Saturday, 2005 February 19, 18:32 — mathematics

the stubborn spiral

Rusin’s disco ball
golden angles
Saff & Kuijlaars

Examples of three algorithms for distributing nodes fairly evenly over a sphere. Those on the middle and right slice the sphere into parallel bands of equal area (much narrower than the white discs), and put one node (center of a disc) somewhere in each band. Saff & Kuijlaars place the nodes along a spiral path across the bands, keeping the distance between turns of the spiral roughly constant. Failing to grok how their rule does that, I approach it from another angle.

In the ideal limit (N approaching infinity), the unit sphere is covered exactly once by a strip of width 2√(π/N) and length 2√(Nπ); the strip is divided into N nearly square pieces, with a node in the center of each. What parametric functions of -1 < t < 1 describe the midline of the strip?

(In what follows, θ and φ have their conventional meanings of longitude and co-latitude, i.e. angular distance from a pole.)

The equal slices rule gives cosφ = t. I seek a function θ(t) such that the derivative along the path has constant magnitude:

ds² = dφ² + sin²φ dθ² = Nπ dt²

I proceed:

–sinφ dφ = dt; dφ = –dt / sinφ = –dt / √(1–t²)

ds² = Nπ dt² = ( dt² / (1–t²) ) + (1–t²) dθ²

(1–t²) dθ² = dt² (Nπ – 1 / (1–t²))
  = dt² (Nπ – Nπt² – 1) / (1–t²)

dθ² = dt² (Nπ – Nπt² – 1) / (1–t²)²

So I ask the Wolfram Integrator to integrate

Sqrt[ n*Pi - n*Pi*x^2 - 1 ] / (1-x^2)

and it says

Now, I do make mistakes; I’ve worked the problem several times now and got subtly different answers. Last time around, in fact, there were no imaginary coefficients; so I made this in PoV-Ray:

Not quite what I’m looking for. Can you spot where I went wrong?

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