#
##
## 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/.
##
"""Basic geometrical operations.
This module defines some basic operations on simple geometrical entities
such as points, lines, planes, vectors, segments, triangles, circles.
The operations include intersection, projection, distance computing.
Many of the functions in this module are exported as methods on the
:class:`Coords` and the :class:`Geometry` derived classes.
Renamed functions::
intersectionPointsLWL -> intersectLineWithLine
intersectionTimesLWL -> intersectLineWithLine
intersectionPointsLWP -> intersectLineWithPlane
intersectionTimesLWP -> intersectLineWithPlane
intersectionTimesPOP -> projectPointOnPlane
intersectionPointsPOP -> projectPointOnPlane
intersectionTimesPOL -> projectPointOnLine
intersectionPointsPOL -> projectPointOnLine
distancesPFL -> distanceFromLine
distancesPFS -> distanceFromLine
Planned renaming of functions::
areaNormals
degenerate
levelVolumes
smallestDirection
distance -> distance
closest
closestPair
projectedArea
polygonNormals
averageNormals
triangleInCircle
triangleCircumCircle
triangleBoundingCircle
triangleObtuse
lineIntersection -> to be removed (or intersect2DLineWithLine)
displaceLines
segmentOrientation
rotationAngle
anyPerpendicularVector
perpendicularVector
projectionVOV -> projectVectorOnVector
projectionVOP -> projectVectorOnPlane
pointsAtLines -> pointOnLine
pointsAtSegments -> pointOnSegment
intersectionTimesSWP -> remove or intersectLineWithLine
intersectionSWP -> intersectSegmentWithPlane
intersectionPointsSWP -> intersectSegmentWithPlane
intersectionTimesLWT -> intersectLineWithTriangle
intersectionPointsLWT -> intersectLineWithTriangle
intersectionTimesSWT -> intersectSegmentWithTriangle
intersectionPointsSWT -> intersectSegmentWithTriangle
intersectionPointsPWP -> intersectThreePlanes
intersectionLinesPWP -> intersectTwoPlanes (or intersectPlaneWithPlane)
intersectionSphereSphere -> intersectTwoSpheres (or intersectSphereWithSphere)
faceDistance ->
edgeDistance ->
vertexDistance ->
baryCoords
insideSimplex
insideTriangle
"""
import numpy as np
from pyformex import arraytools as at
from pyformex import utils
from pyformex import multi
from pyformex.coords import Coords
from pyformex.formex import Formex
[docs]class Lines(object):
"""A collection of lines, half-lines(rays) or segments.
This class is intended as a common interface for a collection of
lines, half-lines or segments. Many functions in the geomtools
module operate on lines (or rays, or segments). Sometimes these
are specified by two points, while in other case they need to be
specified by one point and a vector.
Also the user might store his line data in one of these two ways,
and it is annoying and not effective to have to convert between
these two multiple times.
Then this class comes as a help. It is a lightweight class that
can store lines, rays or segments. The class itself does not
discriminate between these: the user is responsible for the
interpretation. This is further complicated by the fact that the
data structure of a vector is the same as that of a point.
In what follows, a line can mean any of:
- a full twosided infinite line,
- a ray starting at some point and infinite in one direction,
- a finite straight segment between two points.
A Lines instance can be initialized by any of the following data:
- a :term:`coords_like` object with shape (nlines, 2, 3),
containing the coordinates of two points on each of the
``nlines`` lines. This is the natural way to specify segments,
but it can be used for infinite lines and rays as well.
- a tuple of two :term:`coords_like` objects, each with shape
(nlines, 2, 3). The first contains a point on each of the lines,
the second is a vector along the line. The vectors do not need
to be unit length vectors. They typically will not be if this
method is used to specify segments.
- another Lines instance. In this case a shallow copy of the Lines
is created, sharing its data with the input Lines.
This is a convenience allowing the user to pass raw data as well
as data already converted to a Lines instance in places where
line data are expected.
Parameters
----------
data: :term:`coords_like` | tuple | Lines
A :term:`coords_like` argument should have a shape (nlines,2,3)
and defines ``nlines`` lines by the coordinates of two points
on each of the lines.
A tuple should contain two (nlines,3) shaped :term:`coords_like`
structures. The first holds a point on the lines, the second
a vector along the lines.
If a Lines instance is specified as data, a shallow copy sharing
the same data is created.
A Lines instance has the following attributes (onbly the first two are
actually stored):
Attributes
----------
p: array (nlines,3)
The first point of the Lines.
n: array (nlines,3)
The vector from the first to the second point.
coords: Coords (nlines, 2, 3)
A :class:`~coords.Coords` containing both points on the lines.
Examples
--------
Create Lines from two points.
>>> L = Lines([[[2.,0.,0.],[2.,3.,0.]], [[0.,1.,0.],[1.,1.,0.]]])
>>> print(L.p)
[[ 2. 0. 0.]
[ 0. 1. 0.]]
>>> print(L.n)
[[ 0. 3. 0.]
[ 1. 0. 0.]]
>>> L.coords
Coords([[[ 2., 0., 0.],
[ 2., 3., 0.]],
<BLANKLINE>
[[ 0., 1., 0.],
[ 1., 1., 0.]]])
Create the same Lines from one point and a vector.
>>> M = Lines(([[2.,0.,0.],[0.,1.,0.]], [[0.,3.,0.],[1.,0.,0.]]))
>>> print((M.p == L.p).all() and (M.n == L.n).all())
True
>>> print(id(M.p) == id(L.p) and id(M.n) == id(L.n))
False
Create a shallow copy.
>>> M = Lines(L)
>>> print((M.p == L.p).all() and (M.n == L.n).all())
True
>>> print(id(M.p) == id(L.p) and id(M.n) == id(L.n))
True
"""
_exclude_members_ = ['coords']
def __init__(self, data):
"""Initialize the Lines instance."""
if isinstance(data, Lines):
self.p, self.n = data.p, data.n
elif isinstance(data, tuple):
p = at.checkArray(data[0], shape=(-1, 3))
n = at.checkArray(data[1], shape=(p.shape[0], 3))
self.p, self.n = p, n
else: # should have shape (nlines,2,3)
P = at.checkArray(data, shape=(-1, 2, 3))
self.p, self.n = P[:, 0], P[:, 1] - P[:, 0]
@property
def coords(self):
"""Return the start and end points."""
return Coords(np.stack([self.p, self.p+self.n], axis=1))
#################### low level functions ###############################
# TODO: Make these into Lines methods?
[docs]def pointsAtLines(q, m, t):
"""Return the points of lines (q,m) at parameter values t.
Parameters
----------
q: float array (..., 3)
Array with (starting) points of a collection of lines.
m: float array (..., 3)
Array with vectors along the lines, broadcast compatible with q.
The vectors do not need to have unit length.
t: float array
Array with parameter values, broadcast compatible with q[...]
and m[...].
Parametric value 0 is at point q, parametric value 1 is at q + m.
Returns
-------
Coords:
A Coords array with the points at parameter values t.
"""
t = t[..., np.newaxis]
return Coords(q + t*m)
[docs]def pointsAtSegments(S, t):
"""Return the points of line segments S at parameter values t.
Parameters
----------
S: float array (..., 2, 3)
A collection of line segments defined by two points.
t: float array
Array with parameter values, broadcast compatible with S[...].
Parametric value 0 is at point 0, parametric value 1 is at point 1.
Returns
-------
Coords:
A Coords array with the points at parameter values t.
"""
q0 = S[..., 0, :]
q1 = S[..., 1, :]
return pointsAtLines(q0, q1-q0, t)
################## intersection #######################
[docs]def intersectLineWithLine(q1, m1, q2, m2, mode='all', times=False):
"""Find the common perpendicular of lines (q1,m1) and lines (q2,m2).
Return the intersection points of lines (q1,m1) and lines (q2,m2)
with the perpendiculars between them. For intersecting lines, the
corresponding points will coincide.
Parameters
----------
q1: float array (nq1, 3)
Points on the first set of lines.
m1: float array (nq1, 3)
Direction vectors of the first set of lines.
q2: float array (nq2, 3)
Points on the second set of lines.
m2: float array (nq2, 3)
Direction vectors of the second set of lines.
mode: 'all' | 'pair'
If 'all', the intersection of all lines (q1,m1) with all lines
(q2,m2) is computed; nq1 and nq2 can be different.
If 'pair', nq1 and nq2 should be equal (or 1) and the
intersection of pairs of lines is computed (using broadcasting
for length 1 data).
times: bool
If True, return the parameter values of the intersection
points instead of the coordinates (see Notes).
Returns
-------
ar1: float array
The intersection points of the common perpendiculars with the first
set of lines. See Notes.
ar2: float array
The intersection points of the common perpendiculars with the second
set of lines. See Notes.
Notes
-----
By default (``times=False``) the returned arrays are the coordinates of
the points, with shape (nq1,nq2,3) for mode 'all' and (nq1,3) for
mode 'pair'.
