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POLAR COORDINATES

Polar Coordinate System. A coordinate system in which the position of a point P(r, θ) is given by its radial distance r from an origin O and the angle θ measured counterclockwise from a horizontal line OX called the polar axis to line OP. See Fig. 1. The line OP from the origin to the point is called the radius vector, the angle θ is called the polar angle, and the origin O is called the pole.

Transformations between polar and rectangular coordinates. The formulas for converting from rectangular to polar coordinates or vice versa are:

x = r cos θ

y = r sin θ

θ = arc tan y/x

Slope in polar coordinates. Whereas in rectangular coordinates dy/dx represents the slope of the curve y = f(x), in polar coordinates dr/dθ does not represent the slope of the curve r = f(θ). It merely represents the rate of change of the radius vector r with respect to the angle θ. In order to determine the slope in polar coordinates we use the relations

x = r cos θ

y = r sin θ .

Thus

and

Angle ψ from the radius vector to the tangent. The angle ψ from the radius vector to the tangent (measured counterclockwise) is an important angle that plays a role in polar coordinates somewhat similar to that of the slope in rectangular coordinates. It is given by

where r' = dr/dθ. See Fig. 2.

Angle of intersection between two curves. The angle of intersection f12 between two curves C1 and C2 meeting at a point P, not the pole, is given by

where ψ1 and ψ2 are the angles from the radius vector OP to the respective tangents to the curves at P and f12 is the angle measured counterclockwise from the tangent of curve C1 to the tangent of curve C2. See Fig. 3.

Differential of arc length. The differential of arc length is given by

which can also be written as

Curvature. The curvature of a curve at a point P is given by

where r' = dr/dθ.

Finding points of intersection. The points of intersection of two curves whose equations are

r = f1(θ)

r = f2(θ)

can often be found by solving

7)         f1(θ) = f2(θ)

Example. Find the points of intersection of r = 1 + sin θ and r = 5 - 3 sin θ.

Solution. Set 1 + sin θ = 5 - 3 sin θ. We get sin θ = 1. Thus θ = π/2 and (2, π/2) is the only point of intersection.

When a pole is a point of intersection, it may not appear among the solutions of 7).