Polar Coordinates Fundamentals

The Polar Coordinate system is an alternate coordinate system (as opposed to the Cartesian Coordinate system).

Building a Polar Coordinate System

1. Draw a point $O$ which will be the origin. The origin is called the pole in the Polar Coordinate system.
2. Draw an initial ray from $O$ to the right (with infinite length). This will be the polar axis.
3. Each point in the plane can now be assigned the polar coordinates $(r, \theta)$.

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Locating Points

Even though a point in a Polar Coordinate system is $(r, \theta)$, you should consider $\theta$ first when locating points.

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Uniqueness of Points

For any point in 3-dimensional space, there are an infinite number of coordinates which will identify the point, in the Polar Coordinate system.

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Converting Polar Coordinates to Rectangular Coordinates

To translate a point in Polar Coordinates to Rectangular (i.e. Cartesian) Coordinates, use the following formulas:

X-Coordinate	Y-Coordinate
$$\cos(\theta) = \frac{x}{r}$$	$$\sin(\theta) = \frac{y}{r}$$
$$r\cdot\cos(\theta) = \frac{x}{r}\cdot r$$	$$r\cdot\sin(\theta) = \frac{y}{r}\cdot r$$
$$\boldsymbol{x=r\cos(\theta)}$$	$$\boldsymbol{y=r\sin(\theta)}$$

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Converting Rectangular Coordinates to Polar Coordinates

To translate a point in Rectangular (i.e. Cartesian) Coordinates to Polar Coordinates, use the following formulas:

r-Coordinate	$\theta$-Coordinate
To find the $r$-coordinate, use the __Pythagorean Theorem:__	To find the $\theta$-coordinate, use $\tan(\theta)$:
$$r^2=x^2+y^2$$	$$\tan(\theta) = \frac{y}{x}$$

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Symmetry in Polar Equations

Symmetry About the X-Axis

Occurs if the point $(r, \theta)$ is on the graph whenever the point $(r, -\theta)$ is on the graph.

Symmetry About the Y-Axis

Occurs if the point $(r,\theta)$ is on the graph whenever $(r, \pi-\theta) = (-r,-\theta)$ is on the graph.

Symmetry About the Origin

Occurs if the point $(r,\theta)$ is on the graph whenever $(-r,\theta)=(r,\theta+\pi)$ is on the graph.

Confusing, right?

Refer the the illustrations below; they depict each of the symmetry scenarios described above:

Note: Whenever you see any two of these symmetries together, the third is implied. That is: if a graph is symmetric about both the $\text{x-axis}$ and the $\text{y-axis}$, then it must be symmetric about the origin.

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Graphing Polar Equations

Stub

TODO: Fill in notes from corresponding textbook sections

Rose Curves

A common curve graphed in polar coordinates is the Rose Curve. This curve is so-named because the shape resembles a flower.

The general equation for a rose curve has some common features:

$$r=\boldsymbol{A}\cos(\boldsymbol{B}\theta)$$

Remember from Trigonometry: the coefficient $A$ defines the curve's amplitude, and the coefficient $B$ defines the curve's period (frequency).

$$\boldsymbol{A =}\textbf{Amplitude} \\ \boldsymbol{B =}\textbf{Period (frequency)}$$

For the rose curve, there are some addition properties of the $\boldsymbol{B}$ coefficient:

$\boldsymbol{B}$ is even $\implies$ Rose has $\boldsymbol{2B}$ petals
$\boldsymbol{B}$ is odd $\implies$ Rose has $\boldsymbol{B}$ petals
($B$ must be an integer)