With ``times=True`` the return values are the parameter values of the
intersection points, and the size opf the arrays is (nq1,nq2) for
mode 'all' and (nq1,) for mode 'pair'.
The coordinates of a point with parameter value t on a line (q,m)
are given by ``q + t * m``.
The intersection of two parallel lines results in NAN-values. These are
not removed from the result. The user has to do it himself if needed.
Examples
--------
>>> q,m = [[0,0,0],[0,0,1],[0,0,3]], [[1,0,0],[1,1,0],[0,1,0]]
>>> p,n = [[2.,0.,0.],[0.,0.,0.]], [[0.,1.,0.],[0.,0.,1.]]
>>> x1,x2 = intersectLineWithLine(q,m,p,n)
>>> print(x1)
[[[ 2. 0. 0.]
[ 0. 0. 0.]]
<BLANKLINE>
[[ 2. 2. 1.]
[ 0. 0. 1.]]
<BLANKLINE>
[[ nan nan nan]
[ 0. 0. 3.]]]
>>> print(x2)
[[[ 2. 0. 0.]
[ 0. 0. 0.]]
<BLANKLINE>
[[ 2. 2. 0.]
[ 0. 0. 1.]]
<BLANKLINE>
[[ nan nan nan]
[ 0. 0. 3.]]]
>>> x1,x2 = intersectLineWithLine(q[:2],m[:2],p,n,mode='pair')
>>> print(x1)
[[ 2. 0. 0.]
[ 0. 0. 1.]]
>>> print(x2)
[[ 2. 0. 0.]
[ 0. 0. 1.]]
>>> t1,t2 = intersectLineWithLine(q,m,p,n,times=True)
>>> print(t1)
[[ 2. -0.]
[ 2. -0.]
[ nan -0.]]
>>> print(t2)
[[ -0. -0.]
[ 2. 1.]
[ nan 3.]]
>>> t1,t2 = intersectLineWithLine(q[:2],m[:2],p,n,mode='pair',times=True)
>>> print(t1)
[ 2. -0.]
>>> print(t2)
[-0. 1.]
"""
if mode == 'all':
q1 = np.asarray(q1).reshape(-1, 1, 3)
m1 = np.asarray(m1).reshape(-1, 1, 3)
q2 = np.asarray(q2).reshape(1, -1, 3)
m2 = np.asarray(m2).reshape(1, -1, 3)
else:
q1 = np.asarray(q1).reshape(-1, 3)
m1 = np.asarray(m1).reshape(-1, 3)
q2 = np.asarray(q2).reshape(-1, 3)
m2 = np.asarray(m2).reshape(-1, 3)
dot11 = at.dotpr(m1, m1)
dot22 = at.dotpr(m2, m2)
dot12 = at.dotpr(m1, m2)
denom = (dot12**2-dot11*dot22)
q12 = q2-q1
dot11 = dot11[..., np.newaxis]
dot22 = dot22[..., np.newaxis]
dot12 = dot12[..., np.newaxis]
with np.errstate(divide='ignore', over='ignore', under='ignore'):
t1 = at.dotpr(q12, m2*dot12-m1*dot22) / denom
t2 = at.dotpr(q12, m2*dot11-m1*dot12) / denom
if times:
return t1, t2
else:
return pointsAtLines(q1, m1, t1), pointsAtLines(q2, m2, t2)
[docs]def intersectLineWithPlane(q, m, p, n, mode='all', times=False):
"""Find the intersection of lines (q,m) and planes (p,n).
Return the intersection points of lines (q,m) and planes (p,n).
Parameters:
- `q`,`m`: (nq,3) shaped arrays of points and vectors (`mode=all`)
or broadcast compatible arrays (`mode=pair`), defining a single line
or a set of lines.
- `p`,`n`: (np,3) shaped arrays of points and normals (`mode=all`)
or broadcast compatible arrays (`mode=pair`), defining a single plane
or a set of planes.
- `mode`: `all` to calculate the intersection of each line (q,m) with
all planes (p,n) or `pair` for pairwise intersections.
Returns a (nq,np) shaped (`mode=all`) array of parameter values t,
such that the intersection points are given by q+t*m.
Notice that the result will contain an INF value for lines that are
parallel to the plane.
Example:
>>> q,m = [[0,0,0],[0,1,0],[0,0,3]], [[1,0,0],[0,1,0],[0,0,1]]
>>> p,n = [[1.,1.,1.],[1.,1.,1.]], [[1.,1.,0.],[1.,1.,1.]]
>>> t = intersectLineWithPlane(q,m,p,n,times=True)
>>> print(t)
[[ 2. 3.]
[ 1. 2.]
[ inf 0.]]
>>> x = intersectLineWithPlane(q,m,p,n)
>>> print(x)
[[[ 2. 0. 0.]
[ 3. 0. 0.]]
<BLANKLINE>
[[ 0. 2. 0.]
[ 0. 3. 0.]]
<BLANKLINE>
[[ nan nan inf]
[ 0. 0. 3.]]]
>>> x = intersectLineWithPlane(q[:2],m[:2],p,n,mode='pair')
>>> print(x)
[[ 2. 0. 0.]
[ 0. 3. 0.]]
"""
q = at.checkArray(q, (-1, 3), 'f', 'i')
m = at.checkArray(m, (-1, 3), 'f', 'i')
p = at.checkArray(p, (-1, 3), 'f', 'i')
n = at.checkArray(n, (-1, 3), 'f', 'i')
with np.errstate(divide='ignore', invalid='ignore'):
if mode == 'all':
t = (at.dotpr(p, n) - np.inner(q, n)) / np.inner(m, n)
elif mode == 'pair':
t = at.dotpr(n, p-q) / at.dotpr(m, n)
if times:
return t
else:
if mode == 'all':
q = q[:, np.newaxis]
m = m[:, np.newaxis]
return pointsAtLines(q, m, t)
[docs]def intersectLineWithTriangle(T, p, p1, method='line', atol=1.e-5):
# TODO: add an optional argument return_index to return the index
# put index rows in order of intersection points
# change parameters p, p1 to single L (nlines,2,3) ???
"""Compute the intersection points with a set of lines.
Parameters
----------
T: :term:`coords_like` (ntri,3,3)
The coordinates of the three vertices of ntri triangles.
p: :term:`coords_like` (nlines,3)
A first point for each of the lines to intersect.
p1: :term:`coords_like` (nlines,3)
The second point defining the lines to intersect.
method: 'line' | 'segment' | ' ray'
Define whether the points ``p`` and ``p1`` define an infinitely
long line, a finite segment p-p1 or a half infinite ray (p->p1).
atol: float
Tolerance to be used in deciding whether an intersection point on
a border edge is inside the surface or not.
Returns
-------
X: Coords (nipts, 3)
A fused set of intersection points
ind: int array (nipts, 3)
An array identifying the related intersection points, lines and
triangle.
Notes
-----
A line laying in the plane of a triangle does not generate
intersections.
This method is faster than the similar function
:func:`intersectionPointsLWT`.
Examples
-------
>>> T = Formex('3:.12.34').coords
>>> L = Coords([[[0.,0.,0.], [0.,0.,1.]],
... [[0.5,0.5,0.5], [0.5,0.5,1.]],
... [[0.2,0.7,0.5], [0.2,0.8,0.5]],
... [[0.2,0.7,0.5], [0.2,0.7,0.2]],
... ])
>>> P, ind = intersectLineWithTriangle(T, L[:,0,:], L[:,1,:])
>>> print(P)
[[ 0. 0. 0. ]
[ 0.5 0.5 0. ]
[ 0.2 0.7 0. ]]
>>> print(ind)
[[0 0 0]
[0 0 1]
[1 1 0]
[1 1 1]
[2 3 1]]
>>> P, ind = intersectLineWithTriangle(T, L[:,0,:], L[:,1,:],
... method='ray')
>>> print(P)
[[ 0. 0. 0. ]
[ 0.2 0.7 0. ]]
>>> print(ind)
[[0 0 0]
[0 0 1]
[1 3 1]]
>>> P, ind = intersectLineWithTriangle(T, L[:,0,:], L[:,1,:],
... method='segment')
>>> print(P)
[[ 0. 0. 0.]]
