Rectification of a Quartic
Date: 12/02/97 at 15:57:05 From: Pat Sharkey Subject: Rectification of a Quartic I've been told that some 4th order equations can be rectified. I'm specifically interested in Cassinian ovals and want to know if there are any Cassinian ovals whose perimeter can be calculated without looking up tables, etc. I think I read somewhere that in the early 17th century there were some 4th order equations that were 'rectifyable' and I was wondering if the Cassinian ovals fell into that category. I see that many of your correspondents have asked you the same question - concerning the rectification of that second-order equation called the ellipse - which incidentally I'm not the slightest bit interested in. I've searched a fair bit of Ireland for the answer to this. Thanking you and have a happy Christmas.
Date: 12/02/97 at 17:10:19 From: Doctor Rob Subject: Re: Rectification of a Quartic Cassinian Ovals have the equation (x^2+y^2+a^2)^2 = b^4 + 4*a^2*x^2. Differentiating implicitly with respect to x, 2*(x^2+y^2+a^2)*(2*x+2*y*dy/dx) = 8*a^2*x, x+y*dy/dx = 2*a^2*x/(x^2+y^2+a^2) so dy/dx = (2*a^2*x - x*[x^2+y^2+a^2])/(y*[x^2+y^2+a^2]), = x*(a^2-x^2-y^2)/(y*[x^2+y^2+a^2]) Arclength is the indefinite integral of Sqrt[1+(dy/dx)^2] with respect to x, so we need to compute that function and decide if it is integrable. My computer algebra program tells me that that this function equals b^2*Sqrt[(-a^2+Sqrt[b^4+4*a^2*x^2])/ (Sqrt[b^4+4*a^2*x^2]*[-a^2-x^2+Sqrt(b^4+4*a^2*x^2)])] At this point, the substitution x = (b^2/[2*a])*Tan[u] would seem like a good idea. After some simplification, the substitution u = ArcCos[(b^2-v^2)/a^2] suggests itself. I have not carried through the entire computation, but I believe that through a series of substitutions like this, one can simplify this function until it can be directly integrated, or else reduced to one of the known nonintegrable forms, such as an elliptic integral. I leave that task to you. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 12/03/97 at 10:54:25 From: Pat Sharkey Subject: Re: Rectification of a Quartic Hello Dr. Math, I wish to convey my sincerest gratitude for a prompt and very helpful reply to my query about the Cassinian ovals. I could never get off the ground before, but now I have from you the tools to get into the ring with the problem. It's a great service that you provide. Long life and good health to all your team. Sincerely, Pat Sharkey, a greatful Irishman
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