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### 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

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

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

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

Associated Topics:
High School Equations, Graphs, Translations

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