θ as a function of one variable: If we rewrite θ(x, y) as π/2 + θ(xy), then the new θ is an odd function of one variable that satisfies θ' + xθ" = cos(θ). It is easy to prove (by successive approximation, for example), that the solution exists as a real analytic function. It then turns out that two sets of equations, called the Gauss and Codazzi equations, are satisfied for the coordinates. This implies that the surface actually exists.I thought it amusing that the existence of this surface followed from an idea in Hilbert's non-existence theorem.
Graphing the surface: What makes this coordinate system easy to work with is that the coordinate curves have curvature that is basically θ' (along the s curve at (s, t) the curvature is tθ'(st)) and the torsion (the rate that the curve twists in 3-space) is identically 1. Along the axes this explains the earlier appearance of the tangent series. Knowing curvature and torsion tells you how to navigate along the curve in three-space.
Power series: Computer calculation gives θ = t − t3/odd(3!)3! + 7 t5/odd(5!)5! − 521 t7/odd(7!)7! + 31139 t9/odd(9!)9! − 18279367 t11/odd(11!)11! + ..., where odd(n) is the largest odd factor of n. These calculations were straightforward because the operator θ → θ' + tθ" takes tn to n2tn−1; we can start with θ = t and then repeatedly substitute into the cosine and find the next coefficient by dividing by n2. The part of the graph of θ that is directly relevant to our surface is the interval where it is between −π/2 and π/2, which is approximately −1.86 < t < 1.86. On this interval θ is quite simple; it is an increasing function, whose derivative is 1 at t = 0 and decreases to a little over ½ at either end. Outside this interval it oscillates around π/2 to the right and −π/2 to the left, with increasing period and decreasing amplitude. Although it is defined over the whole real line, if you look at the coefficients in decimal form, the radius of convergence appears to be around 3. To learn more about the function, I looked at it as a complex solution of θ' + zθ" = cos(θ).
Intro Curvature −1 Coordinates θ Complex θ More Equations Trick General Entire
