Polar Coordinates

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 .

The two polar coordinates ''r'' (the radial coordinate) and ''θ'' (the angular coordinate or polar angle), sometimes represented as ''φ'' or ''t'', are defined in terms of Cartesian Coordinates by
:

x = r \cos 	heta \,

y = r \sin 	heta \,

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.

From those two formulas, conversion formulas to go from polar coordinates to Cartesian coordinates are derived, including
:

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

heta = \arctan rac{y}{x} \,

For example, if you had 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.

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''.

COMMON POLAR EQUATIONS

Line

A Line can be expressed as a polar equation in two different ways, depending on whether it runs through the pole.

If a line does run through the pole, its equation can be represented by the equation
:

heta = arphi \,

, where φ is the angle of elevation of the line, or
:

heta = \arctan(m) \,

, where ''m'' is the slope of the line in the Cartesian Coordinate System .

If a line does not run through the pole, but runs through the point (''r''₀, φ), its equation is
:

r(	heta) = rac{r_0}{\cos(	heta-arphi)} \,

,
which will generate a line perpendicular to the line θ = φ.

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(	heta)=a \,

For a circle with a center at (''r₀'', ''φ'') and radius ''r₀'' the equation is
:

r(	heta)=2r_0 \cos(	heta-arphi) \,

For any circle with a center at (''r₀'', ''φ'') and radius ''a'' the equation is
:

r^2 - 2 r r_0 \cos(	heta - arphi) + r_0^2 = 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(	heta) = a \pm b \cos 	heta \,

OR
:

r(	heta) = a \pm b \sin 	heta \,

Cardioid

A Cardioid is a special Limaçon where ''a'' and ''b'' are equal. It is it given by the equations
:

r(	heta) = a \pm a \cos 	heta \,

OR
:

r(	heta) = a \pm a \sin 	heta \,

Lemniscate

A Lemniscate is a mathematical curve which looks like a Figure Eight . It is it given by the equations
:

r^2 = a \cos 	heta \,

OR
:

r^2 = a \sin 	heta \,

Polar Rose

A Polar Rose is a mathematical curve which looks like a petalled flower. It is given by the equations
:

r(	heta) = a \cos(k	heta) \,

OR
:

r(	heta) = a \sin(k	heta) \,

.
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.

Spiral of Archimedes

The Archimedean Spiral is a spiral that was discovered by Archimedes . It is represented by the equation:
:

r(	heta) = a+b	heta \,

.
Changing the parameter ''a'' will turn the spiral, while ''b'' controls the distance between the arms.

COMPLEX NUMBERS

Complex Numbers , written in rectangular form as

a + bi \,

, can also be expressed in polar form in two different ways:
#

r(\cos	heta+i\sin	heta) \,

, abbreviated

r \mbox{ cis } 	heta \,

(r \angle 	heta) \,

r e^{i	heta} \,

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 	heta \,

b = r \sin 	heta \,

: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: $(r \mbox{ cis } heta) --- (R \mbox{ cis } arphi) = rR \mbox{ cis } ( heta+arphi) \,$

Division: $rac{r \mbox{ cis } heta}{R \mbox{ cis } arphi} = rac{r}{R} \mbox{ cis } ( heta-arphi) \,$

Exponentiation ( De Moivre's Formula ): $(r \mbox{ cis } heta)^n = r^n \mbox{ cis } (n heta) \,$

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Polar Coordinates