#
##
## This file is part of pyFormex 2.0 (Mon Sep 14 12:29:05 CEST 2020)
## pyFormex is a tool for generating, manipulating and transforming 3D
## geometrical models by sequences of mathematical operations.
## Home page: http://pyformex.org
## Project page: http://savannah.nongnu.org/projects/pyformex/
## Copyright 2004-2020 (C) Benedict Verhegghe (benedict.verhegghe@ugent.be)
## Distributed under the GNU General Public License version 3 or later.
##
## This program is free software: you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program. If not, see http://www.gnu.org/licenses/.
##
"""Predefined geometries with a simple shape.
This module contains some functions, data and classes for generating
Formex structures representing simple geometric shapes.
You need to import this module in your scripts to have access to its
contents.
"""
import numpy as np
from pyformex import utils
from pyformex import arraytools as at
from pyformex.coords import Coords
from pyformex.formex import Formex, connect
# A collection of Formex string input patterns to construct some simple
# geometrical shapes
Pattern = {
'line': 'l:1',
'angle': 'l:1+2',
'square': 'l:1234',
'plus': 'l:1+2+3+4',
'cross': 'l:5+6+7+8',
'diamond': 'l:/45678',
'rtriangle': 'l:164',
'cube': 'l:1234I/aI/bI/cI/41234',
'star': 'l:1+2+3+4+5+6+7+8',
'star3d': 'l:1+2+3+4+5+6+7+8+A+B+C+D+E+F+G+H+a+b+c+d+e+f+g+h',
'triade': '2:01020I',
}
[docs]def shape(name):
"""Return a Formex with one of the predefined named shapes.
This is a convenience function returning a plex-2 Formex constructed
from one of the patterns defined in the Pattern dictionary.
Currently, the following pattern names are defined:
'line', 'angle', 'square', 'plus', 'cross', 'diamond', 'rtriangle', 'cube',
'star', 'star3d'.
See the Pattern example.
"""
return Formex(Pattern[name])
[docs]def randomPoints(n, bbox=[[0., 0., 0.], [1., 1., 1.]]):
"""Create n random points in a specified bbox.
Examples
--------
>>> X = randomPoints(10)
>>> X.shape
(10, 3)
>>> np.asarray(X >= 0.).all(), np.asarray(X <= 1.).all()
(True, True)
>>> X = randomPoints(15, bbox=[[1.,1.,1.],[2.,2.,2.]])
>>> X.shape
(15, 3)
>>> np.asarray(X >= 1.).all(), np.asarray(X <= 2.).all()
(True, True)
"""
bbox = at.checkArray(bbox,(2,3),'f')
return Coords(at.randomNoise((n, 3))).scale(bbox[1]-bbox[0]).trl(bbox[0])
[docs]def regularGrid(x0, x1, nx, swapaxes=None):
"""Create a regular grid of points between two points x0 and x1.
Parameters:
- `x0`: n-dimensional float (usually 1D, 2D or 3D).
- `x1`: n-dimensional float with same dimension as `x0`.
- `nx`: n-dimensional int with same dimension as `x0` and `x1`.
The space between `x0` and `x1` is subdivided in `nx[i]` equal parts
along the axis i.
- `swapaxes`: bool. If False(default), the points are number first in
the direction of the 0 axis, then the next axis,...
If True, numbering starts in the direction of the highest axis.
This is the legacy behavior.
Returns a rectangular grid of n-dimensional coordinates in an array with
shape ( nx[0]+1, nx[1]+1, ..., ndim ).
Example:
>>> regularGrid(0.,1.,4)
array([[ 0. ],
[ 0.25],
[ 0.5 ],
[ 0.75],
[ 1. ]])
>>> regularGrid((0.,0.),(1.,1.),(3,2))
array([[[ 0. , 0. ],
[ 0.33, 0. ],
[ 0.67, 0. ]],
<BLANKLINE>
[[ 1. , 0. ],
[ 0. , 0.5 ],
[ 0.33, 0.5 ]],
<BLANKLINE>
[[ 0.67, 0.5 ],
[ 1. , 0.5 ],
[ 0. , 1. ]],
<BLANKLINE>
[[ 0.33, 1. ],
[ 0.67, 1. ],
[ 1. , 1. ]]])
"""
if swapaxes is None:
# We do not use a decorator utils.warning, because
# this function gets called during startup (initialization of elements)
utils.warn("warn_regular_grid")
swapaxes = False
x0 = np.asarray(x0).ravel()
x1 = np.asarray(x1).ravel()
nx = np.asarray(nx).ravel()
if x0.size != x1.size or nx.size != x0.size:
raise ValueError("Expected equally sized 1D arrays x0,x1,nx")
if any(nx < 0):
raise ValueError("nx values should be >= 0")
# First construct a grid with integer coordinates
ndim = x0.size
shape = np.append(tuple(nx + 1), ndim)
if swapaxes:
# we can just use numpy.indices
ind = np.indices(nx + 1)
else:
# we need to reverse the axes for numpy.indices
ind = np.indices(nx[::-1] + 1)[::-1]
ind = ind.reshape((ndim, -1))
# And a grid with the complementary indices
nx[nx == 0] = 1
jnd = nx.reshape((ndim, -1)) - ind
ind = ind.transpose()
jnd = jnd.transpose()
return ((x0 * jnd + x1 * ind) / nx).reshape(shape)
[docs]def point(x=0., y=0., z=0.):
"""Return a Formex which is a point, by default at the origin.
