tag:blogger.com,1999:blog-11295132.post116049245150274333..comments2014-12-04T12:37:01.029-08:00Comments on A Neighborhood of Infinity: Oriented Fish and Hairy BallsDan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-11295132.post-91146342596498958902007-01-08T14:49:00.000-08:002007-01-08T14:49:00.000-08:00panic,
Simply adding time won't solve the problem...panic,<br /><br />Simply adding time won't solve the problem, but time might come into one possible approach to the animation problems I mention. The idea is to make the function f a function not just of some vector, but the history of the vector. But I've no space for details here...sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-8227498420080384442007-01-08T14:27:00.000-08:002007-01-08T14:27:00.000-08:00Could you solve the oriented fish problem by givin...Could you solve the oriented fish problem by giving "time" as an argument to the normal generator, making it a function R^4 -> R^3?panichttp://www.blogger.com/profile/16674377776447649149noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-36763440058570779952007-01-08T07:16:00.000-08:002007-01-08T07:16:00.000-08:00bot builder,
It just dawned on me - quaternions p...bot builder,<br /><br />It just dawned on me - quaternions provide a solution in 4D. but not 3D. If you identify 4-vectors with quaternions in the obvious way then f(x)=i*x will do. You can adapt this to any even dimension.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-64667231182750709402007-01-07T11:45:00.000-08:002007-01-07T11:45:00.000-08:00bot builder,
quaternions don't give a solution to...bot builder,<br /><br />quaternions don't give a solution to this problem, and have some fascinating topological no-go theorems of their own. For example, if you've written a slerp routine you'll know that there are two unit quaternions for each rotation and you have to make a decision about which one to use. This causes discontinuities of its own. But the fact that you have two of these things per rotation relates to the double cover of SO(3) that I briefly mentioned...sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-90087253216504724272007-01-07T11:31:00.000-08:002007-01-07T11:31:00.000-08:00Interesting. I thought quaternions could handle t...Interesting. I thought quaternions could handle this. At least, that's what I always use for interpolation of 3d rotations.Bot Builderhttp://www.blogger.com/profile/02775704516611135021noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-63730400094231549492007-01-07T09:34:00.000-08:002007-01-07T09:34:00.000-08:00You have a cool job ! I'd like to be able to use s...You have a cool job ! I'd like to be able to use some cool math for my job.<br /><br />The hairy ball theorem that I know is : every vector field on S^2 has a singularity and it is related to the Euler characteristic of the sphere.<br /><br />I had not made the connection with the example you give. Interesting ... as always on this blog :-)alpheccarhttp://www.blogger.com/profile/14645433315403867431noreply@blogger.com