Category:Conversion between Cartesian and Polar Coordinates in Plane

This category contains pages concerning Conversion between Cartesian and Polar Coordinates in Plane:

Let $S$ be the plane.

Let a Cartesian plane $\CC$ be applied to $S$.

Let a polar coordinate plane $\PP$ be superimposed upon $\CC$ such that:

$(1): \quad$ The origin of $\CC$ coincides with the pole of $\PP$.

$(2): \quad$ The $x$-axis of $\CC$ coincides with the polar axis of $\PP$.

Let $p$ be a point in $S$.

Let $p$ be specified as $p = \polar {r, \theta}$ expressed in the polar coordinates of $\PP$.

Then $p$ is expressed as $\tuple {r \cos \theta, r \sin \theta}$ in $\CC$.

Contrariwise, let $p$ be expressed as $\tuple {x, y}$ in the cartesian coordinates of $\CC$.

Then $p$ is expressed in $\PP$ as:

$p = \polar {\sqrt {x^2 + y^2}, \arctan \dfrac y x + \pi \sqbrk {x < 0 \text{ or } y < 0} + \pi \sqbrk {x > 0 \text{ and } y < 0} }$

where:

$\sqbrk {\, \cdot \,}$ is Iverson's convention.

$\arctan$ denotes the arctangent function.