How do you graph #r^2= - cos theta#?

Here, I introduce the meaningful interpretations for the presence of

#kr, r^m and m theta#, in polar equations. This is important to

understand equations that have some family characteristics.

Examples:

#r = - cos theta, r^2 = -cos theta, r = - cos 2theta#,

# r^2 = - cos 2theta#, and so on.

On par with scaling x and y in Cartesian frame,

power scaling of r, #r^2# increases/reduces lengths and just

multiplication of #theta# by scalars, like #2theta, theta/2# produce

slower/faster rotations, about the pole.

See the combined graph for #r = - cos theta, r^2 = - cos theta,#

#r = - cos 2 theta and r^2 = - cos 2theta#.

It would be interesting to know, which is which.

The graph of #r^2 = - cos theta# is the elongated circle. Compared

to the circle #r = - cos theta#, the point #(1/sqrt2, 3pi/4)# moves to

# (1/sqrt(sqrt2), 3pi/4)#, in that direction. See the graph, for this r-

scaling effect.

graph{(x^2+y^2+x)((x^2+y^2)^1.5+x)((x^2+y^2)^1.5+x^2-y^2)((x^2+y^2)^2+x^2-y^2)(y+x)((x+0.6)^2+(y-.6)^2 - .0015)((x+0.5)^2+(y-0.5)^2-0.001)=0[-2 2 -1.1 1.1]}