>>> print(ind)
[[0 0 0]
[0 0 1]]
"""
# line vectors
m = p1-p
# triangle bounding spheres
r, C, n = triangleBoundingCircle(T)
# detect candidate lines/triangles
# For a line to intersect a triangle, the distance of the center
# of the triangle to the line should not be more than the radius
# This is a lengthy procedure, so use multiprocessing
it = pointNearLine(C, p, m, r+atol, -1)
il = np.concatenate([np.full_like(iti,i) for i,iti in enumerate(it)])
it = np.concatenate(it)
# intersect candidate lines/triangles
P = intersectLineWithPlane(p[il], m[il], C[it], areaNormals(T)[1][it],
mode='pair')
# remove nan/inf (lines parallel to triangles)
i1 = np.where(~(np.isnan(P).any(axis=-1) + np.isinf(P).any(axis=-1)))[0]
if len(i1)==0:
return Coords(), []
xt, xp, jl = T[it][i1], P[i1], il[i1]
# remove intersections outside triangles
i2 = insideTriangle(xt, xp[:, np.newaxis], atol=atol).reshape(-1)
i = i1[i2]
if method in ['ray', 'segment']:
xp, jl = xp[i2], jl[i2]
if method == 'ray':
i3 = insideRay(p[jl], p1[jl] - p[jl], xp, atol)
elif method == 'segment':
i3 = insideSegment(p[jl], p1[jl], xp, atol)
i = i[i3]
P, j = P[i].fuse()
return P, np.column_stack([j, il[i], it[i]])
[docs]def intersectionTimesSWP(S, p, n, mode='all'):
"""Return the intersection of line segments S with planes (p,n).
This is like intersectionTimesLWP, but the lines are defined by two points
instead of by a point and a vector.
Parameters:
- `S`: (nS,2,3) shaped array (`mode=all`) or broadcast compatible array
(`mode=pair`), defining one or more line segments.
- `p`,`n`: (np,3) shaped arrays of points and normals (`mode=all`)
or broadcast compatible arrays (`mode=pair`), defining a single plane
or a set of planes.
- `mode`: `all` to calculate the intersection of each line segment S with
all planes (p,n) or `pair` for pairwise intersections.
Returns a (nS,np) shaped (`mode=all`) array of parameter values t,
such that the intersection points are given by
`(1-t)*S[...,0,:] + t*S[...,1,:]`.
"""
q0 = S[..., 0, :]
q1 = S[..., 1, :]
return intersectLineWithPlane(q0, q1-q0, p, n, mode, times=True)
[docs]def intersectionSWP(S, p, n, mode='all', return_all=False, atol=0.):
"""Return the intersection points of line segments S with planes (p,n).
Parameters:
- `S`: (nS,2,3) shaped array, defining a single line segment or a set of
line segments.
- `p`,`n`: (np,3) shaped arrays of points and normals, defining a single
plane or a set of planes.
- `mode`: `all` to calculate the intersection of each line segment S with
all planes (p,n) or `pair` for pairwise intersections.
- `return_all`: if True, all intersection points of the lines along the
segments are returned. Default is to return only the points that lie
on the segments.
- `atol`: float tolerance of the points inside the line segments.
Return values if `return_all==True`:
- `t`: (nS,NP) parametric values of the intersection points along the line
segments.
- `x`: the intersection points themselves (nS,nP,3).
Return values if `return_all==False`:
- `t`: (n,) parametric values of the intersection points along the line
segments (n <= nS*nP)
- `x`: the intersection points themselves (n,3).
- `wl`: (n,) line indices corresponding with the returned intersections.
- `wp`: (n,) plane indices corresponding with the returned intersections
"""
S = np.asanyarray(S).reshape(-1, 2, 3)
p = np.asanyarray(p).reshape(-1, 3)
n = np.asanyarray(n).reshape(-1, 3)
# Find intersection parameters
t = intersectionTimesSWP(S, p, n, mode)
if not return_all:
# Find points inside segments
ok = (t >= 0.0-atol) * (t <= 1.0+atol)
t = t[ok]
if mode == 'all':
wl, wt = np.where(ok)
elif mode == 'pair':
S = S[ok]
wl = wt = np.where(ok)[0]
if len(t) > 0:
if mode == 'all':
S = S[:, np.newaxis]
x = pointsAtSegments(S, t)
if x.ndim == 1:
np.reshape(x, (1, 3))
if mode == 'all' and not return_all:
x = x[ok]
else:
# No intersection: return empty Coords
x = Coords()
if return_all:
return t, x
else:
return t, x, wl, wt
[docs]def intersectionPointsSWP(S, p, n, mode='all', return_all=False, atol=0.):
"""Return the intersection points of line segments S with planes (p,n).
This is like :func:`intersectionSWP` but does not return the parameter
values. It is equivalent to::
intersectionSWP(S,p,n,mode,return_all)[1:]
"""
res = intersectionSWP(S, p, n, mode, return_all, atol)
if return_all:
return res[1]
else:
return res[1:]
[docs]def intersectionTimesLWT(q, m, F, mode='all'):
"""Return the intersection of lines (q,m) with triangles F.
Parameters:
- `q`,`m`: (nq,3) shaped arrays of points and vectors (`mode=all`)
or broadcast compatible arrays (`mode=pair`), defining a single line
or a set of lines.
- `F`: (nF,3,3) shaped array (`mode=all`) or broadcast compatible array
(`mode=pair`), defining a single triangle or a set of triangles.
- `mode`: `all` to calculate the intersection of each line (q,m) with
all triangles F or `pair` for pairwise intersections.
Returns a (nq,nF) shaped (`mode=all`) array of parameter values t,
such that the intersection points are given q+tm.
"""
Fn = np.cross(F[..., 1, :]-F[..., 0, :], F[..., 2, :]-F[..., 1, :])
return intersectLineWithPlane(q, m, F[..., 0, :], Fn, mode, times=True)
[docs]def intersectionPointsLWT(q, m, F, mode='all', return_all=False):
"""Return the intersection points of lines (q,m) with triangles F.
Parameters:
- `q`,`m`: (nq,3) shaped arrays of points and vectors, defining a single
line or a set of lines.
- `F`: (nF,3,3) shaped array, defining a single triangle or a set of
triangles.
- `mode`: `all` to calculate the intersection points of each line (q,m) with
all triangles F or `pair` for pairwise intersections.
- `return_all`: if True, all intersection points are returned. Default is
to return only the points that lie inside the triangles.
Returns:
If `return_all==True`, a (nq,nF,3) shaped (`mode=all`) array of
intersection points, else, a tuple of intersection points with shape (n,3)
and line and plane indices with shape (n), where n <= nq*nF.
"""
q = np.asanyarray(q).reshape(-1, 3)
m = np.asanyarray(m).reshape(-1, 3)
F = np.asanyarray(F).reshape(-1, 3, 3)
if not return_all:
# Find lines passing through the bounding spheres of the triangles
r, c, n = triangleBoundingCircle(F)
if mode == 'all':
mode = 'pair'
#
# TODO: check if/why this is slower
# If so, we should move this into distanceFromLine
# this is much slower for large arrays
# d = distanceFromLine(c,(q,m),mode).transpose()
d = np.row_stack([distanceFromLine(c, ([q[i]], [m[i]]), mode)
for i in range(q.shape[0])])
wl, wt = np.where(d<=r)
elif mode == 'pair':
d = distanceFromLine(c, (q, m), mode)
wl = wt = np.where(d<=r)[0]
if wl.size == 0:
return np.empty((0, 3,), dtype=float), wl, wt
q, m, F = q[wl], m[wl], F[wt]
t = intersectionTimesLWT(q, m, F, mode)
if mode == 'all':
#
# TODO:
## !!!!! CAN WE EVER GET HERE? only if return_all??
#
q = q[:, np.newaxis]
m = m[:, np.newaxis]
x = pointsAtLines(q, m, t)
if not return_all:
# Find points inside the faces
ok = insideTriangle(F, x[np.newaxis]).reshape(-1)
return x[ok], wl[ok], wt[ok]
else:
return x
[docs]def intersectionTimesSWT(S, F, mode='all'):
"""Return the intersection of lines segments S with triangles F.
Parameters:
- `S`: (nS,2,3) shaped array (`mode=all`) or broadcast compatible array
(`mode=pair`), defining a single line segment or a set of line segments.
- `F`: (nF,3,3) shaped array (`mode=all`) or broadcast compatible array
(`mode=pair`), defining a single triangle or a set of triangles.
- `mode`: `all` to calculate the intersection of each line segment S with
all triangles F or `pair` for pairwise intersections.
Returns a (nS,nF) shaped (`mode=all`) array of parameter values t,
such that the intersection points are given by
`(1-t)*S[...,0,:] + t*S[...,1,:]`.
"""
Fn = np.cross(F[..., 1, :]-F[..., 0, :], F[..., 2, :]-F[..., 1, :])
return intersectionTimesSWP(S, F[..., 0, :], Fn, mode)
[docs]def intersectionPointsSWT(S, F, mode='all', return_all=False):
"""Return the intersection points of lines segments S with triangles F.
Parameters:
- `S`: (nS,2,3) shaped array, defining a single line segment or a set of
line segments.
- `F`: (nF,3,3) shaped array, defining a single triangle or a set of
triangles.
- `mode`: `all` to calculate the intersection points of each line segment S
with all triangles F or `pair` for pairwise intersections.
- `return_all`: if True, all intersection points are returned. Default is
to return only the points that lie on the segments and inside the
triangles.
