Unit sphere

The set of points in three-dimensional Euclidean space whose distance from the origin is one. It is the boundary of the unit ball.

Euclidean distance: The straight-line distance between points in Euclidean space, computed from the sum of squared coordinate differences. The defining condition selects points whose Euclidean distance from the origin equals one.
Sphere equation: An equation describing points at a fixed distance from a center in Cartesian coordinates. For the unit sphere, the equation is x² + y² + z² = 1.
Unit vector: A vector with magnitude one, representing direction independently of length. Every point on the unit sphere is the endpoint of a unit vector from the origin.
Unit circle: The set of points in a plane at distance one from the origin. It is the two-dimensional analogue of the unit sphere.
Cartesian coordinates: A coordinate system that locates points using signed distances along perpendicular axes. Coordinates turn the unit sphere's distance condition into an equation in three variables.
Spherical coordinates: A coordinate system representing points by radial distance and two angles. Fixing the radial coordinate at one parametrizes the unit sphere.
Directional derivative: The rate at which a function changes in a specified direction at a point. Directions of unit length are points on the unit sphere, making comparisons independent of scale.
Unit 3-sphere: The set of points at distance one from the origin in four-dimensional Euclidean space. It extends the unit-sphere definition by adding a fourth coordinate.
Sphere: The set of points at a fixed distance from a center in Euclidean space. The unit sphere is the special case with center at the origin and radius one.
Latitude and longitude: Angular coordinates that locate positions on a sphere relative to an equator and reference meridian. They provide familiar angular labels for points on the unit sphere.

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