#
##
## 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/.
##
"""Definition of curves in pyFormex.
This module defines classes and functions specialized for handling
one-dimensional geometry in pyFormex. These may be straight lines, polylines,
higher order curves and collections thereof. In general, the curves are 3D,
but special cases may be created for handling plane curves.
"""
import numpy as np
import pyformex as pf
from pyformex import arraytools as at
from pyformex.coords import Coords
from pyformex.coordsys import CoordSys
from pyformex.geometry import Geometry
from pyformex.formex import Formex, connect
from pyformex.mesh import Mesh
from pyformex import geomtools as gt
from pyformex.varray import Varray
[docs]class Curve(Geometry):
"""Base class for curve type classes.
This is a virtual class intended to be subclassed.
It defines the common definitions for all curve types.
The subclasses should at least define the following attributes and methods
or override them if the defaults are not suitable.
Attributes:
- `coords`: coordinates of points defining the curve
- `nparts`: number of parts (e.g. straight segments of a polyline)
- `closed`: is the curve closed or not
- `range`: [min,max], range of the parameter: default 0..1
Methods:
- `sub_points(t,j)`: returns points at parameter value t,j
- `sub_directions(t,j)`: returns direction at parameter value t,j
- `pointsOn()`: the defining points placed on the curve
- `pointsOff()`: the defining points placeded off the curve (control points)
- `parts(j,k)`:
- `approx(nseg=None,ndiv=None,chordal=None)`:
Furthermore it may define, for efficiency reasons, the following methods:
- `sub_points_2`
- `sub_directions_2`
"""
def __init__(self):
Geometry.__init__(self)
self.prop = None
# Required implementation of abstract methods
[docs] def nelems(self):
return 1
def _select(self, sel, compact=False):
return self
# Other methods
def pointsOn(self):
return self.coords
def pointsOff(self):
return Coords()
def ncoords(self):
return self.coords.shape[0]
def npoints(self):
return self.pointsOn().shape[0]
[docs] def endPoints(self):
"""Return start and end points of the curve.
Returns a Coords with two points, or None if the curve is closed.
"""
if self.closed:
return None
else:
return self.coords[[0, -1]]
[docs] def sub_points(self, t, j):
"""Return the points at values t in part j
t can be an array of parameter values, j is a single segment number.
"""
raise NotImplementedError
[docs] def sub_points_2(self, t, j):
"""Return the points at values,parts given by zip(t,j)
t and j can both be arrays, but should have the same length.
"""
raise NotImplementedError
[docs] def sub_directions(self, t, j):
"""Return the directions at values t in part j
t can be an array of parameter values, j is a single segment number.
"""
raise NotImplementedError
[docs] def sub_directions_2(self, t, j):
"""Return the directions at values,parts given by zip(t,j)
t and j can both be arrays, but should have the same length.
"""
raise NotImplementedError
def lengths(self):
raise NotImplementedError
def charLength(self):
try:
return self.lengths().mean()
except Exception:
return self.bbox().dsize()
[docs] def localParam(self, t):
"""Split global parameter value in part number and local parameter
Parameter values are floating point values. Their integer part
is interpreted as the curve segment number, and the decimal part
goes from 0 to 1 over the segment.
Returns a tuple of arrays i,t, where i are the (integer) part
numbers and t the local parameter values (between 0 and 1).
"""
# Do not use np.asarray here! We change it, so need a copy!
t = np.array(t).astype(at.Float).ravel()
ti = np.floor(t).clip(min=0, max=self.nparts-1)
t -= ti
i = ti.astype(int)
return i, t
[docs] def pointsAt(self, t, return_position=False):
"""Return the points at parameter values t.
Parameter values are floating point values. Their integer part
is interpreted as the curve segment number, and the decimal part
goes from 0 to 1 over the segment.
Returns a Coords with the coordinates of the points.
If return_position is True, also returns the part numbers on which
the point are lying and the local parameter values.
"""
t = np.array(t)
i, t = self.localParam(t)
try:
allX = self.sub_points_2(t, i)
except Exception:
allX = np.concatenate([self.sub_points(tj, ij) for tj, ij in zip(t, i)])
X = Coords(allX)
if return_position:
return X, i, t
else:
return X
[docs] def directionsAt(self, t):
"""Return the directions at parameter values t.
Parameter values are floating point values. Their integer part
is interpreted as the curve segment number, and the decimal part
goes from 0 to 1 over the segment.
"""
i, t = self.localParam(t)
try:
allX = self.sub_directions_2(t, i)
except Exception:
allX = np.concatenate([self.sub_directions(tj, ij) for tj, ij in zip(t, i)])
return Coords(allX)
[docs] def subPoints(self, div=10, extend=[0., 0.]):
"""Return a sequence of points on the Curve.
- `div`: int or a list of floats (usually in the range [0.,1.])
If `div` is an integer, a list of floats is constructed by dividing
the range [0.,1.] into `div` equal parts.
The list of floats then specifies a set of parameter values for which
points at in each part are returned. The points are returned in a
single Coords in order of the parts.
The extend parameter allows to extend the curve beyond the endpoints.
The normal parameter space of each part is [0.0 .. 1.0]. The extend
parameter will add a curve with parameter space [-extend[0] .. 0.0]
for the first part, and a curve with parameter space
[1.0 .. 1 + extend[0]] for the last part.
The parameter step in the extensions will be adjusted slightly so
that the specified extension is a multiple of the step size.
If the curve is closed, the extend parameter is disregarded.
"""
# Subspline parts (without end point)
if at.isInt(div):
u = np.arange(div) / float(div)
else:
u = np.array(div).ravel()
div = len(u)
parts = [self.sub_points(u, j) for j in range(self.nparts)]
if not self.closed:
nstart, nend = np.round(np.asarray(extend)*div, 0).astype(at.Int)
# Extend at start
if nstart > 0:
u = np.arange(-nstart, 0) * extend[0] / float(nstart)
parts.insert(0, self.sub_points(u, 0))
# Extend at end
if nend > 0:
u = 1. + np.arange(0, nend+1) * extend[1] / float(nend)
else:
# Always extend at end to include last point
u = np.array([1.])
parts.append(self.sub_points(u, self.nparts-1))
X = np.concatenate(parts, axis=0)
return Coords(X)
[docs] def split(self, split=None):
"""Split a curve into a list of partial curves
split is a list of integer values specifying the node numbers
where the curve is to be split. As a convenience, a single int may
be given if the curve is to be split at a single node, or None
to split at all nodes.
Returns a list of open curves of the same type as the original.
"""
if split is None:
split = np.arange(1, self.nparts)
else:
if at.isInt(split):
split = [split]
split = np.unique(split)
if len(split) == 0:
return [self.copy()]
if self.closed:
if split[-1] != split[0]+self.nparts:
split = np.concatenate([split, [split[0]]])
else:
if split[0] != 0:
split = np.concatenate([[0], split])
if split[-1] != self.nparts:
split = np.concatenate([split, [self.nparts]])
return [self.parts(j, k) for j, k in zip(split[:-1], split[1:])]
[docs] def length(self):
"""Return the total length of the curve.
This is only available for curves that implement the 'lengths'
method.
"""
return self.lengths().sum()
[docs] def atApproximate(self, nseg=None, ndiv=None, equidistant=False, npre=None):
"""Return parameter values for approximating a Curve with a PolyLine.
Parameters:
- `nseg`: number of segments of the resulting PolyLine. The number
of returned parameter values is `nseg` if the curve is closed,
else `nseg+1`. Only used if `ndiv` is not specified.
- `ndiv`: positive integer or a list thereof. If a single integer,
it specifies the number of straight segments in each part of the
curve, and is thus equivalent with `nseg = ndiv * self.nparts`.
If a list of integers, its length should be equal to the number
of parts in the curve and each integer in the list specifies the
number of segments the corresponding part.
- `equidistant`: bool, only used if ndiv is not specified.
If True, the points are spaced almost equidistantly
over the curve. If False (default), the points are spread
equally over the parameter space.
- `npre`: integer: only used when `equidistant` is True:
number of segments per part of the curve used in the
pre-approximation. This pre-approximation is currently
required to compute curve lengths.