Returns:
If `return_all==True`, a (nS,nF,3) shaped (`mode=all`) array of
intersection points, else, a tuple of intersection points with shape (n,3)
and line and plane indices with shape (n), where n <= nS*nF.
"""
S = np.asanyarray(S).reshape(-1, 2, 3)
F = np.asanyarray(F).reshape(-1, 3, 3)
if not return_all:
# Find lines passing through the bounding spheres of the triangles
r, c, n = triangleBoundingCircle(F)
if mode == 'all':
#
# this is much slower for large arrays
# TODO: check why
# d = distanceFromLine(c,S,mode).transpose()
mode = 'pair'
d = np.row_stack([distanceFromLine(c, S[i], mode)
for i in range(S.shape[0])])
wl, wt = np.where(d<=r)
elif mode == 'pair':
d = distanceFromLine(c, S, mode)
wl = wt = np.where(d<=r)[0]
if wl.size == 0:
return np.empty((0, 3,), dtype=float), wl, wt
S, F = S[wl], F[wt]
t = intersectionTimesSWT(S, F, mode)
if mode == 'all':
S = S[:, np.newaxis]
x = pointsAtSegments(S, t)
if not return_all:
# Find points inside the segments and faces
ok = (t >= 0.0) * (t <= 1.0) * insideTriangle(F, x[np.newaxis]).reshape(-1)
return x[ok], wl[ok], wt[ok]
else:
return x
[docs]def intersectionPointsPWP(p1, n1, p2, n2, p3, n3, mode='all'):
"""Return the intersection points of planes (p1,n1), (p2,n2) and (p3,n3).
Parameters:
- `pi`,`ni` (i=1...3): (npi,3) shaped arrays of points and normals
(`mode=all`)
or broadcast compatible arrays (`mode=pair`), defining a single plane
or a set of planes.
- `mode`: `all` to calculate the intersection of each plane (p1,n1) with
all planes (p2,n2) and (p3,n3) or `pair` for pairwise intersections.
Returns a (np1,np2,np3,3) shaped (`mode=all`) array of intersection points.
"""
if mode == 'all':
p1 = np.asanyarray(p1).reshape(-1, 1, 1, 3)
n1 = np.asanyarray(n1).reshape(-1, 1, 1, 3)
p2 = np.asanyarray(p2).reshape(1, -1, 1, 3)
n2 = np.asanyarray(n2).reshape(1, -1, 1, 3)
p3 = np.asanyarray(p3).reshape(1, 1, -1, 3)
n3 = np.asanyarray(n3).reshape(1, 1, -1, 3)
dot1 = at.dotpr(p1, n1)[..., np.newaxis]
dot2 = at.dotpr(p2, n2)[..., np.newaxis]
dot3 = at.dotpr(p3, n3)[..., np.newaxis]
cross23 = np.cross(n2, n3)
cross31 = np.cross(n3, n1)
cross12 = np.cross(n1, n2)
denom = at.dotpr(n1, cross23)[..., np.newaxis]
return (dot1*cross23+dot2*cross31+dot3*cross12)/denom
[docs]def intersectionLinesPWP(p1, n1, p2, n2, mode='all'):
"""Return the intersection lines of planes (p1,n1) and (p2,n2).
Parameters:
- `pi`,`ni` (i=1...2): (npi,3) shaped arrays of points and normals (`mode=all`)
or broadcast compatible arrays (`mode=pair`), defining a single plane
or a set of planes.
- `mode`: `all` to calculate the intersection of each plane (p1,n1) with
all planes (p2,n2) or `pair` for pairwise intersections.
Returns a tuple of (np1,np2,3) shaped (`mode=all`) arrays of intersection
points q and vectors m, such that the intersection lines are given by
``q+t*m``.
"""
if mode == 'all':
p1 = np.asanyarray(p1).reshape(-1, 1, 3)
n1 = np.asanyarray(n1).reshape(-1, 1, 3)
p2 = np.asanyarray(p2).reshape(1, -1, 3)
n2 = np.asanyarray(n2).reshape(1, -1, 3)
m = np.cross(n1, n2)
q = intersectionPointsPWP(p1, n1, p2, n2, p1, m, mode='pair')
return q, m
[docs]def intersectionSphereSphere(R, r, d):
"""Intersection of two spheres (or two circles in the x,y plane).
Computes the intersection of two spheres with radii R, resp. r, having
their centres at distance d <= R+r. The intersection is a circle with
its center on the segment connecting the two sphere centers at a distance
x from the first sphere, and having a radius y. The return value is a
tuple x,y.
"""
if d > R+r:
raise ValueError("d (%s) should not be larger than R+r (%s)" % (d, R+r))
dd = R**2-r**2+d**2
d2 = 2*d
x = dd/d2
y = np.sqrt(d2**2*R**2 - dd**2) / d2
return x, y
########## projection #############################################
[docs]def projectPointOnPlane(X, p, n, mode='all'):
"""Return the projection of points X on planes (p,n).
Parameters:
- `X`: a (nx,3) shaped array of points.
- `p`, `n`: (np,3) shaped arrays of points and normals defining `np` planes.
- `mode`: 'all' or 'pair:
- if 'all', the projection of all points on all planes is computed;
`nx` and `np` can be different.
- if 'pair': `nx` and `np` should be equal (or 1) and the projection
of pairs of point and plane are computed (using broadcasting for
length 1 data).
Returns a float array of size (nx,np,3) for mode 'all', or size (nx,3)
for mode 'pair'.
Example:
>>> X = Coords([[0.,1.,0.],[3.,0.,0.],[4.,3.,0.]])
>>> p,n = [[2.,0.,0.],[0.,1.,0.]], [[1.,0.,0.],[0.,1.,0.]]
>>> print(projectPointOnPlane(X,p,n))
[[[ 2. 1. 0.]
[ 0. 1. 0.]]
<BLANKLINE>
[[ 2. 0. 0.]
[ 3. 1. 0.]]
<BLANKLINE>
[[ 2. 3. 0.]
[ 4. 1. 0.]]]
>>> print(projectPointOnPlane(X[:2],p,n,mode='pair'))
[[ 2. 1. 0.]
[ 3. 1. 0.]]
"""
X = np.asarray(X).reshape(-1, 3)
p = np.asarray(p).reshape(-1, 3)
n = np.asarray(n).reshape(-1, 3)
t = projectPointOnPlaneTimes(X, p, n, mode)
if mode == 'all':
X = X[:, np.newaxis]
return pointsAtLines(X, n, t)
[docs]def projectPointOnPlaneTimes(X, p, n, mode='all'):
"""Return the projection of points X on planes (p,n).
This is like :meth:`projectPointOnPlane` but instead of returning
the projected points, returns the parametric values t along the
lines (X,n), such that the projection points can be computed from
X+t*n.
Parameters: see :meth:`projectPointOnPlane`.
Returns a float array of size (nx,np) for mode 'all', or size (nx,)
for mode 'pair'.
Example:
>>> X = Coords([[0.,1.,0.],[3.,0.,0.],[4.,3.,0.]])
>>> p,n = [[2.,0.,0.],[0.,1.,0.]], [[1.,0.,0.],[0.,1.,0.]]
>>> print(projectPointOnPlaneTimes(X,p,n))
[[ 2. 0.]
[-1. 1.]
[-2. -2.]]
>>> print(projectPointOnPlaneTimes(X[:2],p,n,mode='pair'))
[ 2. 1.]
"""
if mode == 'all':
return (at.dotpr(p, n) - np.inner(X, n)) / at.dotpr(n, n)
elif mode == 'pair':
return (at.dotpr(p, n) - at.dotpr(X, n)) / at.dotpr(n, n)
[docs]def projectPointOnLine(X, p, n, mode='all'):
"""Return the projection of points X on lines (p,n).
Parameters:
- `X`: a (nx,3) shaped array of points.
- `p`, `n`: (np,3) shaped arrays of points and vectors defining `np` lines.
- `mode`: 'all' or 'pair:
- if 'all', the projection of all points on all lines is computed;
`nx` and `np` can be different.
- if 'pair': `nx` and `np` should be equal (or 1) and the projection
of pairs of point and line are computed (using broadcasting for
length 1 data).
Returns a float array of size (nx,np,3) for mode 'all', or size (nx,3)
for mode 'pair'.
Example:
>>> X = Coords([[0.,1.,0.],[3.,0.,0.],[4.,3.,0.]])
>>> p,n = [[2.,0.,0.],[0.,1.,0.]], [[0.,2.,0.],[1.,0.,0.]]
>>> print(projectPointOnLine(X,p,n))
[[[ 2. 1. 0.]
[ 0. 1. 0.]]
<BLANKLINE>
[[ 2. 0. 0.]
[ 3. 1. 0.]]
<BLANKLINE>
[[ 2. 3. 0.]
[ 4. 1. 0.]]]
>>> print(projectPointOnLine(X[:2],p,n,mode='pair'))
[[ 2. 1. 0.]
[ 3. 1. 0.]]
