sfepy.linalg.geometry module

sfepy.linalg.geometry.barycentric_coors(coors, s_coors)

Get barycentric (area in 2D, volume in 3D) coordinates of points with coordinates coors w.r.t. the simplex given by s_coors.

Returns:

bc : array

The barycentric coordinates. Then reference element coordinates xi = dot(bc.T, ref_coors).

sfepy.linalg.geometry.flag_points_in_polygon2d(polygon, coors)

Test if points are in a 2D polygon.

Parameters:

polygon : array, (:, 2)

The polygon coordinates.

coors: array, (:, 2)

The coordinates of points.

Returns:

flag : bool array

The flag that is True for points that are in the polygon.

Notes

This is a semi-vectorized version of [1].

[1] PNPOLY - Point Inclusion in Polygon Test, W. Randolph Franklin (WRF)

sfepy.linalg.geometry.get_coors_in_ball(coors, centre, radius, inside=True)

Return indices of coordinates inside or outside a ball given by centre and radius.

Notes

All float comparisons are done using <= or >= operators, i.e. the points on the boundaries are taken into account.

sfepy.linalg.geometry.get_coors_in_tube(coors, centre, axis, radius_in, radius_out, length, inside_radii=True)

Return indices of coordinates inside a tube given by centre, axis vector, inner and outer radii and length.

Parameters:

inside_radii : bool, optional

If False, select points outside the radii, but within the tube length.

Notes

All float comparisons are done using <= or >= operators, i.e. the points on the boundaries are taken into account.

sfepy.linalg.geometry.get_face_areas(faces, coors)

Get areas of planar convex faces in 2D and 3D.

Parameters:

faces : array, shape (n, m)

The indices of n faces with m vertices into coors.

coors : array

The coordinates of face vertices.

Returns:

areas : array

The areas of the faces.

sfepy.linalg.geometry.get_perpendiculars(vec)

For a given vector, get a unit vector perpendicular to it in 2D, or get two mutually perpendicular unit vectors perpendicular to it in 3D.

sfepy.linalg.geometry.get_simplex_circumcentres(coors, force_inside_eps=None)

Compute the circumcentres of n_s simplices in 1D, 2D and 3D.

Parameters:

coors : array

The coordinates of the simplices with n_v vertices given in an array of shape (n_s, n_v, dim), where dim is the space dimension and 2 <= n_v <= (dim + 1).

force_inside_eps : float, optional

If not None, move the circumcentres that are outside of their simplices or closer to their boundary then force_inside_eps so that they are inside the simplices at the distance given by force_inside_eps. It is ignored for edges.

Returns:

centres : array

The circumcentre coordinates as an array of shape (n_s, dim).

sfepy.linalg.geometry.inverse_element_mapping(coors, e_coors, eval_base, ref_coors, suppress_errors=False)

Given spatial element coordinates, find the inverse mapping for points with coordinats X = X(xi), i.e. xi = xi(X).

Returns:

xi : array

The reference element coordinates.

sfepy.linalg.geometry.make_axis_rotation_matrix(direction, angle)

Create a rotation matrix

System Message: WARNING/2 (\ull{R})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

corresponding to the rotation around a general axis

System Message: WARNING/2 (\ul{d})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

by a specified angle

System Message: WARNING/2 (\alpha)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

.

System Message: WARNING/2 (\ull{R} = \ul{d}\ul{d}^T + \cos(\alpha) (I - \ul{d}\ul{d}^T) + \sin(\alpha) \skewop(\ul{d}) )

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

Parameters:

direction : array

The rotation axis direction vector

System Message: WARNING/2 (\ul{d})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

.

angle : float

The rotation angle

System Message: WARNING/2 (\alpha)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

.

Returns:

mtx : array

The rotation matrix

System Message: WARNING/2 (\ull{R})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

.

Notes

The matrix follows the right hand rule: if the right hand thumb points along the axis vector

System Message: WARNING/2 (\ul{d})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

the fingers show the positive angle rotation direction.

Examples

Make transformation matrix for rotation of coordinate system by 90 degrees around ‘z’ axis.

>>> mtx = make_axis_rotation_matrix([0., 0., 1.], nm.pi/2)
>>> mtx
array([[ 0.,  1.,  0.],
       [-1.,  0.,  0.],
       [ 0.,  0.,  1.]])

Coordinates of vector

System Message: WARNING/2 ([1, 0, 0]^T)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

w.r.t. the original system in the rotated system. (Or rotation of the vector by -90 degrees in the original system.)

>>> nm.dot(mtx, [1., 0., 0.])
>>> array([ 0., -1.,  0.])

Coordinates of vector

System Message: WARNING/2 ([1, 0, 0]^T)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

w.r.t. the rotated system in the original system. (Or rotation of the vector by +90 degrees in the original system.)

>>> nm.dot(mtx.T, [1., 0., 0.])
>>> array([ 0.,  1.,  0.])

sfepy.linalg.geometry.points_in_simplex(coors, s_coors, eps=1e-08)

Test if points with coordinates coors are in the simplex given by s_coors.

sfepy.linalg.geometry.rotation_matrix2d(angle)

Construct a 2D (plane) rotation matrix corresponding to angle.

sfepy.linalg.geometry.transform_bar_to_space_coors(bar_coors, coors)

Transform barycentric coordinates bar_coors within simplices with vertex coordinates coors to space coordinates.