Examples
--------
>>> PL = PolyLine([[0,0,0],[1,0,0],[1,1,0]])
>>> print(PL.atApproximate(nseg=6))
[ 0. 0.33 0.67 1. 1.33 1.67 2. ]
>>> print(PL.atApproximate(ndiv=3))
[ 0. 0.33 0.67 1. 1.33 1.67 2. ]
>>> print(PL.atApproximate(ndiv=(2,4)))
[ 0. 0.5 1. 1.25 1.5 1.75 2. ]
>>> print(PL.atApproximate(ndiv=(2,)))
[ 0. 0.5 1. ]
"""
if ndiv is not None:
if at.isInt(ndiv):
nseg = ndiv * self.nparts
ndiv = None
equidistant = False
if equidistant:
if npre is None:
npre = 100
S = self.approx(ndiv=npre)
atl = S.atLength(nseg) / npre
elif ndiv is not None:
atl = np.concatenate([[0.]] + [i + at.unitDivisor(n)[1:]
for i, n in enumerate(ndiv)])
else:
if nseg is None:
nseg = self.nparts
atl = np.arange(nseg+1) * float(self.nparts) / nseg
return atl
[docs] def atChordal(self, chordal, atl=None):
"""Return parameter values to approximate within given chordal error.
Parameters:
- chordal: relative tolerance on the distance of the chord to the
curve. The tolerance is relative to the curve's charLength().
- atl: list of floats: list of parameter values that need to be
included in the result. The list should contain increasing values
in the curve's parameter range. If not specified, a default is
set assuring that the curve is properly approximated. If you
specify this yourself, you may end up with bad approximations due
to bad choice of the initial values.
Returns a list of parameter values that create a PolyLine
approximate for the curve, such that the chordal error is
everywhere smaller than the give value. The chordal error
is defined as the distance from a point of the curve to the
chord.
"""
# Create first approximation
if atl is None:
if hasattr(self, 'degree'):
ndiv = self.degree
else:
ndiv = 1
atl = self.atApproximate(nseg=ndiv*self.nparts)
charlen = self.approxAt(atl).charLength()
# Split in handled and to-be-handled
atl, bt = list(atl[:1]), list(atl[1:])
# Handle segment by segment
#
# THIS SHOULD BE CHANGED:
# insert NO points of degree 1 (always correct)
# insert 1 point at 1/2 for degree 2 (current implementation)
# insert 2 points at 1/3 and 2/3 for degree 3
# etc...
#
while len(bt) > 0:
c0, c2 = atl[-1], bt[0]
if c2 > c0:
c1 = 0.5*(c0+c2)
X = self.pointsAt([c0, c1, c2])
XM = 0.5*(X[0]+X[2])
d = pf.arraytools.length(X[1]-XM) / charlen
if d >= chordal:
# need refinement: put new point in front of todo list
bt = [c1] + bt
continue
# move on
atl.append(bt.pop(0))
return atl
[docs] def approxAt(self, atl):
"""Create a PolyLine approximation with specified parameter values.
Parameters:
- `atl`: a list of parameter values in the curve parameter range.
The Curve points at these parameter values are connected with
straight segments to form a PolyLine approximation.
"""
X = self.pointsAt(atl)
PL = PolyLine(X, closed=self.closed)
return PL.setProp(self.prop)
[docs] def approx(self, nseg=None, ndiv=None, chordal=0.02, equidistant=False, npre=None):
"""Approximate a Curve with a PolyLine of n segments
If neither `nseg` nor `ndiv` are specified (default), a chordal method
is used limiting the chordal distance of the curve to the PolyLine
segments.
Parameters:
- `nseg`: integer: number of straight segments of the resulting
PolyLine. Only used if `ndiv` is not specified.
- `ndiv`: positive integer or a list thereof. If a single integer,
it specifies the number of straight segments in each part of the
curve, and is thus equivalent with `nseg = ndiv * self.nparts`.
If a list of integers, its length should be equal to the number
of parts in the curve and each integer in the list specifies the
number of segments the corresponding part.
- `chordal`: float: accuracy of the approximation when using the
'chordal error' method. This is the case if neither `nseg` nor
`ndiv` are specified (the default). The value is relative to the
curve's :func:`charLength`.
- `equidistant`: bool, only used if `nseg` is specified and `ndiv`
is not. If True the `nseg+1` points are spaced almost equidistantly
over the curve. If False (default), the points are spread equally
over the parameter space.
- `npre`: integer, only used if the `chordal` method is used or if
`nseg` is not None and `equidistant` is True: the number of segments
per part of the curve (like `ndiv`) used in a pre-approximation.
If not specified, it is set to the degree of the curve for the
`chordal` method (1 for PolyLine), and to 100 in the `equidistant`
method (where the pre-approximation is currently used to compute
accurate curve lengths).
"""
if nseg or ndiv or equidistant:
atl = self.atApproximate(nseg, ndiv, equidistant, npre)
else:
atl = self.atChordal(chordal)
if self.closed and atl[-1] == float(self.nparts):
atl = atl[:-1]
return self.approxAt(atl)
# For compatibility/convenience
approximate = approx
[docs] def frenet(self, ndiv=None, nseg=None, chordal=0.01, upvector=None,
avgdir=True, compensate=False):
"""Return points and Frenet frame along the curve.
A PolyLine approximation for the curve is constructed, using the
:meth:`Curve.approx()` method with the arguments `ndiv`, `nseg`
and `chordal`.
Then Frenet frames are constructed with :meth:`PolyLine._movingFrenet`
using the remaining arguments.
The resulting PolyLine points and Frenet frames are returned.
Parameters:
- `upvector`: (3,) vector: a vector normal to the (tangent,normal)
plane at the first point of the curve. It defines the binormal at
the first point. If not specified it is set to the shorted distance
through the set of 10 first points.
- `avgdir`: bool or array.
If True (default), the tangential vector is set to
the average direction of the two segments ending at a node.
If False, the tangent vectors will be those of the line segment
starting at the points.
The tangential vector can also be set by the user by specifying
an array with the matching number of vectors.
- `compensate`: bool: If True, adds a compensation algorithm if the
curve is closed. For a closed curve the moving Frenet
algorithm can be continued back to the first point. If the resulting
binormial does not coincide with the starting one, some torsion is
added to the end portions of the curve to make the two binormals
coincide.
This feature is off by default because it is currently experimental
and is likely to change in future.
It may also form the base for setting the starting as well as the
ending binormal.
Returns:
- `X`: a Coords with `npts` points on the curve
- `T`: normalized tangent vector to the curve at `npts` points
- `N`: normalized normal vector to the curve at `npts` points
- `B`: normalized binormal vector to the curve at `npts` points
"""
PL = self.approx(ndiv=ndiv, nseg=nseg, chordal=chordal)
X = PL.coords
T, N, B = PL._movingFrenet(upvector=upvector, avgdir=avgdir,
compensate=compensate)
return X, T, N, B
[docs] def position(self, geom, csys=None):
""" Position a Geometry object along a path.
Parameters:
- `geom`: Geometry or Coords.
- `csys`: CoordSys.
For each point of the curve, a copy of the Geometry/Coords object
is created, and positioned thus that the specified csys (default
the global axes) coincides with the curve's frenet axes at that
point.
Returns a list of Geometry/Coords objects.
"""
X, T, N, B = self.frenet()
if csys:
geom = geom.fromCS(csys)
return [geom.fromCS(CoordSys(rot=np.row_stack([t, n, b]), trl=x))
for x, t, n, b in zip(X, T, N, B)]
[docs] def sweep(self, mesh, eltype=None, csys=None):
"""Sweep a mesh along the curve, creating an extrusion.
Parameters:
- `mesh`: Mesh-like object. This is usually a planar object
that is swept in the direction normal to its plane.
- `eltype`: string. Name of the element type on the
returned Meshes.
Returns a Mesh obtained by sweeping the given Mesh over a path.
The returned Mesh has double plexitude of the original.
If `path` is a closed Curve connect back to the first.
.. note:: Sweeping nonplanar objects and/or sweeping along very curly
curves may result in physically impossible geometries.
See Also
--------
sweep2
"""
loop = self.closed
mesh = mesh.toMesh()
seq = self.position(mesh.coords, csys)
return mesh.connect(seq, eltype=eltype, loop=loop)
[docs] def sweep2(self, coords, origin=(0., 0., 0.), scale=None, **kargs):
""" Sweep a Coords object along a curve, returning a series of copies.
At each point of the curve a copy of the coords is created,
possibly scaled, and rotated to keep same orientation with
respect to the curve.
Parameters
----------
coords: Coords object
The Coords object to be copied, possibly scaled and positioned
at each point of the curve.
origin: float :term:`array_like` (3,)
The local origin in the Coords. This is the point that will
be positioned at the curve's points. It is also the center of
scaling if `scale` is provided.
scale: float :term:`array_like`, optional
If provided, it should have the shape (npts,3). For each point
of the curve, it specifies the three scaling factors to be used
in the three coordinate directions on the `coords` for the copy
that is to be plavced at that point.
