antipodal

Antipodal describes a pair of points that lie on opposite sides of a sphere, so that the line through one point and the sphere’s center extends to the other point. On a sphere, the antipode of a point p is the point opposite p such that the center is the midpoint of the segment joining them. The concept is widely used in geometry, geography, and related fields.

In geography, the antipode of a location on Earth is the point on the Earth's surface that is diametrically opposite. If a point has latitude φ and longitude λ, its antipode has latitude −φ and longitude λ±180 degrees (modulo 360). Because land and sea distribution is uneven, most true land antipodes lie in the ocean.

In mathematics, antipodal points are central to the study of spheres and related spaces. The antipodal map on the unit sphere S^n is the function that sends x to −x, pairing each point with its opposite. This map is a symmetry of the sphere and is used in various geometric and topological arguments. The concept also appears in more advanced topics, such as the degree of the antipodal map and its role in certain fixed-point and continuity theorems.

The term antipodal comes from the Greek anti- “opposite” and pod “foot.” Beyond geometry, it appears in physics and other disciplines whenever opposite-side or opposite-point relationships on a spherical surface are relevant.