exact, are pretty well represented themselves by low-order finite Fourier series.

In higher dimensions, the obvious generalization is a tensor-product Fourier series; this would be reasonable over a rectangular domain. Slightly less obvious, over the interior of a spherical domain, a better thing to do would be spherical harmonic series based on Y_{nm}(θ,φ)j_{n}(λ_{p}r). On the surface, drop the half-order Bessel functions. Must investigate numerics.