**kargs: optional keyword arguments
Extra keyword arguments that are passed to
:func:`positionCoordsObj`.
Returns
-------
A sequence of the transformed Coords objects.
See Also
--------
sweep
"""
if scale is not None:
scale = at.checkArray(scale, shape=(self.ncoords(), 3), kind='f', allow='i')
# translate input coords to origin
base = coords.translate(-Coords(origin))
# create the scaled copies
if scale is not None:
sequence = (base.scale(*sc) for sc in scale)
else:
sequence = (base, ) * self.ncoords()
# position the copies at the curve's points
sequence = positionCoordsObj(sequence, self, **kargs)
return sequence
[docs] def toNurbs(self):
"""Convert a curve to a NurbsCurve.
This is currently only implemented for BezierSpline and
PolyLine
"""
from pyformex.plugins.nurbs import NurbsCurve
if isinstance(self, (BezierSpline, PolyLine)):
return NurbsCurve(self.coords, degree=self.degree,
closed=self.closed, blended=False)
else:
raise ValueError("Can not convert a curve of type "
f"{self.__class__.__name__} to NurbsCurve")
[docs] def setProp(self, p=None):
"""Create or destroy the property number for the Curve.
A curve can have a single integer as property number.
If it is set, derived curves and approximations will inherit it.
Use this method to set or remove the property.
Parameters:
- `p`: integer or None. If a value None is given, the property is
removed from the Curve.
"""
try:
self.prop = int(p)
except Exception:
self.prop = None
return self
##############################################################################
#
# Curves that can be transformed by Coords Transforms
#
##############################################################################
#
[docs]class PolyLine(Curve):
"""A class representing a series of straight line segments.
coords is a (npts,3) shaped array of coordinates of the subsequent
vertices of the polyline (or a compatible data object).
If closed == True, the polyline is closed by connecting the last
point to the first. This does not change the vertex data.
The `control` parameter has the same meaning as `coords` and is added
for symmetry with other Curve classes. If specified, it will override
the `coords` argument.
"""
degree = 1
def __init__(self, coords=[], control=None, closed=False):
"""Initialize a PolyLine from a coordinate array."""
Curve.__init__(self)
if control is not None:
coords = control
if isinstance(coords, Formex):
if coords.nplex() == 1:
coords = coords.points()
elif coords.nplex() == 2:
coords = Coords.concatenate([coords.coords[:, 0, :],
coords.coords[-1, 1, :]])
else:
raise ValueError("Only Formices with plexitude "
"1 or 2 can be converted to PolyLine")
else:
# This allows any Coords initialization, e.g. with a pattern string!
coords = Coords(coords)
if coords.ndim != 2 or coords.shape[1] != 3 or coords.shape[0] < 2:
raise ValueError("Expected an (npoints,3) shaped coordinate array "
f"with npoints >= 2, got shape {coords.shape}")
self.coords = coords
self.nparts = self.coords.shape[0]
if not closed:
self.nparts -= 1
self.closed = closed
[docs] def close(self, atol=0.):
"""Close a PolyLine.
If the PolyLine is already closed, this does nothing.
Else it is closed by one of two methods, depending on the distance
between the start and end points of the PolyLine:
- if the distance is smaller than atol, the last point is removed and
the last segment now connects the penultimate point with the first
one, leaving the number of segments unchanged and the number of points
decreased by one;
- if the distance is not smaller than atol, a new segment is added
connecting the last point with the first, resulting in a curve with
one more segment and the number of points unchanged.
Since the default value for atol is 0.0, this is the default
behavior for all PolyLines.
Returns True if the PolyLine was closed without adding a segment.
The return value can be used to reopen the PolyLine while keeping all
segments (see :meth:`open`).
.. warning:: This method changes the PolyLine inplace.
"""
if not self.closed:
d = at.length(self.coords[0]-self.coords[-1])
if d < atol:
self.coords = self.coords[:-1]
ret = True
else:
self.nparts += 1
ret = False
self.closed = True
return ret
[docs] def open(self, keep_last=False):
"""Open a closed PolyLine.
If the PolyLine is not closed, this does nothing.
Else, the PolyLine is opened in one of two ways:
- if keep_last is True, the PolyLine is opened by adding a last
point equal to the first, and keeping the number of segments
unchanged;
- if False, the PolyLine is opened by removing the last segment,
keeping the number of points unchanged. This is the default
behavior.
There is no return value.
If a closed PolyLine is opened with keep_last=True, the first and
last point will coincide. In order to close it again, a positive
atol value needs to be used in :meth:`close`.
.. warning:: This method changes the PolyLine inplace.
"""
if self.closed:
if keep_last:
self.coords = Coords.concatenate([self.coords, self.coords[0]])
else:
self.nparts -= 1
self.closed = False
[docs] def nelems(self):
return self.nparts
[docs] def toMesh(self):
"""Convert the PolyLine to a plex-2 Mesh.
The returned Mesh is equivalent with the PolyLine.
"""
e1 = np.arange(self.ncoords())
elems = np.column_stack([e1, np.roll(e1, -1)])
if not self.closed:
elems = elems[:-1]
return Mesh(self.coords, elems, eltype='line2').setProp(self.prop)
[docs] def sub_points(self, t, j):
"""Return the points at values t in part j"""
j = int(j)
t = np.asarray(t).reshape(-1, 1)
n = self.coords.shape[0]
X0 = self.coords[j % n]
X1 = self.coords[(j+1) % n]
X = (1.-t) * X0 + t * X1
return X
[docs] def sub_points_2(self, t, j):
"""Return the points at value,part pairs (t,j)"""
j = int(j)
t = np.asarray(t).reshape(-1, 1)
n = self.coords.shape[0]
X0 = self.coords[j % n]
X1 = self.coords[(j+1) % n]
X = (1.-t) * X0 + t * X1
return X
[docs] def sub_directions(self, t, j):
"""Return the unit direction vectors at values t in part j."""
j = int(j)
t = np.asarray(t).reshape(-1, 1)
return self.directions()[j].reshape(len(t), 3)
[docs] def vectors(self):
"""Return the vectors of each point to the next one.
The vectors are returned as a Coords object.
If the curve is not closed, the number of vectors returned is
one less than the number of points.
"""
x = self.coords
if self.closed:
x1 = x
x2 = np.roll(x, -1, axis=0) # NEXT POINT
else:
x1 = x[:-1]
x2 = x[1:]
return x2-x1
[docs] def directions(self, return_doubles=False):
"""Returns unit vectors in the direction of the next point.
This directions are returned as a Coords object with the same
number of elements as the point set.
If two subsequent points are identical, the first one gets
the direction of the previous segment. If more than two subsequent
points are equal, an invalid direction (NaN) will result.
If the curve is not closed, the last direction is set equal to the
penultimate.
If return_doubles is True, the return value is a tuple of the direction
and an index of the points that are identical with their follower.
"""
d = at.normalize(self.vectors())
w = np.where(np.isnan(d).any(axis=-1))[0]
d[w] = d[w-1]
if not self.closed:
d = np.concatenate([d, d[-1:]], axis=0)
if return_doubles:
return d, w
else:
return d
[docs] def avgDirections(self, return_doubles=False):
"""Returns the average directions at points.
For each point the returned direction is the average of the direction
from the preceding point to the current, and the direction from the
current to the next point.
If the curve is open, the first and last direction are equal to the
direction of the first, resp. last segment.
Where two subsequent points are identical, the average directions
are set equal to those of the segment ending in the first and the
segment starting from the last.
"""
d, w = self.directions(True)
d1 = d
d2 = np.roll(d, 1, axis=0) # PREVIOUS DIRECTION
w = np.concatenate([w, w+1])
if not self.closed:
w = np.concatenate([[0, self.npoints()-1], w])
w = np.setdiff1d(np.arange(self.npoints()), w)
d[w] = 0.5 * (d1[w]+d2[w])
if return_doubles:
return d, w
else:
return d
[docs] def cosAngles(self):
"""Return the cosinus of the angles between subsequent segments.
Returns an array of floats in the range [-1.0..1.0]. The value
at index i is the cosinus of the angle between the segment i and
the segment i-1. The number of floats is always equal to the number
of points.
If the curve is not closed, the first value is the cosinus of the
angle between the last and the first segment, while the last value
is always equal to 1.0.
Where a curve has two subsequent coincident points, the value for the
the first point will be 1.0. Where a curve has more than two
subsequent coincident points, NAN values will result.
