By Heinrich W. Guggenheimer

Designed for complex undergraduate or starting graduate research, this article includes an straight forward advent to non-stop teams and differential invariants; an in depth remedy of teams of motions in euclidean, affine, and riemannian geometry; and improvement of the tactic of crucial formulation for worldwide differential geometry.

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The fonnula for a space evolute is £ = ex + ~ N + ;- B where c is a constant. 12 (The Plane Evolute ofa Parabola). Suppose a parabola y2 = 2ax is given with parametrization ex(t) = (t 2 /(2a), t). Then ex'(t) = (t/a, 1), v = lex'i = Ja 2 + t 2 /a and, therefore, T = (t/Ja 2 + t 2 , a/Ja 2 + t 2 ). 4. 20. Neil's parabola Hence, K = a 2 /(a 2 + t 2 )3/2 and, putting this in the expression for T', N = (a/Ja 2 + t 2 , -t/Ja 2 + t 2 ). ). 13. 20). 14. Show that the evolute of the ellipse a(t) = (a cos t, b sin t) is the astroid 2 2 2 .

Verify the final step by differentiation and compute T. 7 for a Maple approach. 15 (The Astroid). 8). The definition of the astroid (which was discovered by people searching for the best form of gear teeth) is very similar to that of the cycloid. For the astroid, however, a circle is rolled, not on a line, but inside another circle. More precisely, let a circle of radius a/4 roll inside a large circle of radius a (centered at (0,0) say). For concreteness, suppose we start the little circle at (a, 0) and follow the path of the point originally in contact with (a, 0) as the circle rolls up.

Then (as we noted above) the plane is determined by a point P and a normal vector n =1= O. Since fJ lies in the plane, for all s. (fJ(s)-P)·n=O By differentiating twice, we obtain two equations: fJ' (s) . n = 0 and fJ// (s) . n = O. That is, T . n = . n = O. These equations say that n is perpendicular to both T and N. Thus n is a multiple of Band o and K N ± nllnl = B. Hence B is constant and B' = O. The Frenet Formulas then give T = O. o So now we see that curvature measures the deviation of a curve from being a line and torsion the deviation of a curve from being contained in a plane.