Each of the coordinates can be specified and is zero by default.
"""
return Formex([[[x, y, z]]])
[docs]def line(p1=[0., 0., 0.], p2=[1., 0., 0.], n=1):
"""Return a Formex which is a line between two specified points.
p1: first point, p2: second point
The line is split up in n segments.
"""
return Formex([[p1, p2]]).subdivide(n)
[docs]def rect(p1=[0., 0., 0.], p2=[1., 0., 0.], nx=1, ny=1):
"""Return a Formex which is a the circumference of a rectangle.
p1 and p2 are two opposite corner points of the rectangle.
The edges of the rectangle are in planes parallel to the z-axis.
There will always be two opposite edges that are parallel with the x-axis.
The other two will only be parallel with the y-axis if both points
have the same z-value, but in any case they will be parallel with the
y-z plane.
The edges parallel with the x-axis are divide in nx parts, the other
ones in ny parts.
"""
p1 = Coords(p1)
p2 = Coords(p2)
p12 = Coords([p2[0], p1[1], p1[2]])
p21 = Coords([p1[0], p2[1], p2[2]])
return Formex.concatenate([
line(p1, p12, nx),
line(p12, p2, ny),
line(p2, p21, nx),
line(p21, p1, ny)
])
[docs]def rectangle(nx=1, ny=1, b=None, h=None, bias=0., diag=None):
"""Return a Formex representing a rectangular surface.
The rectangle has a size(b,h) divided into (nx,ny) cells.
The default b/h values are equal to nx/ny, resulting in a modular grid.
The rectangle lies in the (x,y) plane, with one corner at [0,0,0].
By default, the elements are quads. By setting diag='u','d' of 'x',
diagonals are added in /, resp. \\ and both directions, to form triangles.
"""
if diag == 'x':
base = Formex(
[[[0.0, 0.0, 0.0], [1.0, -1.0, 0.0], [1.0, 1.0, 0.0]]]
).rosette(4, 90.).translate([-1.0, -1.0, 0.0]).scale(0.5)
else:
base = Formex({'u': '3:012934', 'd': '3:016823'}.get(diag, '4:0123'))
if b is None:
sx = 1.
else:
sx = float(b) / nx
if h is None:
sy = 1.
else:
sy = float(h) / ny
return base.replic2(nx, ny, bias=bias).scale([sx, sy, 0.])
[docs]def Cube():
"""Create the surface of a cube
Returns a TriSurface representing the surface of a unit cube.
Each face of the cube is represented by two triangles.
"""
from pyformex.trisurface import TriSurface
back = Formex('3:012934')
fb = back.reverse() + back.translate(2, 1)
faces = fb + fb.rollAxes(1) + fb.rollAxes(2)
return TriSurface(faces)
[docs]def circle(a1=2., a2=0., a3=360., r=None, n=None, c=None, eltype='line2',
angle_spec=at.DEG):
"""A polygonal approximation of a circle or arc.
All points generated by this function lie on a circle with unit radius at
the origin in the x-y-plane.
- `a1`: the angle enclosed between the start and end points of each line
segment (dash angle).
- `a2`: the angle enclosed between the start points of two subsequent line
segments (module angle). If ``a2==0.0``, `a2` will be taken equal to `a1`.
- `a3`: the total angle enclosed between the first point of the first
segment and the end point of the last segment (arc angle).
All angles are given in degrees and are measured in the direction from
x- to y-axis. The first point of the first segment is always on the x-axis.
The default values produce a full circle (approximately).
If $a3 < 360$, the result is an arc.
Large values of `a1` and `a2` result in polygons. Thus
`circle(120.)` is an equilateral triangle and `circle(60.)`
is regular hexagon.
