Types of Polar Graphs

Types of Polar Graphs:

Types of Polar Graphs

Graphing Equations:

Graphing Equations To determine the type of graph, all you have to do is look at the equation. To graph it, simply plug in values for Θ and plot the resultant points. There are five kinds of common polar graphs. The graphs are symmetrical over either the polar axis (x-axis) or the vertical axis (y-axis). If you have an equation with sine, it will be symmetrical over the vertical axis. If you have an equation with cosine, it will be symmetrical over the polar axis.

Circles:

Circles The polar equation for a circle is either: If the equation has cosine, the circle will be on the x-axis; if it has sine, it will be on the y-axis. The circle’s diameter is the absolute value of a. If a > 0, the circle will be on the positive side of the axis. If a < 0, the circle will be on the negative side of the axis.

Limaçons:

Limaçons Pronounced “ lim -uh-sawn” There are four types of limaçons : inner looped, cardioid , dimpled, and convex. A glance at the equation will tell you which type it is. Like the circle, if the equation contains cosine, the limaçon will be on the x-axis; if the equation contains sine, the limaçon will be on the y-axis.

Limaçon Equations:

Limaçon Equations To tell which type of limaçon you have, look at a and b. a < b, you have a limaçon with an inner loop a = b, you have a cardioid limaçon b < a < 2b, you have a dimpled limaçon a ≥ 2b, you have a convex limaçon

More on the equations:

More on the equations The diameter of an inner loop limaçon is equal to the value of a+b . The diameter of a cardioid limaçon is equal to 2a. The diameter of a dimpled limaçon is equal to the value of a+b . The diameter of a convex limaçon is equal to the value of a+b .

Limaçon Types:

Limaçon Types Inner loop Cardioid Dimpled Convex

Roses:

Roses A rose is given that name because it looks like a flower. How original. There are a couple of catches. First, you must have n ≥ 2. If n was 1, you would have the same equation as a circle. Second, n needs to be an integer. You can still make a graph with a decimal, but you won’t have a complete rose. It would have petals that are only half-drawn. If n is odd, you will have n petals. If n is even, you will have 2n petals. The length of the petal is the absolute value of a.

Roses:

Roses 2cos(3 Θ ) 3cos(4 Θ ) 4sin(5 Θ ) 2sin(2 Θ )

Lemniscates:

Lemniscates A lemniscate looks like the infinity symbol. It is symmetric around the pole. The length of the petal is the absolute value of a.

Spirals of Archimedes:

Spirals of Archimedes These are famous spirals that also appear in nature. Although named for Archimedes, he did not discover it; however, he did use it in a 225 B.C.E. treatise about spirals. If Θ > 0, the spiral starts at the pole and spirals away to the left. If Θ < 0, the spirals starts at the pole and spirals away to the right. (i.e. if a is positive, go left; if a is negative, go right)