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Chapter Contents

• Plane Analytical Geometry
• 1. Basic Definitions
• 2. The Straight Line
• Perpendicular Distance from a Point to a Line
• 3. The Circle
• 4. The Parabola
• 5. The Ellipse
• 6. The Hyperbola
• 7. Polar Coordinates
• 8. Curves in Polar Coordinates
• Equi-angular Spiral

• Comments, Questions?

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8. Curves in Polar Coordinates

We'll plot the graphs in this section using a computer. You'll also learn how to sketch some of them on paper because it helps you understand how graphs in polar coordinates work.

Don't worry about all the difficult-looking algebra in the second part of the answers - it's just there to demonstrate that polar coordinates are much simpler than rectangular coordinates for these graphs. We convert them using what we learned in the last section, Polar Coordinates.

Helpful Background

Curves in polar coordinates work very similarly to vectors. See:

Vector concepts

Examples

Sketch each of the following functions using polar coordinates, and then convert each to an equation in rectangular coordinates.

(1) r = 2 + 3 sin θ

(This polar graph is called a limacon from the Latin word for "snail".)

Answer

(2) r = 3 cos 2θ

Answer

(3) r = sin θ - 1

(This one is called a cardioid because it is heart-shaped. It is a special case of the limacon.)

Answer

(4) r = 2.5

Answer

(5) r = 0.2 θ

This is an interesting spiral. As θ increases, so does r.

Answer

See also Equiangular Spiral.

(6) r = sin (2^θ) − 1.7

This is the face I drew at the top of this page. We're not even going to try to find the equivalent in rectangular coordinates!

Answer

Application

Check out Polar Coordinates and Cardioid Microphones for an application of polar coordinates.