Remark that the default a2 == a1 produces a continuous line,
while a2 > a1 results in a dashed line.
Three optional arguments can be added to scale and position the circle
in 3D space:
- `r`: the radius of the circle
- `n`: the normal on the plane of the circle
- `c`: the center of the circle
Examples
--------
>>> circle(90., 90.).coords
Coords([[[ 1., 0., 0.],
[ 0., 1., 0.]],
<BLANKLINE>
[[ 0., 1., 0.],
[-1., 0., 0.]],
<BLANKLINE>
[[-1., 0., 0.],
[-0., -1., 0.]],
<BLANKLINE>
[[-0., -1., 0.],
[ 1., -0., 0.]]])
>>> circle(90., 45.).coords
Coords([[[ 1. , 0. , 0. ],
[ 0. , 1. , 0. ]],
<BLANKLINE>
[[ 0.71, 0.71, 0. ],
[-0.71, 0.71, 0. ]],
<BLANKLINE>
[[ 0. , 1. , 0. ],
[-1. , 0. , 0. ]],
<BLANKLINE>
[[-0.71, 0.71, 0. ],
[-0.71, -0.71, 0. ]],
<BLANKLINE>
[[-1. , 0. , 0. ],
[-0. , -1. , 0. ]],
<BLANKLINE>
[[-0.71, -0.71, 0. ],
[ 0.71, -0.71, 0. ]],
<BLANKLINE>
[[-0. , -1. , 0. ],
[ 1. , -0. , 0. ]],
<BLANKLINE>
[[ 0.71, -0.71, 0. ],
[ 0.71, 0.71, 0. ]]])
"""
a1 = a1 * angle_spec
a2 = a1 if a2 == 0.0 else a2 * angle_spec
a3 = a3 * angle_spec
ns = int(round(a3/a2))
if eltype=='line2':
F = Formex(
[[[1., 0., 0.],
[np.cos(a1), np.sin(a1), 0.]]]
).rosette(ns, a2, axis=2, around=[0., 0., 0.], angle_spec=at.RAD)
elif eltype=='line3':
F = Formex(
[[[1., 0., 0.],
[np.cos(a1/2.), np.sin(a1/2.), 0.],
[np.cos(a1), np.sin(a1), 0.]]], eltype=eltype
).rosette(ns, a2, axis=2, around=[0., 0., 0.], angle_spec=at.RAD)
if r is not None:
F = F.scale(r)
if n is not None:
F = F.rotate(at.rotMatrix2([0., 0., 1.], n))
if c is not None:
F = F.trl(c)
return F
[docs]def polygon(n):
"""A regular polygon with n sides.
Creates the circumference of a regular polygon with $n$ sides,
inscribed in a circle with radius 1 and center at the origin.
The first point lies on the axis 0. All points are in the (0,1) plane.
The return value is a plex-2 Formex.
This function is equivalent to circle(360./n).
"""
return circle(360. / n)
[docs]def polygonSector(n):
"""Create one sector of a regular polygon with n sides"""
if n < 3:
raise ValueError("n should be at least 3")
P0 = Coords([0., 0., 0.])
P1 = Coords([1., 0., 0.])
P2 = P1.rotate(360./n)
return Formex([[P0, P1, P2]])
[docs]def triangle():
"""An equilateral triangle with base [0,1] on axis 0.
Returns an equilateral triangle with side length 1.
The first point is the origin, the second points is on the axis 0.
The return value is a plex-3 Formex.
"""
return Formex([[[0., 0., 0.], [1., 0., 0.], [0.5, 0.5 * np.sqrt(3.), 0.]]])
[docs]def quadraticCurve(x=None, n=8):
"""Create a collection of curves.
x is a (3,3) shaped array of coordinates, specifying 3 points.
Return an array with 2*n+1 points lying on the quadratic curve through
the points x. Each of the intervals [x0,x1] and [x1,x2] will be divided
in n segments.
"""
x = at.checkArray(x, (3, 3), kind='f')
# Interpolation functions in normalized coordinates (-1..1)
h = [lambda x: x*(x-1)/2, lambda x: (1+x)*(1-x), lambda x: x*(1+x)/2]
t = np.arange(-n, n+1) / float(n)
H = np.column_stack([hi(t) for hi in h])
return np.dot(H, x)
[docs]def sphere(ndiv=6, base='icosa', equiv='max'):
"""Create a triangulated approximation of a spherical surface.