"""
X = np.asarray(X).reshape(-1, 3)
p = np.asarray(p).reshape(-1, 3)
n = np.asarray(n).reshape(-1, 3)
t = projectPointOnLineTimes(X, p, n, mode)
return pointsAtLines(p, n, t)
[docs]def projectPointOnLineTimes(X, p, n, mode='all'):
"""Return the projection of points X on lines (p,n).
This is like :meth:`projectPointOnLine` but instead of returning
the projected points, returns the parametric values t along the
lines (X,n), such that the projection points can be computed from
p+t*n.
Parameters: see :meth:`projectPointOnLine`.
Returns a float array of size (nx,np) for mode 'all', or size (nx,)
for mode 'pair'.
Example:
>>> X = Coords([[0.,1.,0.],[3.,0.,0.],[4.,3.,0.]])
>>> p,n = [[2.,0.,0.],[0.,1.,0.]], [[0.,1.,0.],[1.,0.,0.]]
>>> print(projectPointOnLineTimes(X,p,n))
[[ 1. 0.]
[ 0. 3.]
[ 3. 4.]]
>>> print(projectPointOnLineTimes(X[:2],p,n,mode='pair'))
[ 1. 3.]
"""
if mode == 'all':
return (np.inner(X, n) - at.dotpr(p, n)) / at.dotpr(n, n)
elif mode == 'pair':
return (at.dotpr(X, n) - at.dotpr(p, n)) / at.dotpr(n, n)
#################### distance ##############################
# TODO: we should extend the method of Coords.distanceFromLine
[docs]def distanceFromLine(X, lines, mode='all'):
"""Return the distance of points X from lines (p,n).
Parameters
----------
X: :term:`coords_like` (nx,3)
A collection of points.
lines: :term:`line_like`
One of the following definitions of the line(s):
- a tuple (p,n), where both p and n are (np,3) shaped arrays of
respectively points and vectors defining `np` lines;
- an (np,2,3) shaped array containing two points of each line.
mode: 'all' or 'pair:
If 'all', the distance of all points to all lines is computed;
`nx` and `np` can be different.
If 'pair': `nx` and `np` should be equal (or 1) and the distance
of pairs of point and line are computed (using broadcasting for
length 1 data).
Returns
-------
float array
A float array of size (nx,np) for mode 'all', or size (nx)
for mode 'pair', with the distances between the points and
the lines.
Examples
--------
>>> X = Coords([[0.,1.,0.],[3.,0.,0.],[4.,3.,0.]])
>>> L = Lines(([[2.,0.,0.],[0.,1.,0.]], [[0.,3.,0.],[1.,0.,0.]]))
>>> print(distanceFromLine(X,L))
[[ 2. 0.]
[ 1. 1.]
[ 2. 2.]]
>>> print(distanceFromLine(X[:2],L,mode='pair'))
[ 2. 1.]
>>> L = Lines(([[[2.,0.,0.],[2.,2.,0.]], [[0.,1.,0.],[1.,1.,0.]]]))
>>> print(distanceFromLine(X,L))
[[ 2. 0.]
[ 1. 1.]
[ 2. 2.]]
>>> print(distanceFromLine(X[:2],L,mode='pair'))
[ 2. 1.]
"""
lines = Lines(lines)
if mode == 'all':
return np.column_stack(X.distanceFromLine(p, n)
for p, n in zip(lines.p, lines.n))
else:
Y = projectPointOnLine(X, lines.p, lines.n, mode)
return at.length(Y-X)
[docs]def pointNearLine(X, p, n, atol, nproc=1):
"""Find the points from X that are near to lines (p,n).
Finds the points from X that are closer than atol to any of the lines (p,n).
Parameters
----------
X: :term:`coords_like` (npts,3)
An array of points.
p: :term:`coords_like` (nlines,3)
An array of points which together with n define the lines.
n: :term:`coords_like` (nlines,3)
An array of direction vectors which together with p define the lines.
atol: float or float array (npts,)
A global or pointwise tolerance to be used in determining close points.
Returns
-------
list of int arrays
An index array for each of the lines, holding the indices of the points
that are close to that line.
Examples
--------
>>> X = Coords([[0.,1.,0.],[3.,0.,0.],[4.,3.,0.]])
>>> p,n = [[2.,0.,0.],[0.,1.,0.]], [[0.,3.,0.],[1.,0.,0.]]
>>> print(pointNearLine(X,p,n,1.5))
[array([1]), array([0, 1])]
>>> print(pointNearLine(X,p,n,1.5,2))
[array([1]), array([0, 1])]
"""
if at.isFloat(atol):
atol = [atol]*len(X)
atol = at.checkArray1D(atol, kind=None, allow=None, size=len(X))
if nproc == 1:
ip = [np.where(X.distanceFromLine(pi, ni) < atol)[0] for pi, ni in zip(p, n)]
return ip
args = multi.splitArgs((X, p, n, atol), mask=(0, 1, 1, 0), nproc=nproc)
tasks = [(pointNearLine, a) for a in args]
res = multi.multitask(tasks, nproc)
from pyformex import olist
return olist.concatenate(res)
[docs]def faceDistance(X, Fp, return_points=False):
"""Compute the closest perpendicular distance to a set of triangles.
X is a (nX,3) shaped array of points.
Fp is a (nF,3,3) shaped array of triangles.
Note that some points may not have a normal with footpoint inside any
of the facets.
The return value is a tuple OKpid,OKdist,OKpoints where:
- OKpid is an array with the point numbers having a normal distance;
- OKdist is an array with the shortest distances for these points;
- OKpoints is an array with the closest footpoints for these points
and is only returned if return_points = True.
"""
if not Fp.shape[1] == 3:
raise ValueError("Currently this function only works for triangular faces.")
# Compute normals on the faces
Fn = np.cross(Fp[:, 1]-Fp[:, 0], Fp[:, 2]-Fp[:, 1])
# Compute projection of points X on facets F
Y = projectPointOnPlane(X, Fp[:, 0, :], Fn)
# Find intersection points Y inside the facets
inside = insideTriangle(Fp, Y)
pid = np.where(inside)[0]
if pid.size == 0:
if return_points:
return [], [], []
else:
return [], []
# Compute the distances
X = X[pid]
Y = Y[inside]
dist = at.length(X-Y)
# Get the shortest distances
OKpid, OKpos = at.groupArgmin(dist, pid)
OKdist = dist[OKpos]
if return_points:
# Get the closest footpoints matching OKpid
OKpoints = Y[OKpos]
return OKpid, OKdist, OKpoints
return OKpid, OKdist
[docs]def edgeDistance(X, Ep, return_points=False):
"""Compute the closest perpendicular distance of points X to a set of edges.
X is a (nX,3) shaped array of points.
Ep is a (nE,2,3) shaped array of edge vertices.
Note that some points may not have a normal with footpoint inside any
of the edges.
The return value is a tuple OKpid,OKdist,OKpoints where:
- OKpid is an array with the point numbers having a normal distance;
- OKdist is an array with the shortest distances for these points;
- OKpoints is an array with the closest footpoints for these points
and is only returned if return_points = True.
"""
# Compute vectors along the edges
En = Ep[:, 1] - Ep[:, 0]
# Compute intersection points of perpendiculars from X on edges E
t = projectPointOnLineTimes(X, Ep[:, 0], En)
Y = pointsAtLines(Ep[:, 0], En, t)
# Find intersection points Y inside the edges
inside = (t >= 0.) * (t <= 1.)
pid = np.where(inside)[0]
if pid.size == 0:
if return_points:
return [], [], []
else:
return [], []
# Compute the distances
X = X[pid]
Y = Y[inside]
dist = at.length(X-Y)
# Get the shortest distances
OKpid, OKpos = at.groupArgmin(dist, pid)
OKdist = dist[OKpos]
if return_points:
# Get the closest footpoints matching OKpid
OKpoints = Y[OKpos]
return OKpid, OKdist, OKpoints
return OKpid, OKdist
[docs]def vertexDistance(X, Vp, return_points=False):
"""Compute the closest distance of points X to a set of vertices.
X is a (nX,3) shaped array of points.
Vp is a (nV,3) shaped array of vertices.
The return value is a tuple OKdist,OKpoints where:
- OKdist is an array with the shortest distances for the points;
- OKpoints is an array with the closest vertices for the points
and is only returned if return_points = True.
"""
# Compute the distances
dist = at.length(X[:, np.newaxis]-Vp)
# Get the shortest distances
OKdist = dist.min(-1)
if return_points:
# Get the closest points matching X
minid = dist.argmin(-1)
OKpoints = Vp[minid]
return OKdist, OKpoints
return OKdist,
################ other functions #################################
[docs]def areaNormals(x):
"""Compute the area and normal vectors of a collection of triangles.
x is an (ntri,3,3) array with the coordinates of the vertices of ntri
triangles.
Returns a tuple (areas,normals) with the areas and the normals of the
triangles. The area is always positive. The normal vectors are normalized.
