Circles

The most basic circle is r = constant i.e. the set of all points equidistant from the origin.

There are s special circles r = acos(Θ) and r = asin(Θ)

Below we see the animations but why are these circles? This will be one of the
few times we'll compare polar and rectangulat coordinates. x = rcos(Θ) and y = r sin(Θ)

x^{2} + y^{2} = r^{2} and tan(Θ) = y/x .

Suppose r = cos(Θ) . Multiply each side by r : r^{2}
= rcos(Θ). We obtain:

x^{2} + y^{2}
= x rearranging we obtain x^{2} - x + y^{2} = 0. ompleting the square
we obtain

(x-1/2)^{2} + y^{2} = 1/4 which we recognize as
the equation of a circle of radius 1/2

centered at (1/2,0). Show r = sin(Θ) is a circle centered at (0,1/2) of radius 1/2.

Spirals

Functions of the form r = f(Θ) where f(Θ) is not a trig function generate spirals.

To plot spirals simply plot values using the usual suspects : 0,π/2,π,3π/2,2π
etc.

In the following animations r =Θ is a a spiral
where the bands are equally spaced .

With the exponential spiral
r = e ^{.1Θ} the spacings increase and with the logarithmic spiral

r =ln(Θ) the spacings decrease.

Then
we consider two spirals which spiral inward an exponential f =e ^{-.1Θ} and a hyperbolic spiral r = 1/(1+Θ)