Polar and Cylindrical Coordinates

The Cartesian coordinate system can describe the linear motion of an object and its dynamics. However, rotations are simpler to portray using polar coordinates.

In the polar coordinate system, a vector is defined with two scalar components: the radial component and the polar angle.

The radial component specifies the radial distance of that vector from the origin.

The polar angle indicates the angle between the vector and the positive x-axis.

Here, the orthogonal unit vectors are along the radial direction and perpendicular to it.

If the scalar components of a vector are known in polar coordinates, then its components in the Cartesian coordinate system can be obtained.

Cylindrical coordinates are a three-dimensional generalization of the polar coordinates and are convenient for describing systems with cylindrical symmetry.

This system defines a vector using the radial distance, azimuthal angle, and z direction.

The first two scalar components are similar to the ones in polar coordinates, while the third represents the height from the xy plane.

The transformation equations convert a vector in cylindrical coordinates to cartesian coordinates.