boundary and sphere {polarzonoid} | R Documentation |
The Homeomorphism between the Boundary of the Polar Zonoid
in \mathbb{R}^{2n+1}
and the sphere \mathbb{S}^{2n}
Description
This section calculates the homeomorphism from
the space of n
or fewer arcs on the circle
to the boundary of the polar zonoid, and its inverse.
In this version of the package, n
must be
0, 1, 2, or 3.
Usage
spherefromboundary( p )
boundaryfromsphere( x )
Arguments
p |
an vector of length |
x |
a non-zero vector of length |
Details
spherefromboundary()
is simply
central projection of the given point onto the unit sphere
centered at (0,0,...,0,0,\pi
),
which is the center of the zonoid.
In this direction there is no restriction on n
.
boundaryfromsphere()
is much harder.
One must find the intersection of two objects:
1) the ray based at the center of the zonoid in the direction x
,
and 2) the boundary of the zonoid.
To do this, an implicit formula for the boundary has been
programmed, but only when
n
is 0, 1, 2, or 3.
These two functions are inverses of each other.
Value
spherefromboundary()
returns
a unit vector in \mathbb{S}^{2n}
.
In case of error, the function returns NULL
.
boundaryfromsphere()
returns the computed point on the boundary of the zonoid.
Names are assigned indicating the corresponding term
in the trigonometric polynomial.
In case of error, the function returns NULL
.
See Also
spherefromarcs()
,
arcsfromsphere()
,
boundaryfromarcs()
,
arcsfromboundary()