PEPCVF - Polar Coordinates and Graphs

Polar Coordinates and Graphs

Polar Coordinates

Polar coordinates offer an alternative way of locating a point in the plane. Introduced by Newton, the polar coordinate system has a point O in the plane called the pole, which is analogous to the origin in the Cartesian coordinate system. A horizontal ray beginning at the pole and extending to the right is called the polar axis and corresponds to the positive x-axis in Cartesian coordinates. If P is any other point in the plane, its polar coordinates r and  tell respectively, the directed distance from the pole O to P and the directed angle in radians from the polar axis (sometimes referred to as the initial ray) to the ray from O to P. The point P is represented by the ordered pair $\left(r,\:\theta\right)$. Polar angles are considered positive if measured in the counterclockwise direction from the polar axis and negative in the clockwise direction. If P = O, then r = 0 and $\left(0,\:\theta\right)$ represents the pole for any value of $\theta$.

While a given set of polar coordinates determines just one point, a given point has infinitely many sets of polar coordinates. Unlike Cartesian coordinates, the polar coordinate systems fails to establish a one-to-one correspondence between the points of the plane and ordered pairs of real numbers. For example, the coordinates $\left(r,\:\theta\right)\:\text{and}\:\left(r,\:\theta+2\pi\right)$ represent the same point, as do the coordinates $\left(r,\:\theta\right)\text{ and }\left(-r,\theta+\pi\right)$. In general, the point $\left(r,\:\theta\right)$ can be written as $\left(r,\:\theta+2n\pi\right)\text{ or }\left(-r,\theta+\left(2n+1\right)\pi\right)$, where n is any integer.

The presentation below introduces polar coordinates and how to plot them in the polar system.

Coordinate Conversion

The polar coordinates $\left(r,\:\theta\right)$ of a point are related to the Cartesian coordinates (x, y) of the point as follows: $x=rcos\theta,\:y=rsin\theta,\:r^2=x^2+y^2,\:and\:\tan\theta=\frac{y}{x}$

A comparison of polar and rectangular coordinates is the focus of the next presentation.

Converting polar equations to rectangular equations is illustrated below.

Polar Graphs

Many important types of graphs have equations that are much simpler in polar form than in rectangular (Cartesian) form. One example is a circle of radius a and centered at the origin whose polar equation is r = a, which is much simpler than its rectangular equation x² + y² = a². View the presentation below illustrating polar graphs of circles and a line and the polar equation's relationship to its corresponding Cartesian equation.

Other special types of polar graphs and how to graph them using a graphing calculator are introduced in the next presentation.

Just as the path of a point can be traced based on how the parameter varies in parametric equations, a polar trace as the angle increases can provide the location and direction traveled when polar equations are used. An animation involving graphs of polar equations follows.

Because many polar curves possess symmetry, it is often helpful to use this property in predicting and understanding the behavior of a curve. Sufficient conditions for symmetry are as follows:

1. If a polar equation is unchanged when $\theta$ is replaced by $-\theta$, the curve is symmetric about the polar axis.

2. If a polar equation is unchanged when r is replaced by -r, or when $\theta$ is replaced by $\theta+\pi$, the curve is symmetric about the pole.

3. If a polar equation is unchanged when $\theta$ is replaced by $\pi-\theta$, the curve is symmetric about the vertical line $\theta=\frac{\pi}{2}$.

General formulas for special polar graphs are often useful in predicting the behavior of polar curves.

Polar Coordinates and Graphs Practice

Polar Coordinates and Graphs: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of polar coordinates and graphs.

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