01polar.md

Polar Coordinates

In polar coordinates, points on a plane are defined by radial distance $r$ and angular coordinate $\theta$ (). 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.

:::{figure} #my-mafs-polar :name: #fig-my-mafs-polar :width: 200px An interacitve comparison of Cartesian and polar coordinates. :::

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

\begin{equation} \label{eq:trans} \begin{bmatrix}r \ \theta \end{bmatrix}

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

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

$$ \label{eq:transback} \begin{bmatrix}x \ y \end{bmatrix} = r \cdot \begin{bmatrix} \cos(\theta) \\ \sin(\theta) \end{bmatrix} $$

:::{figure} #polar_static :name: polar_overview :width: 50px Polar coordinate system :::

::::{note} Animations

:::{figure} #polar_radial :name: fig-polar_radial :width: 50px Radial direction :::

:::{figure} #polar_angular :name: fig-polar_angular :width: 50px Angular direction :::

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