Examples:
>>> C1 = PolyLine('01567')
>>> C2 = PolyLine('01567',closed=True)
>>> C3 = PolyLine('015674')
>>> print(C1.cosAngles())
[-0.71 0.71 0. 0. 1. ]
>>> print(C2.cosAngles())
[ 0. 0.71 0. 0. 0.71]
>>> print(C3.cosAngles())
[ 0. 0.71 0. 0. 0.71 1. ]
"""
d = self.directions()
return at.vectorPairCosAngle(np.roll(d, 1, axis=0), d)
[docs] def splitByAngle(self, cosangle=0.0, angle=None, angle_spec=at.DEG):
"""Split a curve at points with high change of direction.
Parameters:
- `cosangle`: float: threshold value for the cosinus of the angle
between subsequent segment vectors. The curve will be split at
all points where the value of :meth:`cosAngles` is (algebraicallly)
lower than or equal to this threshold value. The default value will
split the curve where the direction changes with more than 90
degrees. A value 1.0 will split the curve at all points. A value
of -1.0 will only split where the curve direction exactly reverses.
- `angle`: float: if specified, cosangle will be computed from
at.cosd(angle,angle_spec)
- `angle_spec`: float: units used for the angle. Thiis is the number
of radians for 1 unit.
Returns a list of PolyLines, where each PolyLine has limited direction
changes. Major changes in direction occur between the PolyLines.
"""
if at.isFloat(angle):
cosangle = at.cosd(angle, angle_spec)
w = np.where(self.cosAngles() <= cosangle)[0]
return self.split(w)
def _compensate(self, end=0, cosangle=0.5):
"""_Compensate an end discontinuity over a piece of curve.
An end discontinuity in the curve is smeared out over a part of
the curve where no sharp angle occur with cosangle < 0.5
"""
cosa = self.cosAngles()
print("COSA", cosa)
L = self.lengths()
if end < 0:
L = L[::-1]
cosa = np.roll(cosa, -1)[::-1]
print(len(L), L)
for i in range(1, len(L)):
if cosa[i] <= cosangle:
break
print("HOLDING AT i=%s" % i)
L = L[:i]
print(len(L), L)
L = L[::-1].cumsum()
comp = L / L[-1]
comp = comp[::-1]
print("COMP", comp)
return comp
def _movingFrenet(self, upvector=None, avgdir=True, compensate=False):
"""Return a Frenet frame along the curve.
..note : The recommended way to use this method is through the
:meth:`frenet` method.
The Frenet frame consists of a system of three orthogonal vectors:
the tangent (T), the normal (N) and the binormal (B).
These vectors define a coordinate system that re-orients while walking
along the polyline.
The _movingFrenet method tries to minimize the twist angle.
Parameters:
- `upvector`: (3,) vector: a vector normal to the (tangent,normal)
plane at the first point of the curve. It defines the binormal at
the first point. If not specified it is set to the shorted distance
through the set of 10 first points.
- `avgdir`: bool or array.
If True (default), the tangential vector is set to
the average direction of the two segments ending at a node.
If False, the tangent vectors will be those of the line segment
starting at the points.
The tangential vector can also be set by the user by specifying
an array with the matching number of vectors.
- `compensate`: bool: If True, adds a compensation algorithm if the
curve is closed. For a closed curve the moving Frenet
algorithm can be continued back to the first point. If the resulting
binormial does not coincide with the starting one, some torsion is
added to the end portions of the curve to make the two binormals
coincide.
This feature is off by default because it is currently experimental
and is likely to change in future.
It may also form the base for setting the starting as well as the
ending binormal.
Returns:
- `T`: normalized tangent vector to the curve at `npts` points
- `N`: normalized normal vector to the curve at `npts` points
- `B`: normalized binormal vector to the curve at `npts` points
"""
if isinstance(avgdir, np.ndarray):
T = at.checkArray(avgdir, (self.ncoords(), 3), 'f')
elif avgdir:
T = self.avgDirections()
else:
T = self.directions()
B = np.zeros(T.shape)
if upvector is None:
upvector = gt.smallestDirection(self.coords[:10])
B[-1] = at.normalize(upvector)
for i, t in enumerate(T):
B[i] = np.cross(t, np.cross(B[i-1], t))
if np.isnan(B[i]).any():
# if we can not compute binormal, keep previous
B[i] = B[i-1]
T = at.normalize(T)
B = at.normalize(B)
if self.closed and compensate:
from pyformex.gui.draw import drawVectors
print(len(T))
print(T[0], B[0])
print(T[-1], B[-1])
P0 = self.coords[0]
if avgdir:
T0 = T[0]
else:
T0 = T[-1]
B0 = np.cross(T0, np.cross(B[-1], T0))
N0 = np.cross(B0, T0)
drawVectors(P0, T0, size=2., nolight=True, color='magenta')
drawVectors(P0, N0, size=2., nolight=True, color='yellow')
drawVectors(P0, B0, size=2., nolight=True, color='cyan')
print("DIFF", B[0], B0)
if at.length(np.cross(B[0], B0)) < 1.e-5:
print("NO NEED FOR COMPENSATION")
else:
PM, BM = gt.intersectionLinesPWP(P0, T[0], P0, T0, mode='pair')
PM = PM[0]
BM = BM[0]
if at.length(BM) < 1.e-5:
print("NORMAL PLANES COINCIDE")
BM = (B0 + B[0])/2
print("COMPENSATED B0: %s" % BM)
print("Compensate at start")
print(BM, B[0])
a = at.vectorPairAngle(B[0], BM) / at.DEG
print("ANGLE to compensate: %s" % a)
sa = np.sign(at.projection(np.cross(B[0], BM), T[0]))
print("Sign of angle: %s" % sa)
a *= sa
torsion = a * self._compensate(0)
print("COMPENSATION ANGLE", torsion)
for i in range(len(torsion)):
B[i] = Coords(B[i]).rotate(torsion[i], axis=T[i])
print("Compensate at end")
print(BM, B0)
a = at.vectorPairAngle(B0, BM) / at.DEG
print("ANGLE to compensate: %s" % a)
sa = np.sign(at.projection(np.cross(BM, B0), T[-1]))
print("Sign of angle: %s" % sa)
a *= sa
torsion = a * self._compensate(-1)
print("COMPENSATION ANGLE", torsion)
for i in range(len(torsion)):
B[-i] = Coords(B[-i]).rotate(torsion[i], axis=T[-i])
N = np.cross(B, T)
return T, N, B
[docs] def roll(self, n):
"""Roll the points of a closed PolyLine."""
if self.closed:
return PolyLine(np.roll(self.coords, n, axis=0), closed=True)
else:
raise ValueError("""Can only roll a closed PolyLine""")
[docs] def lengths(self):
"""Return the length of the parts of the curve."""
return at.length(self.vectors())
[docs] def cumLengths(self):
"""Return the cumulative length of the curve for all vertices."""
return np.concatenate([[0.], self.lengths().cumsum()])
[docs] def relLengths(self):
"""Return the relative length of the curve for all vertices."""
cumlen = self.cumLengths()
return cumlen / cumlen[-1]
[docs] def atLength(self, div):
"""Returns the parameter values at given relative curve length.
``div`` is a list of relative curve lengths (from 0.0 to 1.0).
As a convenience, a single integer value may be specified,
in which case the relative curve lengths are found by dividing
the interval [0.0,1.0] in the specified number of subintervals.
The function returns a list with the parameter values for the points
at the specified relative lengths.
"""
rlen = self.relLengths()
if at.isInt(div):
div = np.arange(div+1) / float(div)
else:
div = np.asarray(div)
z = rlen.searchsorted(div)
# we need interpolation
wi = np.where(z>0)[0]
zw = z[wi]
L0 = rlen[zw-1]
L1 = rlen[zw]
ai = zw + (div[wi] - L1) / (L1-L0)
atl = np.zeros(len(div))
atl[wi] = ai
return atl
[docs] def reverse(self):
"""Return the same curve with the parameter direction reversed."""
return PolyLine(at.reverseAxis(self.coords, axis=0), closed=self.closed)
[docs] def parts(self, j, k):
"""Return a PolyLine containing only segments j to k (k not included).
j and k should be in the range [0:nparts]. For an open curve j < k.
For a closed curve a value k<=j is allowed, to get parts spanning
the point 0 as a single (open) PolyLine.
The resulting PolyLine is always open.
"""
start = j
end = k + 1
if start < end:
coords = self.coords[start:end]
else:
if self.closed:
coords = np.concatenate([self.coords[start:], self.coords[:end]])
else:
raise ValueError("For open curves k>j is required")
return PolyLine(control=coords, closed=False)
[docs] def cutWithPlane(self, p, n, side=''):
"""Return the parts of the PolyLine at one or both sides of a plane.
