a region on the sphere bounded by small circles: all directions v with dot(v, N_i) >= alpha_i for every normal N_i

Syntax

sR = sphericalRegion()

The whole sphere.

sR = sphericalRegion(N)

The intersection of the hemispheres about N.

sR = sphericalRegion(N, alpha)

The intersection of the caps dot(v, N_i) >= alpha_i.

sR = sphericalRegion(N, antipodal=True)

A region of axes, v or -v inside.

sR = sphericalRegion('upper')

The hemisphere about z, ‘lower’ about -z.

sR = sphericalRegion(N, frame=cs)

A region given in a reference frame.

sR = sphericalRegion.byVertices(V)

The polygon through the vertices.

Description

An alpha of 0 makes the boundary a great circle, a positive alpha a cap smaller than a hemisphere. The fundamental sectors of the point groups are spherical regions: fundamentalSector, dirichletSector. sR.checkInside(v) tests directions, sR.volume() gives the fraction of the sphere the region covers.

Input

`N`	vector3d, the normals of the bounding circles, or ‘upper’ or ‘lower’
`alpha`	array, the cosines of the bounding circles, one for all or one per normal; by default 0

Keywords

keyword	default	description
`frame`		the referenceFrame the region is given in
`antipodal`		a region of axes: a direction is inside when v or -v is

Output

`sR`	sphericalRegion

Members

boundary	the boundary as arcs, one vector3d per arc: each circle is cut at the corners on it and the pieces whose midpoint lies inside are kept
byVertices	the region bounded by the great circles through consecutive vertices V, taken counterclockwise
center	the centre (MATLAB @sphericalRegion/center): the pole of a single circle, the bisector of the two boundaries furthest apart for a lune, the barycentre of the vertices of a triangle, and the area centroid of anything larger
checkInside	whether directions lie in the region, up to a small tolerance at the boundary
eq	Return self==value
isLower	whether the region lies within the hemisphere into the screen of the convention pC
isUpper	whether the region lies within the hemisphere out of the screen of the convention pC
plot	the outline of a spherical region on the charts of the current figure, or of a new one
polarCoordinates	polar coordinates of a region (MATLAB @sphericalRegion/polarCoordinates), which is what a colour key is built on: the radius is one at the centre and zero on the boundary, taken along the great circle through the centre, and the azimuth is measured about the centre from a reference direction and then equalised, so that a wide part of the sector takes as much of the turn as a narrow one and the vertices sit at equal angles
restrict2Lower	the hemisphere opposite to a direction, z by default, added to the region (MATLAB’s sphericalRegion.restrict2Lower)
restrict2Upper	the hemisphere seen from a direction added to the region (MATLAB’s sphericalRegion.restrict2Upper)
rhoRange	the azimuth interval of a region: the whole circle when a pole lies inside, otherwise the complement of the widest gap between the azimuths of its boundary
rotate	rotate(x, rot) is x.rotate(rot), or rot * x for an object without that method
sphericalRegion	a region on the sphere bounded by small circles: all directions v with dot(v, N_i) >= alpha_i for every normal N_i
thetaInterval	the polar interval (min, max) of the region along the meridian rho, None where it misses it
thetaIntervals	the polar intervals of the region along the meridian rho: every circle dot(v, n) >= alpha cuts the meridian in R cos(theta - phi) >= alpha, whose solutions are arcs
vertices	the corners of the region: the intersections of its circles that lie inside
volume	the fraction of the sphere the region covers, by a quadrature on a fine grid

MATLAB’s sphericalRegion_index

source

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/sphericalRegion_index_py.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX

@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.

The class sphericalRegion