"""
x = x.reshape(-1, 3, 3)
area, normals = at.vectorPairAreaNormals(x[:, 1]-x[:, 0], x[:, 2]-x[:, 1])
area *= 0.5
return area, normals
[docs]def degenerate(area, normals):
"""Return a list of the degenerate faces according to area and normals.
area,normals are equal sized arrays with the areas and normals of a
list of faces, such as the output of the :func:`areaNormals` function.
A face is degenerate if its area is less or equal to zero or the
normal has a nan (not-a-number) value.
Returns a list of the degenerate element numbers as a sorted array.
"""
return np.unique(np.concatenate([np.where(area<=0)[0],
np.where(np.isnan(normals))[0]]))
[docs]def hexVolume(x):
"""Compute the volume of hexahedrons.
Parameters:
- `x`: float array (nelems,8,3)
Returns a float array (nelems) withe the approximate volume of the
hexahedrons formed by each 8-tuple of vertices. The volume is obained
by dividing the hexahedron in 24 tetrahedrons and using the formulas
from http://www.osti.gov/scitech/servlets/purl/632793
Example:
>>> from pyformex.elements import Hex8
>>> X = Coords(Hex8.vertices).reshape(-1,8,3)
>>> print(hexVolume(X))
[ 1.]
"""
x = at.checkArray(x, shape=(-1, 8, 3), kind='f')
x71 = x[..., 6, :] - x[..., 1, :]
x60 = x[..., 7, :] - x[..., 0, :]
x72 = x[..., 6, :] - x[..., 3, :]
x30 = x[..., 2, :] - x[..., 0, :]
x50 = x[..., 5, :] - x[..., 0, :]
x74 = x[..., 6, :] - x[..., 4, :]
return (at.vectorTripleProduct(x71+x60, x72, x30)
+ at.vectorTripleProduct(x60, x72+x50, x74)
+ at.vectorTripleProduct(x71, x50, x74+x30)) / 12
[docs]def levelVolumes(x):
"""Compute the level volumes of a collection of elements.
x is an (nelems,nplex,3) array with the coordinates of the nplex vertices
of nelems elements, with nplex equal to 2, 3 or 4.
If nplex == 2, returns the lengths of the straight line segments.
If nplex == 3, returns the areas of the triangle elements.
If nplex == 4, returns the signed volumes of the tetrahedron elements.
Positive values result if vertex 3 is at the positive side of the plane
defined by the vertices (0,1,2). Negative volumes are reported for
tetrahedra having reversed vertex order.
For any other value of nplex, raises an error.
If successful, returns an (nelems,) shaped float array.
"""
nplex = x.shape[1]
if nplex == 2:
return at.length(x[:, 1]-x[:, 0])
elif nplex == 3:
return at.vectorPairArea(x[:, 1]-x[:, 0], x[:, 2]-x[:, 1]) / 2
elif nplex == 4:
return at.vectorTripleProduct(x[:, 1]-x[:, 0], x[:, 2]-x[:, 1],
x[:, 3]-x[:, 0]) / 6
else:
raise ValueError("Plexitude should be one of 2, 3 or 4; got %s" % nplex)
# TODO: move to Coords, rename to principalSizes(order=False)
[docs]def inertialDirections(x):
"""Return the directions and dimension of a Coords based of inertia.
- `x`: a Coords-like array
Returns a tuple of the principal direction vectors and the sizes along
these directions, ordered from the smallest to the largest direction.
"""
I = x.inertia() # noqa: E741
Iprin, Iaxes = I.principal()
C = I.ctr
X = x.trl(-C).rot(Iaxes.T)
sizes = X.sizes()
i = np.argsort(sizes)
return Iaxes[i], sizes[i]
[docs]def smallestDirection(x, method='inertia', return_size=False):
"""Return the direction of the smallest dimension of a Coords.
- `x`: a Coords-like array
- `method`: one of 'inertia' or 'random'
- return_size: if True and `method` is 'inertia', a tuple of a direction
vector and the size along that direction and the cross directions;
else, only return the direction vector.
"""
x = x.reshape(-1, 3)
if method == 'inertia':
# The idea is to take the smallest dimension in a coordinate
# system aligned with the global axes.
N, sizes=inertialDirections(x)
if return_size:
return N[0], sizes[0]
else:
return N[0]
elif method == 'random':
# Take the mean of the normals on randomly created triangles
from pyformex.trisurface import TriSurface
n = x.shape[0]
m = 3 * (n // 3)
e = np.arange(m)
np.random.shuffle(e)
if n > m:
e = np.concatenate([e, [0, 1, n-1]])
S = TriSurface(x, e.reshape(-1, 3))
A, N = S.areaNormals()
ok = np.where(np.isnan(N).sum(axis=1) == 0)[0]
N = N[ok]
N = N*N
N = N.mean(axis=0)
N = np.sqrt(N)
N = at.normalize(N)
return N
[docs]def largestDirection(x, return_size=False):
"""Return the direction of the largest dimension of a Coords.
- `x`: a Coords-like array
- return_size: if True and `method` is 'inertia', a tuple of a direction
vector and the size along that direction and the cross directions;
else, only return the direction vector.
"""
x = x.reshape(-1, 3)
N, sizes = inertialDirections(x)
if return_size:
return N[2], sizes[2]
else:
return N[2]
# TODO: add parameter mode = 'all' or 'pair'
[docs]def distance(X, Y):
"""Return the distance of all points of X to those of Y.
Parameters:
- `X`: (nX,3) shaped array of points.
- `Y`: (nY,3) shaped array of points.
Returns an (nX,nT) shaped array with the distances between all points of
X and Y.
"""
X = np.asarray(X).reshape(-1, 3)
Y = np.asarray(Y).reshape(-1, 3)
return at.length(X[:, np.newaxis]-Y)
[docs]def closest(X, Y=None, return_dist=False):
"""Find the point of Y closest to each of the points of X.
Parameters:
- `X`: (nX,3) shaped array of points
- `Y`: (nY,3) shaped array of points. If None, Y is taken equal to X,
allowing to search for the closest point in a single set. In the latter
case, the point itself is excluded from the search (as otherwise
that would obviously be the closest one).
- `return_dist`: bool. If True, also returns the distances of the closest
points.
Returns:
- `ind`: (nX,) int array with the index of the closest point in Y to the
points of X
- `dist`: (nX,) float array with the distance of the closest point. This
is equal to length(X-Y[ind]). It is only returned if return_dist is True.
"""
if Y is None:
dist = distance(X, X) # Compute all distances
ar = np.arange(X.shape[0])
dist[ar, ar] = dist.max()+1.
else:
dist = distance(X, Y) # Compute all distances
ind = dist.argmin(-1) # Locate the smallest distances
if return_dist:
return ind, dist[np.arange(dist.shape[0]), ind]
else:
return ind
[docs]def closestPair(X, Y):
"""Find the closest pair of points from X and Y.
Parameters:
- `X`: (nX,3) shaped array of points
- `Y`: (nY,3) shaped array of points
Returns a tuple (i,j,d) where i,j are the indices in X,Y identifying
the closest points, and d is the distance between them.
"""
dist = distance(X, Y) # Compute all distances
ind = dist.argmin() # Locate the smallest distances
i, j = divmod(ind, Y.shape[0])
return i, j, dist[i, j]
[docs]def projectedArea(x, dir):
"""Compute projected area inside a polygon.
Parameters:
- `x`: (npoints,3) Coords with the ordered vertices of a
(possibly nonplanar) polygonal contour.
- `dir`: either a global axis number (0, 1 or 2) or a direction vector
consisting of 3 floats, specifying the projection direction.
Returns a single float value with the area inside the polygon projected
in the specified direction.
Note that if the polygon is planar and the specified direction is that
of the normal on its plane, the returned area is that of the planar
figure inside the polygon. If the polygon is nonplanar however, the area
inside the polygon is not defined. The projected area in a specified
direction is, since the projected polygon is a planar one.
"""
if x.shape[0] < 3:
return 0.0
if at.isInt(dir):
dir = at.unitVector(dir)
else:
dir = at.normalize(dir)
x1 = np.roll(x, -1, axis=0)
area = at.vectorTripleProduct(Coords(dir), x, x1)
return 0.5 * area.sum()
[docs]def polygonNormals(x):
"""Compute normals in all points of polygons in x.
x is an (nel,nplex,3) coordinate array representing nel (possibly nonplanar)
polygons.
The return value is an (nel,nplex,3) array with the unit normals on the
two edges ending in each point.
"""
if x.shape[1] < 3:
# raise ValueError("Cannot compute normals for plex-2 elements"
n = np.zeros_like(x)
n[:, :, 2] = -1.
return n
ni = np.arange(x.shape[1])
nj = np.roll(ni, 1)
nk = np.roll(ni, -1)
v1 = x-x[:, nj]
v2 = x[:, nk]-x
n = at.vectorPairNormals(v1.reshape(-1, 3),
v2.reshape(-1, 3)).reshape(x.shape)
return n
[docs]def averageNormals(coords, elems, atNodes=False, treshold=None):
"""Compute average normals at all points of elems.
coords is a (ncoords,3) array of nodal coordinates.
elems is an (nel,nplex) array of element connectivity.
