Polar coordinate system

From Exampleproblems

The polar coordinate system is a two-dimensional coordinate system in which points are given by an angle and a distance from the pole, called the origin in the Cartesian coordinate system.

File:Polar graph.gif

A sheet of polar graph paper.

The polar coordinates r (the radial coordinate) and θ (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by

$x = r \cos \theta \,$

$y = r \sin \theta \,$

and therefore $r = \sqrt{x^2 + y^2} \,$

where r is the radial distance from the pole, and θ is the counterclockwise angle from the 0° ray, which is the section of the Cartesian x-axis from the origin eastward.

For example, if you have the coordinates (3, 60°), the point would be plotted 3 units from the origin on the 60° ray. If you had the coordinates (-3, 240°), the point would be in the same location, because -3 units on the 240° ray is the same as 3 units on the 60° ray.

File:CircularCoordinates.png

The equation of a curve expressed in polar coordinates is known as a polar equation, and is usually written with r as a function of θ.

Polar equations of common figures

File:Circle r=1.PNG

A circle with equation r(θ) = 1.

File:Limacon r=.75+1.5cos(theta).PNG

A limacon with equation r(θ) = 3/4 + 3/2 cosθ

File:Cardioid r=1-sin(t).PNG

A cardioid with equation r(θ) = 1 - sinθ.

File:Lemniscate r=sqrt(cos(2theta)).PNG

A lemniscate with equation r² = cos2θ.

File:Rose r=2sin(4theta).PNG

A polar rose with equation r(θ) = 2sin4θ.

Circle

The are several ways to write the polar equation of a circle, which conform to circles at different locations and of different sizes.

For a circle with a center at the pole and radius a the equation is

$r(\theta)=a \,$

For a circle with a center at (r₁, θ₁) and radius |r₁| the equation is

$r(\theta)=2r_1 \cos(\theta-\theta_1) \,$

For any circle with a center at (r₁, θ₁) and radius a the equation is

$r^2 - 2 r r_1 \cos(\theta - \theta_1) + r_1^2 = a^2 \,$

Limaçon

A limaçon (pronounced leem-ah-son), also known as a limaçon of Pascal, is a heart-shaped mathematical curve. It is given by the equations

$r(\theta) = a \pm b \cos \theta \,$ OR

$r(\theta) = a \pm b \sin \theta \,$

Cardioid

A cardioid is a special limaçon where a and b are equal. It is it given by the equations

$r(\theta) = a \pm a \cos \theta \,$ OR

$r(\theta) = a \pm a \sin \theta \,$

Lemniscate

A lemniscate is a mathematical curve which looks like a figure eight. It is it given by the equations

$r^2 = a \cos \theta \,$ OR

$r^2 = a \sin \theta \,$

Polar Rose

A polar rose is a mathematical curve which looks like a petalled flower. It is given by the equations

$r(\theta) = a \cos(k\theta) \,$ OR

$r(\theta) = a \sin(k\theta) \,$

These equations will produce a k-petalled rose if k is odd, or a 2k-petalled rose if k is even. Note that it is impossible to make a rose with 2 more than a multiple of 4 (2, 6, 10, etc.) petals.

Complex numbers

Complex numbers, written in rectangular form as $a + bi \,$ , can also be expressed in polar form in two different ways:

$r(\cos\theta+i\sin\theta) \,$ , abbreviated $r \mbox{ cis } \theta \,$ or $(r \angle \theta) \,$
$r e^{i\theta} \,$

of which both are equivalent as per Euler's formula. To convert between rectangular and polar complex numbers, the following conversion formulas are used:

$a = r \cos \theta \,$

$b = r \sin \theta \,$

and therefore $r = \sqrt{a^2 + b^2} \,$

For the operations of multiplication, division, and exponentiation, and finding roots of complex numbers, it is much easier to use polar complex numbers than rectangular complex numbers. In abbreviated form:

Multiplication: $(r \mbox{ cis } \theta) * (R \mbox{ cis } \phi) = rR \mbox{ cis } (\theta+\phi) \,$
Division: $\frac{r \mbox{ cis } \theta}{R \mbox{ cis } \phi} = \frac{r}{R} \mbox{ cis } (\theta-\phi) \,$
Exponentiation (De Moivre's formula): $(r \mbox{ cis } \theta)^n = r^n \mbox{ cis } (n\theta) \,$