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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. If the plane passes through the center of the sphere (forming a great circle ), so that the height of the cap is equal to the radius of the sphere, the spherical ...
In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The surface of the spherical segment (excluding the bases) is called spherical zone. Geometric parameters for spherical ...
Dually, if v is a vertex of P, then there is a cone that has its apex at v and that is tangent to O in a circle; this circle forms the boundary of a spherical cap within which the sphere's surface is visible from the vertex. That is, the circle is the horizon of the midsphere, as viewed from the vertex. The circles formed in this way are ...
A number of special regions can be defined for a ball: cap, bounded by one plane; sector, bounded by a conical boundary with apex at the center of the sphere; segment, bounded by a pair of parallel planes; shell, bounded by two concentric spheres of differing radii; wedge, bounded by two planes passing through a sphere center and the surface of ...
Provided neither plane is tangent to the sphere, this forms a spherical segment of two bases. Also called a spherical frustum. If one plane is tangent, then a spherical cap is formed. If both are tangent, then we recover the sphere.
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 cap.
The spherical coordinates of a point P then are defined as follows: The radius or radial distance is the Euclidean distance from the origin O to P. The inclination (or polar angle) is the signed angle from the zenith reference direction to the line segment OP. (Elevation may be used as the polar angle instead of inclination; see below.)
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...