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Symmetry; Zero-dimensional ... or the n-dimensional surface of higher dimensional spheres. ... a great circle is the intersection of the sphere with any plane through ...
There are two possibilities: if =, the spheres coincide, and the intersection is the entire sphere; if , the spheres are disjoint and the intersection is empty. When a is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres.
Lunes can be defined on higher dimensional spheres as well. In 4-dimensions a 3-sphere is a generalized sphere. It can contain regular digon lunes as {2} θ,φ, where θ and φ are two dihedral angles. For example, a regular hosotope {2,p,q} has digon faces, {2} 2π/p,2π/q, where its vertex figure is a spherical platonic solid, {p,q}. Each ...
The disk bounded by a great circle is called a great disk: it is the intersection of a ball and a plane passing through its center. In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space R n + 1.
Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty. [18] Great circles are the intersection of the sphere with a plane passing through the center of a sphere: others are called small circles.
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 ...
1. No intersection. 2. Point intersection. 3. Two point intersection. In analytic geometry, a line and a sphere can intersect in three ways: No intersection at all; Intersection in exactly one point; Intersection in two points.
This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as A is (conventionally) termed the pole of A and it is denoted by A'. The points B' and C' are defined similarly.