Examples of Polar Coordinates Help (page 2)

Examples of Polar Coordinates—Circles

To see 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. 5-3.

Fig. 5-3. Polar graph of a circle centered at the origin, with radius a.

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. 5-4) is as follows:

r = 2r ₀ cos ( θ – θ ₀ )

Fig. 5-4. Polar graph of a circle passing through the origin, with center at ( θ ₀ , r ₀ ) and radius r ₀ .

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. 5-5. The values a and 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. 5-5. Polar graph of an ellipse centered at the origin, with semi-axes a and b.

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. 5-6). 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 a and 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. 5-6. Polar graph of a hyperbola centered at the origin, with semi-axes a and b.

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 2θ ) ^1/2

where a is a real-number constant greater than 0, representing the maximum radius. This is illustrated in Fig. 5-7.

Fig. 5-7. Polar graph of a lemniscate centered at the origin, with radius a.