A (possibly high quality) approximation of a spherical surface is
constructed as follows. First a simple base triangulated surface
is created. Its triangular facets are subdivided by dividing all
edges in `ndiv` parts. The resulting mesh is then projected on a
sphere with unit radius. The higher `ndiv` is taken,
the better the approximation. For `ndiv=1`, the base surface is
returned.
Parameters:
- `ndiv`: number of divisions along the edges of the base surface.
- `base`: the type of base surface. One of the following:
- 'icosa': icosahedron (20 faces): this offers the highest
quality with triangles of almost same size ans shape.
- 'octa': octahedron (8 faces): this model will have the same
mesh on each of the quadrants. The coordinate planes do not
cut any triangle. This model is this fit to be subdivided along
coordinate planes.
Returns a TriSurface, representing a triangulated approximation of a
spherical surface with radius 1 and center at the origin.
"""
from pyformex import elements
from pyformex.trisurface import TriSurface
base = getattr(elements, base.capitalize())
M = TriSurface(base.vertices, base.faces)
M = M.subdivide(ndiv).fuse()
M = M.projectOnSphere()
return M
[docs]def sphere3(nx, ny, r=1, bot=-90., top=90.):
"""Return a sphere consisting of surface triangles
A sphere with radius r is modeled by the triangles formed by a regular
grid of nx longitude circles, ny latitude circles and their diagonals.
The two sets of triangles can be distinguished by their property number:
1: horizontal at the bottom, 2: horizontal at the top.
The sphere caps can be cut off by specifying top and bottom latitude
angles (measured in degrees from 0 at north pole to 180 at south pole.
"""
base = Formex([[[0, 0, 0], [1, 0, 0], [1, 1, 0]],
[[1, 1, 0], [0, 1, 0], [0, 0, 0]]],
[1, 2])
grid = base.replicm((nx, ny), (1., 1.))
s = float(top - bot) / ny
return grid.translate([0, bot / s, 1]).spherical(scale=[360. / nx, s, r])
[docs]def sphere2(nx, ny, r=1, bot=-90, top=90):
"""Return a sphere consisting of line elements.
A sphere with radius r is modeled by a regular grid of nx
longitude circles, ny latitude circles and their diagonals.
The 3 sets of lines can be distinguished by their property number:
1: diagonals, 2: meridionals, 3: horizontals.
The sphere caps can be cut off by specifying top and bottom latitude
angles (measured in degrees from 0 at north pole to 180 at south pole.
"""
base = Formex('l:543', [1, 2, 3]) # single cell
d = base.select([0]).replicm((nx, ny)) # all diagonals
m = base.select([1]).replicm((nx, ny)) # all meridionals
h = base.select([2]).replicm((nx, ny + 1)) # all horizontals
grid = m + d + h
s = float(top - bot) / ny
return grid.translate([0, bot / s, 1]).spherical(scale=[360. / nx, s, r])
[docs]@utils.deprecated_by('simple.connectCurves', 'Mesh.connect')
def connectCurves(curve1, curve2, n):
"""Connect two curves to form a surface.
curve1, curve2 are plex-2 Formices with the same number of elements.
The two curves are connected by a surface of quadrilaterals, with n
elements in the direction between the curves.
See Also
--------
Mesh.connect
"""
return curve1.toMesh().connect(curve2.toMesh(), n).toFormex()
[docs]def sector(r, t, nr, nt, h=0., diag=None):
"""Constructs a Formex which is a sector of a circle/cone.
A sector with radius r and angle t is modeled by dividing the
radius in nr parts and the angle in nt parts and then creating
straight line segments.
If a nonzero value of h is given, a conical surface results with its
top at the origin and the base circle of the cone at z=h.
The default is for all points to be in the (x,y) plane.
By default, a plex-4 Formex results. The central quads will collapse
into triangles.
If diag='up' or diag = 'down', all quads are divided by an up directed
diagonal and a plex-3 Formex results.
"""
r = float(r)
t = float(t)
p = Formex(regularGrid([0., 0., 0.], [r, 0., 0.], [nr, 0, 0],
swapaxes=True).reshape(-1, 3))
if h != 0.:
p = p.shear(2, 0, h / r)
q = p.rotate(t / nt)
if isinstance(diag, str):
diag = diag[0]
if diag == 'u':
F = connect([p, p, q], bias=[0, 1, 1]) + \
connect([p, q, q], bias=[1, 2, 1])
elif diag == 'd':
F = connect([q, p, q], bias=[0, 1, 1]) + \
connect([p, p, q], bias=[1, 2, 1])
else:
F = connect([p, p, q, q], bias=[0, 1, 1, 0])
F = Formex.concatenate([F.rotate(i * t / nt) for i in range(nt)])
return F
[docs]def cylinder(D, L, nt, nl, D1=None, angle=360., bias=0., diag=None):
"""Create a cylindrical, conical or truncated conical surface.
