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Nevertheless, because its applications and pedagogy are tied to solid geometry, and because the generalization loses some important properties of lines in the plane, spherical geometry ordinarily does not use the term "line" at all to refer to anything on the sphere itself. If developed as a part of solid geometry, use is made of points ...
The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar result of Apollonius of Perga for the circle. This second part also holds for the plane. The contours and plane sections of the sphere are circles. This property defines the sphere uniquely.
If the sphere is isometrically embedded in Euclidean space, the sphere's intersection with a plane is a circle, which can be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant Euclidean distance (the extrinsic radius) from a point in the plane (the extrinsic center). A great circle lies ...
An example of a spherical cap in blue (and another in red) In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane.It is also a spherical segment of one base, i.e., bounded by a single plane.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
In geometry, a spherical sector, [1] also known as a spherical cone, [2] is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the
The cotangent, or four-part, formulae relate two sides and two angles forming four consecutive parts around the triangle, for example (aCbA) or BaCb). In such a set there are inner and outer parts: for example in the set (BaCb) the inner angle is C, the inner side is a, the outer angle is B, the outer side is b.
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. A polyhedron whose vertices are equidistant from its center can be conveniently studied by projecting its edges onto the sphere to obtain a corresponding ...