If side is '+' or '-', return a list of PolyLines with the parts at
the positive or negative side of the plane.
For any other value, returns a tuple of two lists of PolyLines,
the first one being the parts at the positive side.
p is a point specified by 3 coordinates.
n is the normal vector to a plane, specified by 3 components.
"""
n = np.asarray(n)
p = np.asarray(p)
d = self.coords.distanceFromPlane(p, n)
t = d > 0.0
cut = t != np.roll(t, -1)
if not self.closed:
cut = cut[:-1]
w = np.where(cut)[0] # segments where side switches
res = [[], []]
i = 0 # first point to include in next part
if t[0]:
sid = 0 # start at '+' side
else:
sid = 1 # start at '-' side
Q = Coords() # new point from previous cutting (None at start)
for j in w:
pts = self.coords.take(range(j, j+2), axis=0, mode='wrap')
P = gt.intersectionPointsSWP(pts, p, n, mode='pair')[0]
x = Coords.concatenate([Q, self.coords[i:j+1], P])
res[sid].append(PolyLine(x))
sid = 1-sid
i = j+1
Q = P
# Remaining points
x = Coords.concatenate([Q, self.coords[i:]])
if not self.closed:
# append the remainder as an extra PolyLine
res[sid].append(PolyLine(x))
else:
# append remaining points to the first
if len(res[sid]) > 0:
x = Coords.concatenate([x, res[sid][0].coords])
res[sid][0] = PolyLine(x)
else:
res[sid].append(PolyLine(x))
if len(res[sid]) == 1 and len(res[1-sid]) == 0:
res[sid][0].closed = True
# Do not use side in '+-', because '' in '+-' returns True
if side in ['+', '-']:
return res['+-'.index(side)]
else:
return res
[docs] def append(self, PL, fuse=True, smart=False, **kargs):
"""Concatenate two open PolyLines.
This combines two open PolyLines into a single one. Closed PolyLines
cannot be concatenated.
Parameters:
- `PL`: an open PolyLine, to be appended at the end of the current.
- `fuse`: bool. If True, the last point of `self` and the first point
of `PL` will be fused to a single point if they are close. Extra
parameters may be added to tune the fuse operation. See the
:meth:`Coords.fuse` method. The `ppb` parameter is not allowed.
- `smart`: bool. If True, `PL` will be connected to `self` by the
endpoint that is closest to the last point of self, thus possibly
reversing the sense of PL.
The same result (with the default parameter values) can also be
achieved using operator syntax: `PolyLine1 + PolyLine2`.
"""
if self.closed or PL.closed:
raise RuntimeError("Closed PolyLines cannot be concatenated.")
if smart:
d = PL.coords[[0, -1]].distanceFromPoint(self.coords[-1])
print("Dist: %s" % d)
if d[1] < d[0]:
PL = PL.reverse()
X = PL.coords
if fuse:
x = Coords.concatenate([self.coords[-1], X[0]])
x, e = x.fuse(ppb=3) # !!! YES ! > 2 !!!
if e[0] == e[1]:
X = X[1:]
return PolyLine(Coords.concatenate([self.coords, X]))
# allow syntax PL1 + PL2
__add__ = append
[docs] @staticmethod
def concatenate(PLlist, **kargs):
"""Concatenate a list of :class:`PolyLine` objects.
Parameters:
- `PLlist`: a list of open PolyLines.
- Other parameters are like in :meth:`append`
Returns a PolyLine which is the concatenation of all the PolyLines
in `PLlist`
"""
PL = PLlist[0]
for pl in PLlist[1:]:
PL = PL.append(pl, **kargs)
return PL
[docs] def insertPointsAt(self, t, return_indices=False):
"""Insert new points at parameter values t.
Parameter values are floating point values. Their integer part
is interpreted as the curve segment number, and the decimal part
goes from 0 to 1 over the segment.
Returns a PolyLine with the new points inserted. Note that the
parameter values of the points will have changed.
If return_indices is True, also returns the indices of the
inserted points in the new PolyLine.
"""
t = sorted(t)
X, i, t = self.pointsAt(t, return_position=True)
Xc = self.coords.copy()
if return_indices:
ind = -np.ones(len(Xc))
# Loop in descending order to avoid recomputing parameters
for j in range(len(i))[::-1]:
Xi, ii = X[j].reshape(-1, 3), i[j]+1
Xc = Coords.concatenate([Xc[:ii], Xi, Xc[ii:]])
if return_indices:
ind = np.concatenate([ind[:ii], [j], ind[ii:]])
PL = PolyLine(Xc, closed=self.closed)
if return_indices:
ind = ind.tolist()
ind = [ind.index(i) for i in np.arange(len(i))]
return PL, ind
else:
return PL
[docs] def splitAt(self, t):
"""Split a PolyLine at parametric values t
Parameter values are floating point values. Their integer part
is interpreted as the curve segment number, and the decimal part
goes from 0 to 1 over the segment.
Returns a list of open Polylines.
"""
PL, t = self.insertPointsAt(t, return_indices=True)
return PL.split(t)
[docs] def splitAtLength(self, L):
"""Split a PolyLine at relative lenghts L.
This is a convenience function equivalent with::
self.splitAt(self.atLength(L))
"""
return self.splitAt(self.atLength(L))
[docs] def refine(self, maxlen):
"""Insert extra points in the PolyLine to reduce the segment length.
Parameters:
- `maxlen`: float: maximum length of the segments. The value is
relative to the total curve length.
Returns a PolyLine which is geometrically equivalent to the
input PolyLine.
"""
maxlen *= self.length()
ndiv = np.ceil(self.lengths() / maxlen).astype(at.Int)
return self.approx(ndiv=ndiv)
[docs] def vertexReductionDP(self, tol, maxlen=None, keep=[0, -1]):
"""Douglas-Peucker vertex reduction.
For any subpart of the PolyLine, if the distance of all its vertices to
the line connecting the endpoints is smaller than `tol`, the internal
points are removed.
Returns a bool array flagging the vertices to be kept in the reduced
form.
"""
x = self.coords
n = len(x)
_keep = np.zeros(n, dtype=bool)
_keep[keep] = 1
def decimate(i, j):
"""Recursive vertex decimation.
This will remove all the vertices in the segment i-j that
are closer than tol to any chord.
"""
# Find point farthest from line through endpoints
d = x[i+1:j].distanceFromLine(x[i], x[j]-x[i])
k = np.argmax(d)
dmax = d[k]
k += i+1
if dmax > tol:
# Keep the farthest vertex, split the polyline at that vertex
# and recursively decimate the two parts
_keep[k] = 1
if k > i+1:
decimate(i, k)
if k < j-1:
decimate(k, j)
def reanimate(i, j):
"""Recursive vertex reanimation.
This will revive vertices in the segment i-j if the
segment is longer than maxlen.
"""
if j > i+1 and at.length(x[i]-x[j]) > maxlen:
# too long: revive a point
d = at.length(x[i+1:j]-x[i])
w = np.where(d>maxlen)[0]
if len(w) > 0:
k = w[0] + i+1
_keep[k] = 1
reanimate(k+1, j)
# Compute vertices to keep
decimate(0, n-1)
# Reanimate some vertices if maxlen specified
if maxlen is not None:
w = np.where(_keep)[0]
for i, j in zip(w[:-1], w[1:]):
reanimate(i, j)
return _keep
[docs] def coarsen(self, tol=0.01, maxlen=None, keep=[0, -1]):
""" Reduce the number of points of the PolyLine.
Parameters:
- `tol`: maximum relative distance of vertices to remove from the
chord of the segment. The value is relative to the total curve
length.
- `keep`: list of vertices to keep during the coarsening process.
(Not implemented yet).
Retuns a Polyline with a reduced number of points.
"""
atol = tol*self.length()
if maxlen:
maxlen *= self.length()
keep = self.vertexReductionDP(tol=atol, maxlen=maxlen, keep=keep)
return PolyLine(self.coords[keep], closed=self.closed)
##############################################################################
#
[docs]class Line(PolyLine):
"""A Line is a PolyLine with exactly two points.
Parameters:
- `coords`: compatible with (2,3) shaped float array
"""
def __init__(self, coords):
"""Initialize the Line."""
PolyLine.__init__(self, coords)
if self.coords.shape[0] != 2:
raise ValueError("Expected exactly two points, got %s" % coords.shape[0])
def dxftext(self):
return "Line(%s,%s,%s,%s,%s,%s)" % tuple(self.coords.ravel().tolist())
##############################################################################
#
[docs]class BezierSpline(Curve):
"""A class representing a Bezier spline curve of degree 1, 2 or 3.
