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 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 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).