The default return value is an (nel,nplex,3) array with the averaged
unit normals in all points of all elements.
If atNodes == True, a more compact array with the unique averages
at the nodes is returned.
"""
if treshold is not None:
# TODO: AVERAGE NORMALS THRESHOLD IS NOT IMPLEMENTED!
pass
normals = polygonNormals(coords[elems])
normals, cnt = at.nodalSum(normals, elems, coords.shape[0])
# No need to take average, since we are going to normalize anyway
normals = at.normalize(normals)
if not atNodes:
normals = normals[elems]
return normals
[docs]def triangleInCircle(x):
"""Compute the incircles of the triangles x.
The incircle of a triangle is the largest circle that can be inscribed
in the triangle.
x is a Coords array with shape (ntri,3,3) representing ntri triangles.
Returns a tuple r,C,n with the radii, Center and unit normals of the
incircles.
Example:
>>> X = Formex(Coords([1.,0.,0.])).rosette(3,120.)
>>> print(X)
{[1.0,0.0,0.0], [-0.5,0.866025,0.0], [-0.5,-0.866025,0.0]}
>>> radius, center, normal = triangleInCircle(X.coords.reshape(-1,3,3))
>>> print(radius)
[ 0.5]
>>> print(center)
[[ 0. 0. 0.]]
"""
at.checkArray(x, shape=(-1, 3, 3))
# Edge vectors
v = np.roll(x, -1, axis=1) - x
v = at.normalize(v)
# create bisecting lines in x0 and x1
b0 = v[:, 0]-v[:, 2]
b1 = v[:, 1]-v[:, 0]
# find intersection => center point of incircle
center = intersectLineWithLine(x[:, 0], b0, x[:, 1], b1, mode='pair')[0]
# find distance to any side => radius
radius = distanceFromLine(center, (x[:, 0], v[:, 0]), mode='pair')
# normals
normal = np.cross(v[:, 0], v[:, 1])
normal /= at.length(normal).reshape(-1, 1)
return radius, center, normal
[docs]def triangleCircumCircle(x, bounding=False):
"""Compute the circumcircles of the triangles x.
x is a Coords array with shape (ntri,3,3) representing ntri triangles.
Returns a tuple r,C,n with the radii, Center and unit normals of the
circles going through the vertices of each triangle.
If bounding=True, this returns the triangle bounding circle.
"""
at.checkArray(x, shape=(-1, 3, 3))
# Edge vectors
v = x - np.roll(x, -1, axis=1)
vv = at.dotpr(v, v)
# Edge lengths
lv = np.sqrt(vv)
n = np.cross(v[:, 0], v[:, 1])
nn = at.dotpr(n, n)
# Radius
N = np.sqrt(nn)
r = np.asarray(lv.prod(axis=-1) / N / 2)
# Center
w = -at.dotpr(np.roll(v, 1, axis=1), np.roll(v, 2, axis=1))
a = w * vv
C = a.reshape(-1, 3, 1) * np.roll(x, 1, axis=1)
C = C.sum(axis=1) / nn.reshape(-1, 1) / 2
# Unit normals
n = n / N.reshape(-1, 1)
# Bounding circle
if bounding:
# Modify for obtuse triangles
for i, j, k in [[0, 1, 2], [1, 2, 0], [2, 0, 1]]:
obt = vv[:, i] >= vv[:, j]+vv[:, k]
r[obt] = 0.5 * lv[obt, i]
C[obt] = 0.5 * (x[obt, i] + x[obt, j])
return r, C, n
[docs]def triangleBoundingCircle(x):
"""Compute the bounding circles of the triangles x.
The bounding circle is the smallest circle in the plane of the triangle
such that all vertices of the triangle are on or inside the circle.
If the triangle is acute, this is equivalent to the triangle's
circumcircle. It the triangle is obtuse, the longest edge is the
diameter of the bounding circle.
x is a Coords array with shape (ntri,3,3) representing ntri triangles.
Returns a tuple r,C,n with the radii, Center and unit normals of the
bounding circles.
"""
return triangleCircumCircle(x, bounding=True)
[docs]def triangleObtuse(x):
"""Check for obtuse triangles.
x is a Coords array with shape (ntri,3,3) representing ntri triangles.
Returns an (ntri) array of True/False values indicating whether the
triangles are obtuse.
"""
at.checkArray(x, shape=(-1, 3, 3))
# Edge vectors
v = x - np.roll(x, -1, axis=1)
vv = at.dotpr(v, v)
return ((vv[:, 0] > vv[:, 1]+vv[:, 2])
+ (vv[:, 1] > vv[:, 2]+vv[:, 0])
+ (vv[:, 2] > vv[:, 0]+vv[:, 1]))
[docs]@utils.deprecated_by('geomtools.lineIntersection', 'geomtools.intersectLineWithLine')
def lineIntersection(P0, N0, P1, N1):
"""Find the intersection of 2 (sets of) lines.
This relies on the lines being pairwise coplanar.
"""
Y0, Y1 = intersectLineWithLine(P0, N0, P1, N1, mode='pair')
return Y0
[docs]def displaceLines(A, N, C, d):
"""Move all lines (A,N) over a distance a in the direction of point C.
A,N are arrays with points and directions defining the lines.
C is a point.
d is a scalar or a list of scalars.
All line elements of F are translated in the plane (line,C)
over a distance d in the direction of the point C.
Returns a new set of lines (A,N).
"""
v = at.normalize(N)
w = C - A
vw = (v*w).sum(axis=-1).reshape((-1, 1))
Y = A + vw*v
v = at.normalize(C-Y)
return A + d*v, N
[docs]def segmentOrientation(vertices, vertices2=None, point=None):
"""Determine the orientation of a set of line segments.
vertices and vertices2 are matching sets of points.
point is a single point.
All arguments are Coords objects.
Line segments run between corresponding points of vertices and vertices2.
If vertices2 is None, it is obtained by rolling the vertices one position
foreward, thus corresponding to a closed polygon through the vertices).
If point is None, it is taken as the center of vertices.
The orientation algorithm checks whether the line segments turn
positively around the point.
Returns an array with +1/-1 for positive/negative oriented segments.
"""
if vertices2 is None:
vertices2 = np.roll(vertices, -1, axis=0)
if point is None:
point = vertices.center()
w = np.cross(vertices, vertices2)
orient = np.sign(at.dotpr(point, w)).astype(at.Int)
return orient
[docs]def rotationAngle(A, B, m=None, angle_spec=at.DEG):
"""Return rotation angles and vectors for rotations of A to B.
A and B are (n,3) shaped arrays where each line represents a vector.
This function computes the rotation from each vector of A to the
corresponding vector of B.
If m is None, the return value is a tuple of an (n,) shaped array with
rotation angles (by default in degrees) and an (n,3) shaped array with
unit vectors along the rotation axis.
If m is a (n,3) shaped array with vectors along the rotation axis, the
return value is a (n,) shaped array with rotation angles. The returned
angle is then the angle between the planes formed by the axis and the
vectors.
Specify angle_spec=RAD to get the angles in radians.
"""
A = np.asarray(A).reshape(-1, 3)
B = np.asarray(B).reshape(-1, 3)
if m is None:
A = at.normalize(A)
B = at.normalize(B)
n = np.cross(A, B) # vectors perpendicular to A and B
t = at.length(n) == 0.
if t.any(): # some vectors A and B are parallel
if A.shape[0] >= B.shape[0]:
temp = A[t]
else:
temp = B[t]
n[t] = anyPerpendicularVector(temp)
n = at.normalize(n)
c = at.dotpr(A, B)
angle = at.arccosd(c.clip(min=-1., max=1.), angle_spec)
return angle, n
else:
m = np.asarray(m).reshape(-1, 3)
# project vectors on plane
A = projectionVOP(A, m)
B = projectionVOP(B, m)
angle, n = rotationAngle(A, B, angle_spec=angle_spec)
# check sign of the angles
m = at.normalize(m)
inv = np.isclose(at.dotpr(n, m), [-1.])
angle[inv] *= -1.
return angle
[docs]def anyPerpendicularVector(A):
"""Return arbitrary vectors perpendicular to vectors of A.
A is a (n,3) shaped array of vectors.
The return value is a (n,3) shaped array of perpendicular vectors.
The returned vector is always a vector in the x,y plane. If the original
is the z-axis, the result is the x-axis.
"""
A = np.asarray(A).reshape(-1, 3)
x, y, z = np.hsplit(A, [1, 2])
n = np.zeros(x.shape, dtype=at.Float)
i = np.ones(x.shape, dtype=at.Float)
t = (x==0.)*(y==0.)
