# Sphere

Sphere

In geometry a sphere is the set of all points in three-dimensional space which are at distance $r\,\!$ from a fixed point of that space. The fixed point is called the center of the sphere and $r\,\!$ 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:

$\bold{P}(u,v) = \bold C + | r | \ \sin u \ \bold{\hat{P}} + | r | \ \cos u \ (\cos v \ \bold{\hat{Q}} + \sin v \ \bold{\hat{R}})$

where

 $\bold C$ is the center position (centre), $\bold{\hat{P}}$ is the pole direction (pole_dir), $\bold{\hat{Q}}$ is the direction to the origin of the parameter space, from the center (uv_oridir), $\bold{\hat{R}}$ $= \bold{\hat{P}} \times \bold{\hat{Q}}$, which is negated when reverse_v is TRUE, and $r\,\!$ is the sphere radius (radius).
A Convex Sphere with a Left-Handed uv Coordinate System
(reverse_v is FALSE)

The $u\,\!$-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 $u\,\!$-parameter runs from $-\tfrac{\pi}{2}$ at the south pole, through 0.0 at the equator, and to $\tfrac{\pi}{2}$ at the north pole.

The $v\,\!$-parameter is the longitude metric, and is the azimuth angle running from $-\pi\,\!$ to $\pi\,\!$, with 0.0 on the meridian containing uv_oridir. The $v\,\!$-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 $v\,\!$ with period $2\pi\,\!$ and parameter range [$-\pi\,\!$, $\pi\,\!$), but not periodic in $u\,\!$.
• Spheres are closed in $v\,\!$ but not in $u\,\!$.
• Spheres are singular in $u\,\!$ at the poles; non-singular everywhere else.