A Bezier spline of degree `d` is a continuous curve consisting of `nparts`
successive parts, where each part is a Bezier curve of the same degree.
Currently pyFormex can model linear, quadratic and cubic BezierSplines.
A linear BezierSpline is equivalent to a PolyLine, which has more
specialized methods than the BezierSpline, so it might be more
sensible to use a PolyLine instead of the linear BezierSpline.
A Bezier curve of degree `d` is determined by `d+1` control points,
of which the first and the last are on the curve, while the intermediate
`d-1` points are not.
Since the end point of one part is the begin point of the next part,
a BezierSpline is described by `ncontrol=d*nparts+1` control points if the
curve is open, or `ncontrol=d*nparts` if the curve is closed.
The constructor provides different ways to initialize the full set of
control points. In many cases the off-curve control points can be
generated automatically.
Parameters:
- `coords` : :term:`array_like` (npoints,3)
The points that are on the curve. For an open curve, npoints=nparts+1,
for a closed curve, npoints = nparts.
If not specified, the on-curve points should be included in the
`control` argument.
- `deriv` : :term:`array_like` (npoints,3) or (2,3) or a list of 2 values
one of which can be None and the other is a shape(3,) arraylike.
If specified, it gives the direction of the curve at all points or at
the endpoints only for a shape (2,3) array or only at one of the
endpoints for a list of shape(3,) arraylike and a None type.
For points where the direction is left unspecified or where the
specified direction contains a `NaN` value, the direction
is calculated as the average direction of the two
line segments ending in the point. This will also be used
for points where the specified direction contains a value `NaN`.
In the two endpoints of an open curve however, this average
direction can not be calculated: the two control points in these
parts are set coincident.
- `curl` : float
The curl parameter can be set to influence the curliness of the curve
in between two subsequent points. A value curl=0.0 results in
straight segments. The higher the value, the more the curve becomes
curled.
- `control` : array(nparts,d-1,3) or array(ncontrol,3)
If `coords` was specified and d > 1, this should be a (nparts,d-1,3) array with
the intermediate control points, `d-1` for each part.
If `coords` was not specified, this should be the full array of
`ncontrol` control points for the curve.
If not specified, the control points are generated automatically from
the `coords`, `deriv` and `curl` arguments.
If specified, they override these parameters.
- `closed` : boolean
If True, the curve will be continued from the last point back
to the first to create a closed curve.
- `degree`: int (1, 2 or 3)
Specfies the degree of the curve. Default is 3.
- `endzerocurv` : boolean or tuple of two booleans.
Specifies the end conditions for an open curve.
If True, the end curvature will be forced to zero. The default is
to use maximal continuity of the curvature.
The value may be set to a tuple of two values to specify different
conditions for both ends.
This argument is ignored for a closed curve.
"""
def __init__(self, coords=None, deriv=None, curl=1./3., control=None,
closed=False, degree=3, endzerocurv=False):
"""Create a BezierSpline curve."""
Curve.__init__(self)
if not degree > 0:
raise ValueError("Degree of BezierSpline should be >= 0!")
if endzerocurv in [False, True]:
endzerocurv = (endzerocurv, endzerocurv)
if coords is None:
# All control points are given, in a single array
control = Coords(control)
if len(control.shape) != 2 or control.shape[-1] != 3:
raise ValueError(
"If no coords argument given, the control parameter should "
f"have shape (ncontrol,3), but got {control.shape}")
if closed:
control = Coords.concatenate([control, control[:1]])
ncontrol = control.shape[0]
nextra = (ncontrol-1) % degree
if nextra != 0:
nextra = degree - nextra
control = Coords.concatenate([control, ]+[control[:1]]*nextra)
ncontrol = control.shape[0]
nparts = (ncontrol-1) // degree
else:
# Oncurve points are specified separately
if degree > 3:
raise ValueError(
"BezierSpline of degree > 3 can only be specified by "
"a full set of control points")
coords = Coords(coords)
ncoords = nparts = coords.shape[0]
if ncoords < 2:
raise ValueError("Need at least two points to define a curve")
if not closed:
nparts -= 1
if control is None:
if degree == 1:
control = coords[:nparts]
elif degree == 2:
if ncoords < 3:
control = 0.5*(coords[:1] + coords[-1:])
if closed:
control = Coords.concatenate([control, control])
else:
if closed:
P0 = 0.5 * (np.roll(coords, 1, axis=0)
+ np.roll(coords, -1, axis=0))
P1 = 2*coords - P0
Q0 = 0.5*(np.roll(coords, 1, axis=0) + P1)
Q1 = 0.5*(np.roll(coords, -1, axis=0) + P1)
Q = 0.5*(np.roll(Q0, -1, axis=0)+Q1)
control = Q
else:
P0 = 0.5 * (coords[:-2] + coords[2:])
P1 = 2*coords[1:-1] - P0
Q0 = 0.5*(coords[:-2] + P1)
Q1 = 0.5*(coords[2:] + P1)
Q = 0.5*(Q0[1:]+Q1[:-1])
control = Coords.concatenate(
[Q0[:1], Q, Q1[-1:]], axis=0)
elif degree == 3:
P = PolyLine(coords, closed=closed)
ampl = P.lengths().reshape(-1, 1)
if deriv is None:
deriv = np.array([[np.nan, np.nan, np.nan]]*ncoords)
else:
for id, d in enumerate(deriv):
if d is None:
deriv[id] = [np.nan, ]*3
deriv = Coords(deriv)
nderiv = deriv.shape[0]
if nderiv < ncoords:
if nderiv !=2:
raise ValueError(
"Either all or at least the initial and/or "
"the final directions expected (got {nderiv}")
deriv = np.concatenate([
deriv[:1],
[[np.nan] * 3] * (ncoords-2),
deriv[-1:]])
undefined = np.isnan(deriv).any(axis=-1)
if undefined.any():
deriv[undefined] = P.avgDirections()[undefined]
if closed:
p1 = coords + deriv*curl*ampl
p2 = coords - deriv*curl*np.roll(ampl, 1, axis=0)
p2 = np.roll(p2, -1, axis=0)
else:
# Fix the first and last derivs if they were autoset
if undefined[0]:
if endzerocurv[0]:
# option curvature 0:
deriv[0] = [np.nan, ]*3
else:
# option max. continuity
deriv[0] = 2.*deriv[0] - deriv[1]
if undefined[-1]:
if endzerocurv[1]:
# option curvature 0:
deriv[-1] = [np.nan, ]*3
else:
# option max. continuity
deriv[-1] = 2.*deriv[-1] - deriv[-2]
p1 = coords[:-1] + deriv[:-1]*curl*ampl
p2 = coords[1:] - deriv[1:]*curl*ampl
if np.isnan(p1[0]).any():
p1[0] = p2[0]
if np.isnan(p2[-1]).any():
p2[-1] = p1[-1]
control = np.concatenate([p1, p2], axis=1)
if control is not None and degree > 1:
try:
control = np.asarray(control).reshape(nparts, degree-1, 3)
control = Coords(control)
except Exception:
print("coords array has shape %s" % str(coords.shape))
print("control array has shape %s" % str(control.shape))
raise ValueError("Invalid control points for BezierSpline "
f"of degree {degree}")
# Join the coords and controls in a single array
control = Coords.concatenate([
coords[:nparts, np.newaxis, :], control], axis=1).reshape(-1, 3)
if control is not None:
# We now have a multiple of degree coordinates, add the last:
if closed:
last = 0
else:
last = -1
control = Coords.concatenate([control, coords[last]])
self.degree = degree
# TODO: np.matrix is deprecated
self.coeffs = np.matrix(bezierPowerMatrix(degree))
self.coords = control
self.nparts = nparts
self.closed = closed
def report(self):
return f"""\
BezierSpline: degree={self.degree}; nparts={self.nparts}, ncoords={self.coords.shape[0]}
Control points:
{self.coords}
"""
__repr__ = report
__str__ = report
[docs] def pointsOn(self):
"""Return the points on the curve.
This returns a Coords object of shape [nparts+1]. For a closed curve,
the last point will be equal to the first.
"""
return self.coords[::self.degree]
[docs] def pointsOff(self):
"""Return the points off the curve (the control points)
This returns a Coords object of shape [nparts,ndegree-1], or
an empty Coords if degree <= 1.
"""
if self.degree > 1:
return self.coords[:-1].reshape(-1, self.degree, 3)[:, 1:]
else:
return Coords()
[docs] def part(self, j, k=None):
"""Returns the points defining parts [j:k] of the curve.
