Coordinates

A coordinate system is a system assigning a number (or tuple of numbers) to any point lying in a subregion of an entity. The number (or tuple of numbers) assigned to a point is called the coordinate (or coordinates) of the point.

An atlas is a set of coordinate systems, such that their corresponding subregions cover the entity.

Examples

Longitude and latitude are a coordinate system for the surface of the earth.

If one picks two oriented lines on a plane which meet at right angles, then one can use them to define a coordinate system for the plane, called "cartesian" coordinates. The cartesian coordinates of a point $p$ on the plane are two numbers $(x_{p},y_{p})$, determined as follows. First, order the two lines using the right-hand rule: the cross product of the first line with second line should point away from you. Project $p$ onto the first line, and let $x_{p}$ denote the (signed) distance between its projection and the origin. Similarly, project $p$ onto the second line, and let $y_{p}$ denote the (signed) distance between its projection and the origin.

Using a similar procedure, if one picks a ray on a plane, one can use it to define a "polar" coordinate system on the plane.

Entity vs. Manifold

[TODO]