Polar Coordinate System

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)

x² + y² = r² and tan(Θ) = y/x .

Suppose r = cos(Θ) . Multiply each side by r : r² = rcos(Θ). We obtain:

x² + y² = x rearranging we obtain x² - x + y² = 0. ompleting the square we obtain

(x-1/2)² + y² = 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.

r = cos(a)

r = sin(a)

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)

spiral

exponential spiral

logarithmic spiral

inward exponential file

inward hyperbolic spiral

Lecture Notes Cardioids

Animation Cardioid Sines

Animation Cardioid Cosines

The lecture notes on limacons also conrtain a breif discussion of derivatives in Polar coordinates. For a complete discussion see the page Polar Coordinates - Derivatives and Integrals.

Lecture Notes Limacons

Animation Limacon1

Lecture Notes sin(nt) cos(nt)

Animation cos(3t)

Animation sin(2t)

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