arcs-sphere: The Homeomorphism between the Space of n or fewer arcs, and...

In polarzonoid: Compute Maps and Properties of Polar Zonoids

The Homeomorphism between the Space of n or fewer arcs, and the sphere \mathbb{S}^{2n}

Description

This section calculates the natural homeomorphism from the space of n or fewer arcs on the circle, denoted by A_n, to the sphere \mathbb{S}^{2n}, and its inverse.

Usage

spherefromarcs( arcmat, n=NULL, gapmin=0 )
spherefromarcs_plus( arcmat, n=NULL, gapmin=5e-10 )
arcsfromsphere( u )

Arguments

arcmat	an n0 x 2 matrix with an arc definition in each row; so the total number of arcs is n0. The 1st number in the row is the center of the arc, and the 2nd number is the length of the arc; both in radians. The length can be 0 or 2\pi, defining the empty arc and the full circle respectively. For these improper arcs, there must be only 1 row, and the center is ignored.
n	the given set of arcs is taken to be in A_n. If n is NULL, then n is set to n0 = nrow(arcmat), unless arcmat defines an improper arc, when n is set to 0. If n is not NULL, then we must have n \ge n0. The returned vector is in \mathbb{R}^{2n+1}.
gapmin	the minimum gap between arcs in arcmat that is valid. The value gapmin=0 allows abutting arcs, but not overlapping arcs. If one is sure that the arcs are not overlapping, then set gapmin=-Inf and this validation check is skipped, which saves a tiny bit of time. In spherefromarcs_plus() abutting arcs are invalid, so gapmin must be positive.
u	a unit vector in \mathbb{R}^{2N+1} or \mathbb{R}^{2N}. In the latter case (an even-dimensional space), a 0 is appended to make u odd-dimensional.

Details

These first and last functions are inverses of each other.

Let a be a set of n strictly disjoint arcs in A_n, and denote the n complementary arcs by \bar{a}. Let \alpha : \mathbb{S}^{2n} \to \mathbb{S}^{2n} denote the antipodal map. Let h denote spherefromarcs(). Then h( \bar{a} ) = \alpha( h(a) ). A fancier way to say this: the antipodal map \alpha on \mathbb{S}^{2n} and the complementary map a \mapsto \bar{a} on A_n are conjugate.

If s is the full circle, then h(s) = (0,...,0,1). If \phi is the empty arc, then h(\phi) = (0,...,0,-1).

Value

spherefromarcs() maps from A_n to \mathbb{S}^{2n}; it returns a unit vector in \mathbb{R}^{2n+1}. It is simply the composition of boundaryfromarcs() and spherefromboundary(), with a little error checking.
In case of error, it returns NULL.

spherefromarcs_plus() is the same as spherefromarcs(), except it returns additional data. It returns a list with these items:

u	the same unit vector returned by spherefromarcs(). Its length is 2*n+1.
tangent	a 2*n+1 x 2*n0 matrix whose columns are tangent, at u, to the substratum \mathbb{S}^{2 n_0} \hookrightarrow \mathbb{S}^{2n}. This is the jacobian of the map A_{n_0} \to \mathbb{S}^{2 n_0}, when the space of arcs is parameterized by the endpoints of the arcs.
normal	a 2*n+1 x 2*(n-n0) matrix whose columns are normal, at u, to the substratum \mathbb{S}^{2 n_0} \hookrightarrow \mathbb{S}^{2n}. These columns are an orthonormal basis for the subspace. If n==n0 there are 0 columns.

The matrix cbind(u,tangent,normal) is square. When n>n0, the last normal vector is flipped if necessary, so that the determinant of the square matrix is positive.

If m is the length of u, then arcsfromsphere() maps from \mathbb{S}^{m-1} to A_N. It returns an Nx2 matrix defining N arcs as above. Because the space of arcs is stratified, N might be less than expected, which is (m-1)/2. It is simply the composition of boundaryfromsphere() and arcsfromboundary(), with a little error checking. In this version of the package, valid values for m are 1,3,5, and 7.
In case of error, it returns NULL.

See Also

boundaryfromarcs(), spherefromboundary(), boundaryfromsphere(), arcsfromboundary(), complementaryarcs()

polarzonoid documentation built on June 13, 2025, 9:08 a.m.