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Pole

The point in a polar coordinate system used as a reference. Analogous to the origin in a Cartesian coordinate system.

Polar Axis

A fixed ray with origin at the pole, usually horizontal, from which an angle can be established. Analogous to the x-axis in a Cartesian coordinate system.

Corresponding Polar Coordinates

A point with polar coorinates (r, θ), θ in radians, can also be represented by either:

Conversion from Polar to Rectangular Coordinates

If P is a point with polar coordinates (r,θ), the rectangular coordinates (x,y) of P are given by

Conversion from Rectangular to Polar Coordinates

If P is a point with rectangular coordinates (x,y):

Use formulas

Transforming Equations between Polar and Rectangular Forms

Polar Equation

An equation whose variables are polar coordinates.

Graph of a Polar Equation

All points whose polar coordinates satisfy a particular polar equation.

Straight Line Graphs of a Polar Equation

Let a be a nonzero real number. The graph of equation

Circle Graphs of a Polar Equation

Let a be a positive real number. Then

Cardioid

r=a+asinθ

Limaçon (no inner loop)

A polar equation of the form

Limaçon (inner loop)

A polar equation of the form

Rose Curve

A polar equation of the form

Lemniscate

r^2= a^2cos2(theta)

Logarithmic Spiral

A polar equation of the form