Unit sphere in the context of Clairaut's relation

⭐ Core Definition: Unit sphere

In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space. More generally, the unit ⁠ $n$ ⁠-sphere is an ⁠ $n$ ⁠-sphere of unit radius in ⁠ $(n+1)$ ⁠-dimensional Euclidean space; the unit circle is a special case, the unit ⁠ $1$ ⁠-sphere in the plane. An (open) unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center.

A sphere or ball with unit radius and center at the origin of the space is called the unit sphere or the unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of translation and scaling, so the study of spheres in general can often be reduced to the study of the unit sphere.

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👉 Unit sphere in the context of Clairaut's relation

In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if $\gamma$ is a parametrization of a great circle then

where $\rho (P)$ is the distance from a point $P$ on the great circle to the $z$ -axis, and $\psi (P)$ is the angle between the great circle and the meridian through the point $P$ .

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⭐ Core Definition: Unit sphere
👉 Unit sphere in the context of Clairaut's relation
Unit sphere in the context of Solid angle
Unit sphere in the context of Octant of a sphere

Unit sphere in the context of Solid angle

In geometry, a solid angle (symbol: $Ω$ ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.

In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr), which is equal to one square radian, sr = rad. One steradian corresponds to one unit of area (of any shape) on the unit sphere surrounding the apex, so an object that blocks all rays from the apex would cover a number of steradians equal to the total surface area of the unit sphere, $4\pi$ . Solid angles can also be measured in squares of angular measures such as degrees, minutes, and seconds.

View the full Wikipedia page for Solid angle

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Unit sphere in the context of Octant of a sphere

In geometry, an octant of a sphere is a spherical triangle with three right angles and three right sides. It is sometimes called a trirectangular (spherical) triangle. It is one face of a spherical octahedron.

For a sphere embedded in three-dimensional Euclidean space, the vectors from the sphere's center to each vertex of an octant are the basis vectors of a Cartesian coordinate system relative to which the sphere is a unit sphere. The spherical octant itself is the intersection of the sphere with one octant of space.

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Unit sphere in the context of Clairaut's relation

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⭐ Core Definition: Unit sphere

👉 Unit sphere in the context of Clairaut's relation

Unit sphere in the context of Solid angle

Unit sphere in the context of Octant of a sphere