Polar Coordinates

In polar coordinates, points on a plane are defined by radial distance $r$ and angular coordinate $\theta$ (Figure 2). The radial direction measures the point's distance from the origin, while the angular direction corresponds to the counterclockwise angle from the positive x-axis. These two dimensions together specify any point in the plane.

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Assets 1

Figure 1:An interacitve comparison of Cartesian and polar coordinates.

To convert from Cartesian coordinates $(x, y)$ to polar coordinates $(r, \theta)$ :

\begin{bmatrix}r \\ \theta \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2} \\ \text{atan2}(y, x) \end{bmatrix}

(1)

To convert from polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$ :

\begin{bmatrix}x \\ y \end{bmatrix} = r \cdot \begin{bmatrix} \cos(\theta) \\ \sin(\theta) \end{bmatrix}

(2)

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Assets 2

Figure 2:Polar coordinate system

Coordinate Systems

Assets 1