Polar equation in rectangular form?

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Polar equation in rectangular form?

Joe Davison writes:

> For convenience, let's pretend @ represents the Greek character Theta
>
> When working in polar coordinates, an example of an equation yielding
> a rose would be:
>
> r = 2 sin 2@
>
> I was wondering if it was possible to convert such an equation into
> rectangular form- (In terms of x and y.)

The traditional way of approaching this sort of problem in analytic geometry is
to be sly, rather than just forcing the arctangent  of y/x in for @, etc.

Because  sin(2@) is equal to 2*sin(@)*cos(@)  we can make this substitution
and multiply by r*r to get the equation
                                                r*r*r = 4*r*sin(@)*r*cos(@)

Too bad there are _three_ r's on the left side rather than two or four!  Well,
square the whole thing and you get something like

                                                (x*x + y*y)^3 = 16*y*y*x*x

which is a sixth-degree polynomial equation in x and y.

Whoopee, you say.  But this sort of polynomial equation is what the rose curves
require.  And if you find a way to graph such an equation (implicitly, without
solving for y), then you can tinker around with the coefficients to see what
other interesting similar curves you can graph.  This is a rich field for
exploration.  There is an old (pre-computer) book by Coolidge on the topic
(published by Dover).  Being able to mess around with this kind of stuff on a
simple calculator is rather mind-boggling, isn't it?

Another technique for graphing functions of the form  r = f(@)  is to let T
represent theta in parametric mode.  Then you can plot  x(T) = f(T)*cos(T)
and  y(T) = similar.  You've got to mess around with the limits over which
you let T run, and the step size, and like that, though.

BTW, the TI-92 is an excellent tool for doing the kind of trig expansions you
might want to do to work with, say,   r = cos(5@) , in rectangular form.
And how about the graph of something like  r = sin(2@)*tan(3@) ?
Have fun!

RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623

>>>> The plural of mongoose begins with p. <<<<