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From: JEMebius on 20 Dec 2009 19:26 Grover Hughes wrote: > I'd like to have a formula (closed form if possible, series is > acceptable) which gives the numerical value of the perimeter of a > superellipse as defined by the formula > > (x/a)^n + (y/b)^n = 1 > > where n > 2 and a and b are the known semiaxes. I have searched > Wolfram Mathworld and Wikipedia without success. I DO have the area > formula, but need the same thing for the perimeter. Your help will be > much appreciated. > > Grover Hughes Try and integrate the arc element ds = sqrt ( dx(t)^2 + dy(t)^2 ) from t = 0 to t = 2pi, where x = a.cos(t)^(2/n), y = b.sin(t)^(2/n) is the parametric representation of the superellipse. If desired one uses the transformation u = tan (t/2) to replace sin(t) and cos(t) by rational functions of u, and thus to obtain the integral of an algebraic function. In the wee small hours I am too lazy to do the calculations myself now. Good luck: Johan E. Mebius
From: Robert Israel on 20 Dec 2009 20:07
JEMebius <jemebius(a)xs4all.nl> writes: > Grover Hughes wrote: > > I'd like to have a formula (closed form if possible, series is > > acceptable) which gives the numerical value of the perimeter of a Series in powers of what? > > superellipse as defined by the formula > > > > (x/a)^n + (y/b)^n = 1 > > > > where n > 2 and a and b are the known semiaxes. I have searched > > Wolfram Mathworld and Wikipedia without success. I DO have the area > > formula, but need the same thing for the perimeter. Your help will be > > much appreciated. I assume either n is an even integer or you meant to put in absolute values. Otherwise you'll have problems with this when x or y is negative. > > Grover Hughes > > Try and integrate the arc element ds = sqrt ( dx(t)^2 + dy(t)^2 ) from t = > 0 to t = 2pi, > > where x = a.cos(t)^(2/n), y = b.sin(t)^(2/n) is the parametric > representation of the > superellipse. With the caveat above, you want to integrate from 0 to pi/2 and multiply by 4. > If desired one uses the transformation u = tan (t/2) to replace sin(t) and > cos(t) by > rational functions of u, and thus to obtain the integral of an algebraic > function. Algebraic if n is rational. > In the wee small hours I am too lazy to do the calculations myself now. It's pretty complicated at any hour. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada |