Talk:Roulette (curve)
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[edit] Besant
The Besant book is also available on Google Books but I had problems loading all the pages. The book itself is very comprehensive, but notation and methods of proof are somewhat dated, for example infinitesimals are used rather than limits.--RDBury (talk) 14:41, 26 April 2008 (UTC)
[edit] No slippage vs. equal tangents
The common language way of defining a roulette is to say that a curve rolls on another without slipping. (See, for example, ch. 1 of Wheels, Life and Other Mathematical Amusements by Martin Gardner.) This is more or less what Besant assumes though he seems to regard it as implicit in the meaning of 'roll'. In some modern mathematical treatments however (including the current version of this page), this condition is replaced by one stating that a tangent vector of the rolling curve is carried to a tangent vector of the fixed curve. This version seems not only more confusing, but it requires some sort of proof that one condition implies the other. The proof may be fairly trivial, but not to the extent that it can be skipped without mention. --RDBury (talk) 20:35, 11 May 2008 (UTC)
[edit] Possible generalizations
First, there is nothing in definition that would prevent it from working in a non-Euclidean plane. This does not seem to be covered in the literature however. Second, it seems natural to try to extend the definition to 3 or more dimensions. This runs into a problem in that, since there are more degrees of freedom in the congruences of higher dimensional space, the position and orientation of the moving space is not completely determined unless additional constraints are added. These constraints could take a variety of forms any one of which could be seen as a generalization of the 2 dimensional case. So, for example, one could consider a surface rolling on two fixed surfaces or two fixed curves. The generator could be a point, in which case the "roulette" would be a curve in space, or it could be a curve generating a surface. Besant investigates some of these possibilities but doesn't pick any for the title of "roulette in space". --RDBury (talk) 20:36, 11 May 2008 (UTC)
[edit] Cycloid on line
Both Mathworld and 2dcurves.com state incorrectly that a cycloid rolling on a line will produce an ellipse. This seems to be due to some confusion as to the meaning of the word 'cycloid' as some authors use it to mean epicycloid or hypocycloid. The correct statement is at mathcurve.com where the term 'cycloïde à centre' = 'centered cycloid' means 'epicycloid or hypocycloid'.--RDBury (talk) 15:23, 7 June 2008 (UTC)