Returns a Formex representing (an approximation of) a cylindrical or
(possibly truncated) conical surface with its axis along the z-axis.
The resulting surface is actually a prism or pyramid, and only becomes
a good approximation of a cylinder or cone for high values of `nt`.
Parameters:
- `D`: base diameter (at z=0) of the cylinder/cone,
- `L`: length (along z-axis) of the cylinder/cone,
- `nt`: number of elements along the circumference,
- `nl`: number of elements along the length,
- `D1`: diameter at the top (z=L) of the cylinder/cone: if unspecified,
it is taken equal to `D` and a cylinder results.
Setting either `D1` or `D` to zero results in a cone,
other values will create a truncated cone.
- `diag`: by default, the elements are quads. Setting `diag` to 'u' or 'd'
will put in an 'up' or 'down' diagonal to create triangles.
"""
C = rectangle(nl, nt, L, angle, bias=bias, diag=diag).trl(2, D / 2.)
if D1 is not None and D1 != D:
C = C.shear(2, 0, (D1 - D) / L / 2)
return C.cylindrical(dir=[2, 1, 0])
[docs]def boxes(x):
"""Create a set of cuboid boxes.
`x`: Coords with shape (nelems,2,3), usually with x[:,0,:] < x[:,1,:]
Returns a Formex with shape (nelems,8,3) and of type 'hex8',
where each element is the cuboid box which has x[:,0,:]
as its minimum coordinates and x[:,1,:] as the maximum ones.
Note that the elements may be degenerate or reverted if the minimum
coordinates are not smaller than the maximum ones.
This function can be used to visualize the bboxes() of a geometry.
"""
x = Coords(x).reshape(-1, 2, 3)
i = [[0, 0, 0],
[1, 0, 0],
[1, 1, 0],
[0, 1, 0],
[0, 0, 1],
[1, 0, 1],
[1, 1, 1],
[0, 1, 1]]
j = [0, 1, 2]
return Formex(x[:, i, j], eltype='hex8')
[docs]def boxes2d(x):
"""Create a set of rectangular boxes.
Parameters:
- `x`: Coords with shape (nelems,2,3), usually with x[:,0,:] < x[:,1,:]
and x[:,:,2] == 0.
Returns a Formex with shape (nelems,4,3) and of type 'quad4',
where each element is the rectangular box which has x[:,0,:]
as its minimum coordinates and x[:,1,:] as the maximum ones.
Note that the elements may be degenerate or reverted if the minimum
coordinates are not smaller than the maximum ones.
This function is a 2D version of :meth:`bboxes`.
"""
x = Coords(x).reshape(-1, 2, 3)
i = [[0, 0, 0],
[1, 0, 0],
[1, 1, 0],
[0, 1, 0]]
j = [0, 1, 2]
return Formex(x[:, i, j], eltype='quad4')
[docs]def cuboid(xmin=[0., 0., 0.], xmax=[1., 1., 1.], cs=None):
"""Create a rectangular prism.
Create a rectangular prism from two opposite corners. The vertices
are specified in the global or a given coordinate system. The faces
faces are parallel to the coordinate planes.
Parameters:
- `xmin`: float(3): minimum coordinates
- `xmax`: float(3): maximum coordinates
- `cs`: CoordSys: if specified, the cuboid is constructed in this
coordinate system, and then transformed back to global axes.
Returns a single element Formex with eltype 'hex8'.
"""
F = boxes([xmin, xmax])
if cs:
F = F.fromCS(cs)
return F
[docs]def cuboid2d(xmin=[0., 0., 0.], xmax=[1., 1., 0.]):
"""Create a rectangle.
Creates a rectangle with sides parallel to the global y-axis
and global xz-plane, and having the points xmin and xmax as
opposite corner points.
Returns a single element Formex with eltype 'quad4'.
"""
return boxes2d([xmin, xmax])
[docs]def boundingBox(obj, cs=None):
"""Returns a cuboid that is the bounding box of some geometry
The boundingBox is computed in the specified coordinate system.
The default is the global axes.
Returns a single hexahedral Formex object.
"""
if cs:
obj = obj.toCS(cs)
xmin, xmax = obj.bbox()
return cuboid(xmin, xmax, cs)
def principalBbox(obj):
return boundingBox(obj, cs=obj.principalCS())
# End