B = np.where(t, np.column_stack([i, n, n]), np.column_stack([-y, x, n]))
return B
[docs]def perpendicularVector(A, B):
"""Return vectors perpendicular on both A and B."""
return np.cross(A, B)
[docs]def projectionVOV(A, B):
"""Return the projection of vector of A on vector of B."""
L = at.projection(A, B)
B = at.normalize(B)
shape = list(L.shape)
shape.append(1)
return L.reshape(shape)*B
[docs]def projectionVOP(A, n):
"""Return the projection of vector of A on a plane with normal n."""
Aperp = projectionVOV(A, n)
return A-Aperp
#################### barycentric coordinates ###############
[docs]def baryCoords(S, P):
"""Compute the barycentric coordinates of points wrt. simplexes.
An n-simplex is a geometrical structure defined by n+1 vertices and
bordered by the convex hull of those vertices. In practice it is
either:
- 1-simplex: line segment (nplex=2)
- 2-simplex: triangle (nplex=3)
- 3-simplex: tetrahedron (nplex=4)
The barycentric coordinates of a 3d point with respect to a simplex
of a lower order are the barycentric coordinates of the projection
of that point on the simplex.
Parameters
----------
S: :term:`coords_like` (nel, nplex, 3)
A set of nel n-simplexes (n=nplex-1).
P: :term:`coords_like` (nel, npts, 3) or (1, npts, 3)
A set of npts points (for each of the simplexes in S) for which
the barycentric coordinates are to be computed. If the shape is
(1,npts,3), the same points are used with each of the simplexes.
Returns
-------
float array (nel, npts, nplex)
The barycentric coordinates of the points with respect to the
simplexes.
See Also
--------
insideSimplex: test if a (projection) point falls within a simplex
Examples
--------
>>> S = Coords('.1.6.4').reshape(3,2,3)
>>> P = Coords([[[0,0,0],[0.2,0.2,0],[0.5,0.5,0],[0.5,0.7,0]]])
>>> baryCoords(S,P)
array([[[ 1. , 0. ],
[ 0.8, 0.2],
[ 0.5, 0.5],
[ 0.5, 0.5]],
<BLANKLINE>
[[ 0.5, 0.5],
[ 0.5, 0.5],
[ 0.5, 0.5],
[ 0.4, 0.6]],
<BLANKLINE>
[[ 0. , 1. ],
[ 0.2, 0.8],
[ 0.5, 0.5],
[ 0.7, 0.3]]])
>>> S1 = Coords('016').reshape(1,3,3)
>>> baryCoords(S1,P)
array([[[ 1. , 0. , 0. ],
[ 0.6, 0.2, 0.2],
[ 0. , 0.5, 0.5],
[-0.2, 0.5, 0.7]]])
"""
S = at.checkArray(S, shape=(-1,-1,3), kind='f', allow='i')
P = at.checkArray(P, shape=(-1,-1,3), kind='f', allow='i')
if not (P.shape[0] == 1 or P.shape[0] == S.shape[0]):
raise ValueError(
"First dimension of P should 1 or equal to S.shape[0]")
vs = S[:,1:] - S[:,:1]
vp = P - S[:,:1]
A = at.dotpr(vs[:, :, np.newaxis], vs[:, np.newaxis])
B = at.dotpr(vp[:, :, np.newaxis], vs[:, np.newaxis])
# TODO: We could change axis order in solveMany to avoid transposing here
t = at.solveMany(A, B.transpose(0,2,1)).transpose(0,2,1)
t0 = (1. - t.sum(axis=-1))[:,:,np.newaxis]
t = np.concatenate([t0, t], axis=-1)
return t
[docs]def insideSimplex(S, P, atol=0.):
"""Check which points P are inside the simplexes S.
An n-simplex is a geometrical structure defined by n+1 vertices and
bordered by the convex hull of those vertices. In practice it is
either:
- 1-simplex: line segment (nplex=2)
- 2-simplex: triangle (nplex=3)
- 3-simplex: tetrahedron (nplex=4)
A 3d point is 'inside' a simplex of a lower order if its projection
on that simplex falls within the simplex. This is satisfied iff
all the barycentric coordinates of the point with respect to the
simplex are in the range [0.0, 1.0].
Parameters
----------
S: :term:`coords_like` (nel, nplex, 3)
A set of nel n-simplexes (n=nplex-1).
P: :term:`coords_like` (nel, npts, 3) or (1, npts, 3)
A set of npts points that have to be tested against the nel
simplexes. If the shape is (1,npts,3), the same points are
tested against each of the simplexes. With a shape (nel, npts, 3)
each simplex has its own set of points to be tested.
atol: float, optional
If provided, points which are falling outside the simplex within
this tolerance (in parametric space), are also reported as inside.
Returns
-------
bool array (nel, npts)
The array holds True where the (projection of the) points fall
inside the simplex, to the accuracy atol.
See Also
--------
baryCoords: compute the barycentric coordiantes of the points
Examples
--------
>>> S = Coords('.1.6.4').reshape(3,2,3)
>>> P = Coords([[[0.2,0.2,0],[0.5,0.5,0],[0.5,0.7,0],[0.5,1.2,0]]])
>>> insideSimplex(S,P)
array([[ True, True, True, True],
[ True, True, True, True],
[ True, True, True, False]])
>>> T = Coords('132.14').reshape(2,3,3)
>>> insideSimplex(T,P)
array([[ True, True, False, False],
[False, True, True, False]])
"""
# If all bc are >= 0, they are also guaranteed to be all <= 1.0,
# since their sum is 1.0
return (baryCoords(S, P) >= -atol).all(axis=-1)
insideTriangle = insideSimplex
[docs]def insideSegment(Q, Q1, P, atol=0.):
"""Check if projections of points are inside line segments.
Checks which projections of points P on the line segments Q-Q1
are within the line segments.
Parameters
----------
Q: :term:`array_like` (nseg, 3)
First point of ``nseg`` line segments.
Q1: :term:`array_like` (nseg, 3)
Second point of ``nseg`` line segments.
P: :term:`array_like` (nseg, 3)
The points to test against the segments. The testing is done
by pair: one point for each segment. The test pertains to the
projection of the point on the line containing the segment.
atol: float, optional
If provided, points which are falling outside the segment but within
this tolerance are also reported as inside.
Returns
-------
bool array (nseg)
The array holds True where the (projection of the) point falls
inside the corresponding segemnt, within the accuracy ``atol``.
Notes
-----
On large data sets this is (slightly) more efficient than the
equivalent::
S = np.stack([Q, Q1], axis=1)
P = P[:,np.newaxis]
t = geomtools.insideSimplex(S,P).reshape(-1)
See Also
--------
insideSimplex: a general test of (projection of) points inside simplexes
insideRay: a similar test for points on a half-line (ray)
Examples
--------
>>> nseg = 13
>>> Q = Coords(np.random.rand(nseg,3))
>>> Q1 = Coords(np.random.rand(nseg,3))
>>> u = at.uniformParamValues(nseg-1, -0.1, 1.1).reshape(-1,1,1)
>>> P = Q[:,np.newaxis] * (1.-u) + Q1[:, np.newaxis] * u
>>> P = P.reshape(nseg,3)
>>> insideSegment(Q, Q1, P, atol=1.e-5)
array([False, True, True, True, True, True, True, True, True,
True, True, True, False])
>>> insideRay(Q, Q1-Q, P, atol=1.e-5)
array([False, True, True, True, True, True, True, True, True,
True, True, True, True])
"""
v = Q1-Q
L = at.length(v)
Lp = at.projection(P-Q, v)
return (Lp >= -atol) * (Lp <= L + atol)
[docs]def insideRay(Q, v, P, atol=0.):
"""Check if projections of points are inside rays (half-lines).
Checks which projections of points P on the line (Q,v)
are within the positive half-lines (rays).
Parameters
----------
Q: :term:`array_like` (nlines, 3)
Starting point of ``nlines`` rays (half-lines).
v: :term:`array_like` (nlines, 3)
Direction vectors of ``nlines`` rays (half-lines).
P: :term:`array_like` (nlines, 3)
The points to test against the rays. The testing is done
by pair: one point for each segment. The test pertains to the
projection of the point on the line containing the ray.
atol: float, optional
If provided, points which are falling outside the ray but within
this tolerance are also reported as inside.
Returns
-------
bool array (nseg)
The array holds True where the (projection of the) point falls
inside the corresponding ray, within the accuracy ``atol``.
Notes
-----
On large data sets this is (slightly) more efficient than the
equivalent::
S = np.stack([Q, Q+v], axis=1)
P = P[:,np.newaxis]
B = geomtools.baryCoords(S,P).reshape(-1,2)
t = where(B[:,1] >= atol)
See Also
--------
insideSimplex: a general test of (projection of) points inside simplexes
insideRay: a similar test for points on a half-line (ray)
"""
Lp = at.projection(P-Q, v)
return (Lp >= -atol)
# End
