Antipode
Definition
Let a point $p$ on the 2-dimensional unit circle or the 3-dimensional unit sphere be given. The point opposite to $p$ — that is, among the two intersection points of the line through $p$ and the center of the sphere with the sphere, the point other than $p$ — is denoted $-p$ and is called the antipodal point of $p$.
The ordered pair $(p, -p)$ is called antipodes.
Definition in geography
In geography, the antipodal point of a location on Earth is the point on the Earth’s surface that lies diametrically opposite that location.
Explanation
The term “antipodal point” is also used colloquially in phrases like “they are antipodal to each other.” It denotes situations where two opinions are sharply opposed, or where properties are entirely different — i.e., mutually opposite. The mathematical/scientific definition is not different: an antipodal point is the point farthest from a given reference point.
The coordinates of antipodal points in 2D and 3D are as follows. Informally, add $\pi$ to every angular component.
| 2D | reference point | antipodal point |
|---|---|---|
| Cartesian coordinates | $(x, y)$ | $(-x, -y)$ |
| polar coordinates | $(r, \theta)$ | $(r, \theta + \pi)$ |
| 3D | reference point | antipodal point |
|---|---|---|
| Cartesian coordinates | $(x, y, z)$ | $(-x, -y, -z)$ |
| spherical coordinates | $(r, \theta, \phi)$ | $(r, \theta + \pi/2, \phi + \pi)$ |