Greenwood press

POLAR COORDINATES
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Graph of 
r = 4 cos 5θ for θ = 0 to 2π.
Some graphs are easier to represent in 
polar rather than rectangular form, 
especially if they have rotational symmetry.
logarithmic spiral 
r = 5(1.3)
θ
shell of a chambered nautilus
A nautilus resembles the polar graph 
r = ab
θ
.
The chambered nautilus picture from the
previous figure after being distorted with the
polar coordinate filter in Adobe Photoshop.

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Log-polar transformations have been developed to embed copyright infor-
mation in computer-graphic files to preserve the copyright notice from deletion. 
Spherical polar coordinates are useful in simplifying physics equations such
as Schrodinger’s equation and the Maxwell speed equation. In many cases, writ-
ing expressions in polar form simplifies the application of calculus and differen-
tial equation techniques.
Polar coordinates have another important application in mathematics: They
simplify some operations with complex numbers. Multiplication of the complex
numbers
z = a + bi and w = c + di gives (ac − bd) + (ad + bc)i. The corre-
sponding multiplication in polar form would be of the numbers 
z = [r, θ] and
w = [s, φ]. Then zw = [rs, θ + φ]. The polar form simplifies the powers and
roots of complex numbers. In polar form, 
z
n
= [r
n
, nθ], which is known as De-
Moivre’s theorem. (See Complex Numbers and Vectors.)
online sources for further exploration
Polar plotting and graphing




Spherical and cylindrical coordinates



Logarithmic spirals



Spiral.html>
Sun position in polar coordinates

Azimuth and elevation

Computer graphics



Polar coordinates in robotics

Polar distortion filter

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