Talk:Evolute
From Wikipedia, the free encyclopedia
I disagree with Mathworld's page on two points:
- involutes are not unique and thus "The original curve is then said to be the involute of its evolute" is wrong
- what is the evolute of a sequence of circular arcs, if not disconnected points? And what would be an involute of that?
I'm trying to raise Eric by email 142.177.124.178 14:29, 19 Jul 2004 (UTC)
Continuing the "what to do without a natural parametrisation" stuff: it often happens that s'(t)2 is simpler/easier than s'(t). That may be useful.
Let q(t) = s'(t)2. Differentiate once q'(t) = 2s'(t)s''(t). Fit these into the lower expression to get
It so happens all the s's go away—that sqrt can be avoided. Of course we still need | r'' | 2... 142.177.124.178 02:45, 22 Jul 2004 (UTC)
- It can really help! Evolute of ellipse:
- x = (aC,bS) (with shorthand C=cos(t), S=sin(t))
- x' = ( − aS,bC)
- x'' = − x
- q = | x' | 2 = a2S2 + b2C2
- q' = 2SC(a2 − b2)
- N = 2qx'' − q'x' = − 2ab(bC,aS)
- | N | 2 = 4a2b2(b2C2 + a2S2)
- r'' / | r'' | 2 = 2q2N / | N | 2 = − (b2C2 + a2S2)(C / a,S / b)
- evolute = (a2 − b2)(C3 / a, − S3 / b)
- the almost-astroid fell right out 142.177.124.178 15:20, 22 Jul 2004 (UTC)
