| Documentation |
A sphere is a gridded surface given as a
family of circles whose positions vary linearly along the
axis of the sphere, and whise radius varies in proportions to
the cosine function of the central angle. The horizontal
circles resemble lines of constant latitude, and the vertical
arcs resemble lines of constant longitude.
NOTE! If the control points are sorted in terms of increasing
longitude, and increasing latitude, the upNormal of a sphere
is the outward normal.
EXAMPLE If we take a gridded set of latitudes and longitudes
in degrees,(u,v) such as
(-90,-180) (-90,-90) (-90,0) (-90, 90) (-90, 180)
(-45,-180) (-45,-90) (-45,0) (-45, 90) (-45, 180)
( 0,-180) ( 0,-90) ( 0,0) ( 0, 90) ( 0, 180)
( 45,-180) ( 45,-90) ( 45,0) ( 45, -90) ( 45, 180)
( 90,-180) ( 90,-90) ( 90,0) ( 90, -90) ( 90, 180)
And map these points to 3D using the usual equations (where R
is the radius of the required sphere).
z = R sin u
x = (R cos u)(sin v)
y = (R cos u)(cos v)
We have a sphere of Radius R, centred at (0,0), as a gridded
surface. Notice that the entire first row and the entire last
row of the control points map to a single point in each 3D
Euclidean space, North and South poles respectively, and that
each horizontal curve closes back on itself forming a
geometric cycle. This gives us a metrically bounded (of finite
size), topologically unbounded (not having a boundary, a
cycle) surface. |
