## Polar Equations & Graphs

Circle:

The general equation for a circle with a center at (*r*_{0}, φ) and radius *a* is

*r*^{2} -2*r**r _{0}*cos(

*θ -*

*φ) +r*

_{0}^{2 }= a^{2}

This can be simplified in various ways, to conform to more specific cases,
such as the equation

*r*(*θ) = a*

for a circle with a center at the pole and radius *a.*

Line:

*Radial* lines (those running through the pole) are represented by the
equation

*θ = **φ*

where φ is the angle of elevation of the line; that is, φ = arctan *m* where *m* is the
slope of the line in the Cartesian
coordinate system. The non-radial line that crosses the radial line θ = φ perpendicularly at the point
(*r*_{0}, φ) has the equation

*r*(*θ) = **r _{0}* sec(

*θ -*

*φ)*

Polar Rose:

A polar rose is a famous mathematical curve that looks like a petalled flower, and that can be expressed as a simple polar equation,

*r*(*θ) = *a cos(*kθ + **φ _{0})*

for any constant φ_{0} (including 0). If
*k* is an integer, these equations will produce a *k*-petalled rose if
*k* is odd, or a 2*k*-petalled rose if
*k* is even. If *k* is rational but not an integer, a rose-like shape
may form but with overlapping petals. Note that these equations never define a
rose with 2, 6, 10, 14, etc. petals. The variable *a* represents the length of the petals
of the rose.

Archimedean Spiral:

The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed as a simple polar equation. It is represented by the equation

*r*(*θ) = *a + *bθ*

Changing the parameter *a* will turn the spiral, while * b*
controls the distance between the arms, which for a given spiral is always
constant. The Archimedean spiral has two arms, one for θ > 0 and one for
θ < 0. The two arms are smoothly connected at the pole. Taking the mirror
image of one arm across the 90°/270° line will yield the other arm.