{"id":3434,"date":"2015-01-26T21:51:10","date_gmt":"2015-01-27T05:51:10","guid":{"rendered":"http:\/\/bendwavy.org\/wp\/?p=3434"},"modified":"2015-01-31T23:17:26","modified_gmt":"2015-02-01T07:17:26","slug":"elusive-avoidance","status":"publish","type":"post","link":"https:\/\/bendwavy.org\/wp\/?p=3434","title":{"rendered":"elusive avoidance"},"content":{"rendered":"

<\/p>\n

I’ve been<\/a> designing printable models<\/a> of the Lawson-Klein surface
\n

w = cos(u) cos(2v)
\nx = cos(u) sin(2v)
\ny = sin(u) cos(v)
\nz = sin(u) sin(v) <\/center>
\nAs you can plainly see, this figure lives in S3 (positively curved 3-space), so stereographic projection can bring it into E3 (Euclidean 3-space) without adding more self-intersections. (It crosses itself at u=n\u03c0.)<\/p>\n

To minimize the distortion of the projection, I want the projection center to be as far as possible from the surface. One thing I tried was pursuit: starting with an arbitrary point P in S3 and an arbitrary point L(u,v) in the surface, move (u,v) to bring L closer to P while simultaneously moving P away from L. This gets me nowhere so far: either it fails to converge or P converges to the antipodes of L, which is also in the surface (change u by π).<\/p>\n","protected":false},"excerpt":{"rendered":"

I’ve been designing printable models of the Lawson-Klein surface w = cos(u) cos(2v) x = cos(u) sin(2v) y = sin(u) cos(v) z = sin(u) sin(v) As you can plainly see, this figure lives in S3 (positively curved 3-space), so stereographic … 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":[12,70],"tags":[],"class_list":["post-3434","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-merch"],"_links":{"self":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/3434","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=3434"}],"version-history":[{"count":7,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/3434\/revisions"}],"predecessor-version":[{"id":3441,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/3434\/revisions\/3441"}],"wp:attachment":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3434"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3434"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3434"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}