{"id":1585,"date":"2005-02-19T18:32:25","date_gmt":"2005-02-20T02:32:25","guid":{"rendered":"http:\/\/www.ogre.nu\/wp\/?p=1585"},"modified":"2010-05-26T15:12:43","modified_gmt":"2010-05-26T23:12:43","slug":"the-stubborn-spiral","status":"publish","type":"post","link":"https:\/\/bendwavy.org\/wp\/?p=1585","title":{"rendered":"the stubborn spiral"},"content":{"rendered":"\n\n
Rusin’s disco ball
<\/td>\n		 golden angles
<\/td>\n		 Saff & Kuijlaars
<\/td>\n<\/tr>\n<\/table>\n

Examples of three algorithms<\/a> 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.
\n
\nIn 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?<\/p>\n

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

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

\nds² = dφ² + sin²φ dθ² = Nπ dt²\n<\/p><\/blockquote>\n

I proceed:<\/p>\n

\n–sinφ dφ = dt; dφ = –dt \/ sinφ = –dt \/ √(1–t²)<\/p>\n

ds² = Nπ dt² = ( dt² \/ (1–t²) ) + (1–t²) dθ²<\/p>\n

(1–t²) dθ² = dt² (Nπ – 1 \/ (1–t²))
\n  = dt² (Nπ – Nπt² – 1) \/ (1–t²)<\/p>\n

dθ² = dt² (Nπ – Nπt² – 1) \/ (1–t²)²\n<\/p><\/blockquote>\n

So I ask the Wolfram Integrator<\/a> to integrate<\/p>\n

Sqrt[ n*Pi - n*Pi*x^2 - 1 ] \/ (1-x^2) <\/code><\/p><\/blockquote>\n

and it says

<\/center><\/p>\n

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<\/a>:
\n

<\/center><\/p>\n

Not quite what I’m looking for. Can you spot where I went wrong?<\/p>\n","protected":false},"excerpt":{"rendered":"

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 … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"class_list":["post-1585","post","type-post","status-publish","format-standard","hentry","category-mathematics"],"_links":{"self":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/1585","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1585"}],"version-history":[{"count":1,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/1585\/revisions"}],"predecessor-version":[{"id":2531,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/1585\/revisions\/2531"}],"wp:attachment":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1585"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1585"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}