arcs-boundary: The Homeomorphism between the Space of n or fewer arcs, and...
In polarzonoid: Compute Maps and Properties of Polar Zonoids

arcs and boundary

R Documentation

The Homeomorphism between the Space of `n` or fewer arcs, and the boundary of the Polar Zonoid

Description

This section calculates the homeomorphism from the space of n or fewer arcs on the circle, which is denoted by A_n, to the boundary of the polar zonoid, and its inverse.

Usage

boundaryfromarcs( arcmat, n=NULL, gapmin=0 )
arcsfromboundary( p, tol=5.e-9 )

Arguments

`arcmat`	an Nx2 matrix with an arc definition in each row; so the total number of arcs is N. 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 2 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 the number of rows in `arcmat`, unless `arcmat` defines an improper arc, when `n` is set to 0. If `n` is not `NULL`, then we must have `n` `\ge` the default value. The returned vector is in `\mathbb{R}^{2n+1}`.
`gapmin`	the minimum gap between arcs in `arcmat` that is valid. The default `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.
`p`	a vector in `\mathbb{R}^{2N+1}` or `\mathbb{R}^{2N}`. In the latter case (an even-dimensional space), a `\pi` is appended to it to make p odd-dimensional. The vector must be on the boundary of the polar zonoid in `\mathbb{R}^{2N+1}`, which usually means that it is computed by either `boundaryfromarcs()` or `boundaryfromsphere()`. In this version of the package, valid values for `N` are 0, 1, 2, and 3.
`tol`	if the approximate distance from `p` to a sub-stratum of the boundary is less than `tol`, then take `p` to actually be in this substratum. This step is necessary because the boundary is not smooth at the sub-strata.

Details

In boundaryfromarcs(), the calculation of the returned point is a straightforward integration of trigonometric functions over the arcs. The last component of the vector is simply the sum of the arc lengths.
Let h denote boundaryfromarcs(). If s denotes the full circle, then h(s) = (0,...,0,2\pi). If \phi denotes the empty arc, then h(\phi) = (0,...,0,0).

arcsfromboundary() first calculates the outward pointing normal at p. In this version of the package, the implicitization of the boundary, and thus the normal, is only available when N is 0, 1, 2, or 3. This normal determines a trigonometric polynomial whose roots are calculated with trigpolyroot(). These roots are the endpoints of the arcs.

These two functions are inverses of each other.

Value

boundaryfromarcs() maps from A_n to \mathbb{R}^{2n+1}. It 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.

If m is the length of p, then arcsfromboundary() maps from \mathbb{R}^{m} to A_N. It returns an Nx2 matrix defining N arcs as above. Because p might be in a substratum of the boundary, N might be less than expected, which is (m-1)/2.
In case of error, the function returns NULL.

Examples

# make two disjoint arcs
arcmat = matrix( c(pi/4,pi/2,  pi,pi/4), 2, 2, byrow=TRUE ) ; arcmat

##           [,1]      [,2]
## [1,] 0.7853982 1.5707963
## [2,] 3.1415927 0.7853982

plotarcs( arcmat )

#  map to boundary of the zonoid
b = boundaryfromarcs( arcmat ) ;  b

##        x1        y1        x2        y2         L 
## 0.2346331 1.0000000 0.7071068 1.0000000 2.3561945 

#  map b back to arcs, and compare with original
arcsdistance( arcmat, arcsfromboundary(b) )

## [1] 2.220446e-16

# so the round trip returns to original pair of arcs, up to numerical precision

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