Examples using Polar Coordinates Help (page 2)

Introduction to Examples Using Polar Coordinates - Circles

In order to get a good idea of how the polar coordinate system works, let’s look at the graphs of some familiar objects. Circles, ellipses, spirals, and other figures whose equations are complicated in Cartesian coordinates can often be expressed much more simply in polar coordinates. In general, the polar direction θ is expressed in radians. In the examples that follow, the “rad” abbreviation is eliminated, because it is understood that all angles are in radians.

Circle Centered At Origin

The equation of a circle centered at the origin in the polar plane is given by the following formula:

r = a

where a is a real-number constant greater than 0. This is illustrated in Fig. 10-3.

Fig. 10-3 . Polar graph of a circle centered at the origin.

Circle Passing Through Origin

The general form for the equation of a circle passing through the origin and centered at the point ( θ ₀ , r ₀ ) in the polar plane (Fig. 10-4) is as follows:

r = 2 r ₀ cos ( θ – θ ₀ ) 

Fig. 10-4 . Polar graph of a circle passing through the origin.

Ellipse Centered At Origin

The equation of an ellipse centered at the origin in the polar plane is given by the following formula:

r = ab /( a ² sin ² θ + b ² cos ² θ ) ^1/2

where a and b are real-number constants greater than 0.

In the ellipse, a represents the distance from the origin to the curve as measured along the “horizontal” ray θ = 0, and b represents the distance from the origin to the curve as measured along the “vertical” ray θ = π/2. This is illustrated in Fig. 10-5. The values 2 a and 2 b represent the lengths of the semi-axes of the ellipse; the greater value is the length of the major semi-axis , and the lesser value is the length of the minor semi-axis .

Fig. 10-5 . Polar graph of an ellipse centered at the origin.

Hyperbola Centered At Origin

The general form of the equation of a hyperbola centered at the origin in the polar plane is given by the following formula:

r = ab /( a ² sin ² θ − b ² cos ² θ ) ^1/2

where a and b are real-number constants greater than 0.

Let D represent a rectangle whose center is at the origin, whose vertical edges are tangent to the hyperbola, and whose vertices (corners) lie on the asymptotes of the hyperbola (Fig. 10-6). Let a represent the distance from the origin to D as measured along the “horizontal” ray θ = 0, and let b represent the distance from the origin to D as measured along the “vertical” ray θ = π/2. The values 2 a and 2 b represent the lengths of the semi-axes of the hyperbola; the greater value is the length of the major semi-axis , and the lesser value is the length of the minor semi-axis .

Fig. 10-6 . Polar graph of a hyperbola centered at the origin.

Lemniscate

The general form of the equation of a lemniscate centered at the origin in the polar plane is given by the following formula:

r = a (cos 2 θ ) ^1/2

where a is a real-number constant greater than 0. This is illustrated in Fig. 10-7. The area A of each loop of the figure is given by:

A = a ²

Fig. 10-7 . Polar graph of a lemniscate centered at the origin.