If k is None, it is set equal to j+1, resulting in a single
part with degree+1 points.
"""
if k is None:
k = j+1
start = self.degree * j
end = self.degree * k + 1
return self.coords[start:end]
[docs] def sub_points(self, t, j):
"""Return the points at values t in part j."""
P = self.part(j)
C = self.coeffs * P
U = [t**d for d in range(0, self.degree+1)]
U = np.column_stack(U)
X = np.dot(U, C)
return X
[docs] def sub_directions(self, t, j):
"""Return the unit direction vectors at values t in part j."""
P = self.part(j)
C = self.coeffs * P
U = [d*(t**(d-1)) if d >= 1 else np.zeros_like(t)
for d in range(0, self.degree+1)]
U = np.column_stack(U)
T = np.dot(U, C)
T = at.normalize(T)
return T
[docs] def sub_curvature(self, t, j):
"""Return the curvature at values t in part j."""
P = self.part(j)
C = self.coeffs * P
U1 = [d*(t**(d-1)) if d >= 1 else np.zeros_like(t)
for d in range(0, self.degree+1)]
U1 = np.column_stack(U1)
T1 = np.dot(U1, C)
U2 = [d*(d-1)*(t**(d-2)) if d >=2 else np.zeros_like(t)
for d in range(0, self.degree+1)]
U2 = np.column_stack(U2)
T2 = np.dot(U2, C)
K = at.length(np.cross(T1, T2))/(at.length(T1)**3)
return K
[docs] def length_intgrnd(self, t, j):
"""Return the arc length integrand at value t in part j."""
P = self.part(j)
C = self.coeffs * P
U = [d*(t**(d-1)) if d >= 1 else 0. for d in range(0, self.degree+1)]
U = np.array(U)
T = np.dot(U, C)
T = np.array(T).reshape(3)
return at.length(T)
[docs] def lengths(self):
"""Return the length of the parts of the curve."""
try:
from scipy.integrate import quad
except ImportError:
pf.warning("""..
**The **lengths** function is not available.
Most likely because 'python-scipy' is not installed on your system.""")
return
L = [quad(self.length_intgrnd, 0., 1., args=(j,))[0]
for j in range(self.nparts)]
return np.array(L)
[docs] def parts(self, j, k):
"""Return a curve containing only parts j to k (k not included).
The resulting curve is always open.
"""
return BezierSpline(control=self.part(j, k), degree=self.degree,
closed=False)
[docs] def atLength(self, l, approx=20):
"""Returns the parameter values at given relative curve length.
Parameters:
- `l`: list of relative curve lengths (from 0.0 to 1.0).
As a convenience, a single integer value may be specified,
in which case the relative curve lengths are found by dividing
the interval [0.0,1.0] in the specified number of subintervals.
- `approx`: int or None. If not None, an approximate result is
returned obtained by approximating the curve first by a
PolyLine with `approx` number of line segments per curve segment.
This is currently the only implemented method, and specifying
None will fail.
The function returns a list with the parameter values for the points
at the specified relative lengths.
"""
if at.isInt(approx) and approx > 0:
P = self.approx(ndiv=approx)
return P.atLength(l) / approx
else:
raise ValueError("approx should be int and > 0")
[docs] def insertPointsAt(self, t, split=False):
"""Insert new points on the curve at parameter values t.
Parameters:
- `t`: float: parametric value where the new points will be
inserted. Parameter values are floating point values.
Their integer part is interpreted as the curve segment number,
and the decimal part goes from 0 to 1 over the segment.
* Currently there can only be one new point in each segment *
- `split`: bool: if True, this method behaves like the
:func:`split` method. Users should use the latter method instead
of the `split` parameter.
Returns a single BezierSpline (default). The result is equivalent
with the input curve, but has more points on the curve (and more
control points.
If `split` is True, the behavior is that of the :func:`split` method.
"""
from pyformex.plugins.nurbs import splitBezierCurve
# Loop in descending order to avoid recomputing parameters
X = []
k = self.nparts # last full part
points = [] # points of left over parts
for i in np.argsort(t)[::-1]:
j, uu = self.localParam(t[i])
j = j[0]
uu = uu[0]
if j==k:
raise ValueError(
"Currently there is max. one split point per curve part")
# Push the tail beyond current part
if k > j+1:
points[0:1] = self.part(j+1, k).tolist()
L, R = splitBezierCurve(self.part(j), uu)
# Push the right part
points[0:1] = R.tolist()
# Save the new segment
X.insert(0, Coords.concatenate(points))
# Save left part
points = L.tolist()
k = j
if j >0:
points = self.part(0, j).tolist()[:-1]+points
X.insert(0, Coords.concatenate(points))
if split:
return [BezierSpline(control=x, degree=self.degree, closed=False) for x in X]
else:
X = Coords.concatenate([X[0]] + [xi[1:] for xi in X[1:]])
return BezierSpline(control=X, degree=self.degree, closed=self.closed)
[docs] def splitAt(self, t):
"""Split a BezierSpline at parametric values.
Parameters:
- `t`: float: parametric value where the new points will be
inserted. Parameter values are floating point values.
Their integer part is interpreted as the curve segment number,
and the decimal part goes from 0 to 1 over the segment.
* Currently there can only be one new point in each segment *
Returns a list of len(t)+1 open BezierSplines of the same degree
as the input.
"""
return self.insertPointsAt(t, split=True)
[docs] def toMesh(self):
"""Convert the BezierSpline to a Mesh.
For degrees 1 or 2, the returned Mesh is equivalent with the
BezierSpline, and will have element type 'line1', resp. 'line2'.
For degree 3, the returned Mesh will currently be a quadratic
approximation with element type 'line2'.
"""
if self.degree == 1:
return self.approx(ndiv=1).toMesh()
else:
coords = self.subPoints(2)
e1 = 2*np.arange(len(coords)//2)
elems = np.column_stack([e1, e1+1, e1+2])
if self.closed:
elems[-1][-1] = 0
return Mesh(coords, elems, eltype='line3')
# This is not activated (yet) because it would be called for
# drawing curves.
## def toFormex(self):
## """Convert the BezierSpline to a Formex.
## This is notational convenience for::
## self.toMesh().toFormex()
## """
## return self.toMesh().toFormex()
[docs] def extend(self, extend=[1., 1.]):
"""Extend the curve beyond its endpoints.
This function will add a Bezier curve before the first part and/or
after the last part by applying de Casteljau's algorithm on this part.
"""
if self.closed:
return
if extend[0] > 0.:
# get the control points
P = self.part(0)
# apply de Casteljau's algorithm
t = -extend[0]
M = [P]
for i in range(self.degree):
M.append((1.-t)*M[-1][:-1] + t*M[-1][1:])
# compute control points
Q = np.stack([Mi[0] for Mi in M[::-1]], axis=0)
self.coords = Coords.concatenate([Q[:-1], self.coords])
self.nparts += 1
if extend[1] > 0.:
# get the control points
P = self.part(self.nparts-1)
# apply de Casteljau's algorithm
t = 1.+extend[1]
M = [P]
for i in range(self.degree):
M.append((1.-t)*M[-1][:-1] + t*M[-1][1:])
# compute control points
Q = np.stack([Mi[-1] for Mi in M], axis=0)
self.coords = Coords.concatenate([self.coords, Q[1:]])
self.nparts += 1
[docs] def reverse(self):
"""Return the same curve with the parameter direction reversed."""
control = at.reverseAxis(self.coords, axis=0)
return BezierSpline(control=control, closed=self.closed, degree=self.degree)
##############################################################################
#
[docs]class Contour(Curve):
"""A class for storing a contour.
The contour class stores a continuous (usually closed) curve which consists
of a sequence of strokes, each stroke either being a straight segment or a
quadratic or cubic Bezier curve. A stroke is thus defined by 2, 3 or 4
points.
The contour is defined by a list of points and a Varray of element
connectivity. This format is well suited to store contours of scalable
fonts. The contours are in that case usually 2D.
"""
def __init__(self, coords, elems):
Geometry.__init__(self)
self.prop = None
self.coords = Coords(at.checkArray(coords, (-1, 3), 'f'))
self.elems = Varray(elems)
first = self.elems.col(0)
last = self.elems.col(-1)
if (first[1:] != last[:-1]).any():
raise ValueError("elems do not form a continuous contour")
self.nparts = self.elems.shape[0]
self.closed = self.elems[0][0] == self.elems[-1][-1]
def pointsOn(self):
ind = self.elems.col(0)
if not self.closed:
ind = np.concatenate([ind, [self.elems[-1][-1]]])
return self.coords[ind]
def pointsOff(self):
ind = np.unique(np.concatenate([r[1:-1] for r in self.elems]))
return self.coords[ind]
def ncoords(self):
return self.coords.shape[0]
def npoints(self):
return self.pointsOn().shape[0]
[docs] def endPoints(self):
"""Return start and end points of the curve.
