Greenwood press

PLANE
71
A screw uses a rotating inclined
plane to drill into flat surfaces.

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POLAR COORDINATES
Polar coordinates locate a point by indicating a direction and a distance from
a central point (the pole). In the point 
P = [r, θ], the distance r is given first, fol-
lowed by the direction 
θ expressed as an angle of rotation from a fixed line
through the pole called the polar axis. This is different from Cartesian or rectan-
gular coordinates in which points are located by distances from two perpendicu-
lar axes. When the polar axis corresponds to the positive 
x-axis in a Cartesian
plane, the Cartesian coordinates 
(x, y) for P can be computed as x = r cos θ and
y = r sin θ. While Cartesian graphs are rectangular, polar graphs are circular.
Notice that
x
2
+ y
2
= (r cos θ)
2
+ (r sin θ)
2
= r
2
(cos θ)
2
+ r
2
(sin θ)
2
=
r
2
(cos
2
θ + sin
2
θ) = r
2
(1) = r
2
,
which is the same as the standard equation of a circle with the center at the ori-
gin and radius 
r.
Polar coordinates can be used to map the earth. The figure below is a CIA
(Central Intelligence Agency) map showing the northern hemisphere. The North
Pole is in the center of the concentric circles of latitude. The polar axis is on the
great circle of longitude that goes through Greenwich, England. On polar graph
paper, this axis would be placed in the same position as the positive ray of the 
x-
axis in Cartesian coordinates.
Navigators on ships and airplanes use the language of polar coordinates to
specify the direction and speed of travel. Astronomers use polar coordinates to
plot paths of planets and the sun with respect to a viewing position on the earth.
Polar coordinates are useful in mathematics for writing curves that cannot be
written as functions or simple relations in 
x- and y-coordinates. The following
graph of a five-leaf rose would be difficult to express in an equation using only
x–y coordinates.