Sphere

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Sphere

In geometry a sphere is the set of all points in three-dimensional space which are at distance from a fixed point of that space. The fixed point is called the center of the sphere and is a real number called the radius. In ACIS a positive radius indicates an outward pointing surface normal; that is, a convex spherical surface. A negative radius indicates an inward pointing surface normal; that is, a concave or hollow spherical surface.

Parametric Form

The parametric equation for a sphere is:

where

	is the center position (centre),
	is the pole direction (pole_dir),
	is the direction to the origin of the parameter space, from the center (uv_oridir),
	, which is negated when reverse_v is TRUE, and
	is the sphere radius (radius).

A Convex Sphere with a Left-Handed uv Coordinate System
(reverse_v is FALSE)

The -parameter is the latitude metric, and is the angle between the line from the center to the test point and the equatorial plane; negative in the southern hemisphere and positive in the northern. The -parameter runs from at the south pole, through 0.0 at the equator, and to at the north pole.

The -parameter is the longitude metric, and is the azimuth angle running from to , with 0.0 on the meridian containing uv_oridir. The -direction is specified by the right-hand rule around the pole_dir when reverse_v is FALSE. If reverse_v is TRUE, it is in the opposite direction.

This surface parametrization is left-handed for a convex sphere and right-handed for a hollow one when reverse_v is FALSE, and vice versa when reverse_v is TRUE. When the sphere is transformed, the sense of reverse_v is inverted if the transform includes a reflection. No special action is required for a negation.

The parametrization implemented uses conventional latitude and longitude angles. The direction pole_dir specifies the north pole, and uv_oridir gives the zero meridian and equator.

The variables uv_oridir, pole_dir, and reverse_v that define the sphere parametrization are consistent with the same variables used for tori.

Sphere Characteristics

• Spheres are not true parametric surfaces.
• Spheres are periodic in with period and parameter range [, ), but not periodic in .
• Spheres are closed in but not in .
• Spheres are singular in at the poles; non-singular everywhere else.

See Also

• Constructors – contains API functions and Scheme extensions for the creation of spherical surfaces.