To start off, the origin (mathematics) is the reference point we use to define all other points; a fixed point we refer to when noting the geometry of a space. With this in mind, it is often denoted by O, and the coordinates are always zero.

## The Origin in One, Two, and *n* Dimensions

In one dimension we simply write the origin as 0; it’s the point where we start numbering on a number line. With this in mind, you can go in either of two directions. Going left, you would count off negative numbers; going right, you would count off with positive numbers. Either way you can go an infinite distance (to infinity or negative infinity).

In two dimensions, using Cartesian coordinates, we can define an origin as the point where the x and y axes intersect. This point is written as (0, 0).

In three dimensions, the origin is (0, 0, 0) and is defined as the place the x, y, and z axis intersect.

In *n* dimensions the origin will be the place *n* axes intersect, and every coordinate will be zero.

## Polar Coordinate Systems

Following this we get to the polar coordinate system for this the origin is also called a *pole*. Following this, very point is defined in terms of an *angle *and a *ray *(a line that starts at given coordinates and extends to infinity) from the pole. The pole on a polar coordinate system is (0,φ), where 0 is the intersection of the axes and φ is an angle where we rotate to.

## Shifts

An origin is arbitrary, and can be shifted for convenience. If you are working in two dimensional Cartesian coordinates and shift your origin from (0, 0) to a place that was defined as (a, b), you will need to add (-a, -b) to the coordinates of every point.

For example, suppose you decide to change your origin to the point you’ve been calling (2, 4). Next (2, 4) is (0, 0) = O, your former (0, 0) is (-2, -4), and the point (3, 5) is (1, 1).

If you’re working with three dimensional geometry and want to switch to the point previously referred to as (6, 7, 8), you would translate each point from the old system to the new system by adding (-6, -7, -8) to the coordinates of each point. Following this, the point (20,20,20) would become (14, 13, 12).

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