The Roses of Grandi

Date: 11/13/2000 at 10:58:42
From: Tracy Getz
Subject: Roses of Grandi

The Roses of Grandi are similar to the Folium of Descartes, but I 
could not find any information in your archives about them.

Question: The Roses of Grandi are given by the polar equations

     r = a*sin(n*theta)  
     r = a*cos(n*theta)

where n is an integer. Show that the Roses of Grandi are algebraic.

Date: 11/13/2000 at 15:02:53
From: Doctor Rob
Subject: Re: Roses of Grandi

Thanks for writing to Ask Dr. Math, Tracy.

I don't think you need any information about them other than what is 
given to you above.

The idea is to show that sin(n*theta) and cos(n*theta) can be 
expressed as polynomials in sin(theta) and cos(theta). I would do this 
by using the Principle of Mathematical Induction, combined with the 
sine/cosine of a sum trigonometric identity.

Then you can substitute y/r for its equal sin(theta), and x/r for its 
equal cos(theta). That will remove all instances of the variable 
theta. Then multiply through by r^n to clear fractions. Replace powers 
of r by using

     r^(2*k)   = (x^2+y^2)^k
     r^(2*k+1) = r*(x^2+y^2)^k

That leaves r appearing only to the first power, at most. If r does 
appear, then solve for r, square both sides, and substitute x^2 + y^2 
for its equal r^2. Now clear fractions, and you have a polynomial 
equation in x and y, so it is an algebraic curve.

Example: r = a*sin(2*theta)

               r = a*2*sin(theta)*cos(theta)
                 = 2*a*(y/r)*(x/r)
                 = 2*a*x*y/r^2

             r^3 = 2*a*x*y
     r*(x^2+y^2) = 2*a*x*y,
               r = 2*a*x*y/(x^2+y^2)
             r^2 = 4*a^2*x^2*y^2/(x^2+y^2)^2
     (x^2+y^2)^3 = 4*a^2*x^2*y^2

The MacTutor History of Mathematics archive provides a Web page with 
information about them here:

   Rhodonea Curves
   http://www-history.mcs.st-and.ac.uk/history/Curves/Rhodonea.html   

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/