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...

non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of spheres) in...

(n-1)} ⁠-sphere is the boundary of an ⁠ n {\displaystyle n} ⁠-ball. Given a Cartesian coordinate system, the unit ⁠ n {\displaystyle n} ⁠-sphere of radius...

general of orthogonal spheres, and in particular inversion in one of the spheres orthogonal to the unit sphere maps the unit sphere to itself. It also maps...

sphere: their length is twice the radius, d = 2r. Two points on the sphere connected by a diameter are antipodal points of each other. A unit sphere is...

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the...

\mathbf {P} ^{1}.} This is the Bloch sphere, which can be mapped to the Riemann sphere. The Bloch sphere is a unit 2-sphere, with antipodal points corresponding...

unit sphere. But the result was anyway essentially known (in 1935 Andrey Nikolayevich Tychonoff showed that the unit sphere was a retract of the unit...

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. Uniquely...

radian, sr = rad2. 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...

is a rational distance from all four vertices of the unit square. Unit circle Unit cube Unit sphere Horn, Alastair N. (1991), "IFSs and the Interactive...

arc) is measured by the angle that it subtends at the centre. On the unit sphere, this radian measure is numerically equal to the arc length. By convention...

upward onto the unit upper half-sphere, taking the "south-pole" of the unit sphere as the projection center, and then this half-sphere is projected sideways...

often denoted as S1 because it is a one-dimensional unit n-sphere. If (x, y) is a point on the unit circle's circumference, then |x| and |y| are the lengths...

In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space...

a circle or sphere, the intersection between the sphere and a ray in that direction emanating from the sphere's center; the tips of unit vectors emanating...

angle subtended at the centre of a unit sphere by a unit area (of any shape) on its surface. For a general sphere of radius r, any portion of its surface...

Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M), UTM, or SM is the unit sphere bundle for the tangent...

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation...

stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the...

Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It...

^{2}}}\,du\,dv={\sqrt {EG-F^{2}}}\,du\,dv.} A spherical curve on the unit sphere in R3 may be parametrized as X ( u , v ) = [ cos ⁡ u sin ⁡ v sin ⁡ u...

versine of the angle θ, or the squares of half chord of the angle on a unit circle (sphere). To solve for the distance d, apply the archaversine (inverse haversine)...

unit circle. Let ⟨ , ⟩ be the usual scalar product on R3. Let S2 be the unit sphere in R3. The tangent space to S2 at a point m is naturally identified with...

between two random points in a unit cube Tychonoff cube, an infinite-dimensional analogue of the unit cube Unit square Unit sphere Ball, Keith (2010), "High-dimensional...

a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere (shown...

map is a map N: X → S2 (where S2 is the unit sphere) such that for each p in X, the function value N(p) is a unit vector orthogonal to X at p. The Gauss...

)=1\right\}=S^{1}} while the closed unit ball in ( X , d ) {\displaystyle (X,d)} is the union of the open unit ball and the unit sphere centered at this same point:...

sphere-like region or shell. Sphere may also refer to: Armillary sphere, a physical model of the celestial sphere Celestial sphere, the astronomical description...

convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored, see graphic.) This (unit sphere) simplification...