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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
    
Associated Topics:
High School Calculus

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