Returns a Coords with two points, or None if the curve is closed.
"""
if self.closed:
return None
else:
return self.coords[[0, -1]]
[docs] def part(self, i):
"""Returns the points defining part j of the curve."""
return self.coords[self.elems[i]]
[docs] def stroke(self, i):
"""Return curve for part i"""
X = self.part(i)
degree = len(X) - 1
if len(X) == 1:
return PolyLine(X)
else:
return BezierSpline(control=X, degree=degree)
[docs] def sub_points(self, t, j):
"""Return the points at values t in part j
t can be an array of parameter values, j is a single segment number.
"""
j = int(j)
t = np.asarray(t).reshape(-1, 1)
return self.stroke(j).sub_points(t, 0)
[docs] def sub_directions(self, t, j):
"""Return the directions at values t in part j
t can be an array of parameter values, j is a single segment number.
"""
j = int(j)
t = np.asarray(t).reshape(-1, 1)
return self.stroke(j).sub_points(t, 0)
def lengths(self):
return [self.stroke(i).length() for i in range(self.nparts)]
[docs] def atChordal(self, chordal=0.01, atl=None):
# Define specialized preapproximation
if at is None:
atl = self.atApproximate(ndiv=self.elems.lengths)
return Curve.atChordal(self, chordal, atl)
[docs] def toMesh(self):
"""Convert the Contour to a Mesh."""
return [self.stroke(i).toMesh() for i in range(self.nparts)]
###########################################################################
[docs]class CardinalSpline(BezierSpline):
"""A class representing a cardinal spline.
Create a natural spline through the given points.
The Cardinal Spline with given tension is a Bezier Spline with curl
:math: `curl = ( 1 - tension) / 3`
The separate class name is retained for compatibility and convenience.
See CardinalSpline2 for a direct implementation (it misses the end
intervals of the point set).
"""
def __init__(self, coords, tension=0.0, closed=False, endzerocurv=False):
"""Create a natural spline through the given points."""
BezierSpline.__init__(self, coords, curl=(1.-tension)/3.,
closed=closed, endzerocurv=endzerocurv)
##########################################################################
[docs]class CardinalSpline2(Curve):
"""A class representing a cardinal spline."""
def __init__(self, coords, tension=0.0, closed=False):
"""Create a natural spline through the given points.
This is a direct implementation of the Cardinal Spline.
For open curves, it misses the interpolation in the two end
intervals of the point set.
"""
Curve.__init__(self)
coords = Coords(coords)
self.coords = coords
self.nparts = self.coords.shape[0]
if not closed:
self.nparts -= 3
self.closed = closed
self.tension = float(tension)
s = (1.-self.tension)/2.
# TODO: improve this documentation link
# pag.429 of open GL
# TODO: np.matrix is deprecated
self.coeffs = np.matrix([[-s, 2-s, s-2., s], [2*s, s-3., 3.-2*s, -s],
[-s, 0., s, 0.], [0., 1., 0., 0.]])
[docs] def sub_points(self, t, j):
n = self.coords.shape[0]
i = (j + np.arange(4)) % n
P = self.coords[i]
C = self.coeffs * P
U = np.column_stack([t**3., t**2., t, np.ones_like(t)])
X = np.dot(U, C)
return X
##############################################################################
[docs]class NaturalSpline(Curve):
"""A class representing a natural spline."""
def __init__(self, coords, closed=False, endzerocurv=False):
"""Create a natural spline through the given points.
coords specifies the coordinates of a set of points. A natural spline
is constructed through this points.
closed specifies whether the curve is closed or not.
endzerocurv specifies the end conditions for an open curve.
If True, the end curvature will forced to be zero. The default is
to use maximal continuity (up to the third derivative) between
the first two splines. The value may be set to a tuple of two
values to specify different end conditions for both ends.
This argument is ignored for a closed curve.
"""
Curve.__init__(self)
coords = Coords(coords)
if closed:
coords = Coords.concatenate([coords, coords[:1]])
self.nparts = coords.shape[0] - 1
self.closed = closed
if not closed:
if endzerocurv in [False, True]:
self.endzerocurv = (endzerocurv, endzerocurv)
else:
self.endzerocurv = endzerocurv
self.coords = coords
self.compute_coefficients()
def _set_coords(self, coords):
C = self.copy()
C._set_coords_inplace(coords)
C.compute_coefficients()
return C
def compute_coefficients(self):
n = self.nparts
M = np.zeros([4*n, 4*n])
B = np.zeros([4*n, 3])
# constant submatrix
m = np.array([[0., 0., 0., 1., 0., 0., 0., 0.],
[1., 1., 1., 1., 0., 0., 0., 0.],
[3., 2., 1., 0., 0., 0., -1., 0.],
[6., 2., 0., 0., 0., -2., 0., 0.]])
for i in range(n-1):
f = 4*i
M[f:f+4, f:f+8] = m
B[f:f+2] = self.coords[i:i+2]
# the last spline passes through the last 2 points
f = 4*(n-1)
M[f:f+2, f:f+4] = m[:2, :4]
B[f:f+2] = self.coords[-2:]
# add the appropriate end constrains
if self.closed:
# first and second derivatives at starting and last point
# (that are actually the same point) are the same
M[f+2, f:f+4] = m[2, :4]
M[f+2, 0:4] = m[2, 4:]
M[f+3, f:f+4] = m[3, :4]
M[f+3, 0:4] = m[3, 4:]
else:
if self.endzerocurv[0]:
# second derivatives at start is zero
M[f+2, 0:4] = m[3, 4:]
else:
# third derivative is the same between the first 2 splines
M[f+2, [0, 4]] = np.array([6., -6.])
if self.endzerocurv[1]:
# second derivatives at end is zero
M[f+3, f:f+4] = m[3, :4]
else:
# third derivative is the same between the last 2 splines
M[f+3, [f-4, f]] = np.array([6., -6.])
# calculate the coefficients
C = np.linalg.solve(M, B)
self.coeffs = np.array(C).reshape(-1, 4, 3)
[docs] def sub_points(self, t, j):
C = self.coeffs[j]
U = np.column_stack([t**3., t**2., t, np.ones_like(t)])
X = np.dot(U, C)
return X
##############################################################################
[docs]def circle():
"""Create a spline approximation of a circle.
The returned circle lies in the x,y plane, has its center at (0,0,0)
and has a radius 1.
In the current implementation it is approximated by a bezier spline
with curl 0.375058 through 8 points.
"""
pts = Formex([1.0, 0.0, 0.0]).rosette(8, 45.).coords.reshape(-1, 3)
return BezierSpline(pts, curl=0.375058, closed=True)
[docs]class Arc3(Curve):
"""A class representing a circular arc."""
# TODO: approx returns a PolyLine that does not start from coords[0]. Why?
def __init__(self, coords):
"""Create a circular arc.
The arc is specified by 3 non-colinear points.
"""
Curve.__init__(self)
self.nparts = 1
self.closed = False
self._set_coords(Coords(coords))
def _set_coords(self, coords):
C = self.copy()
C._set_coords_inplace(coords)
if self.coords.shape != (3, 3):
raise ValueError("Expected 3 points")
r, C, n = gt.triangleCircumCircle(self.coords.reshape(-1, 3, 3))
self.radius, self.center, self.normal = r[0], C[0], n[0]
self.angles = at.vectorPairAngle(Coords([1., 0., 0.]), self.coords-self.center)
print("Radius %s, Center %s, Normal %s" % (self.radius, self.center, self.normal))
print("ANGLES=%s" % (self.angles))
return C
[docs] def sub_points(self, t, j):
a = t*(self.angles[-1]-self.angles[0])
X = Coords(np.column_stack([np.cos(a), np.sin(a), np.zeros_like(a)]))
X = X.scale(self.radius).rotate(self.angles[0]/at.DEG).translate(self.center)
return X
[docs]class Arc(Curve):
"""A class representing a circular arc.
The arc can be specified by 3 points (begin, center, end)
or by center, radius and two angles. In the latter case, the arc
lies in a plane parallel to the x,y plane.
If specified by 3 colinear points, the plane of the circle will be
parallel to the x,y plane if the points are in such plane, else the
plane will be parallel to the z-axis.
"""
def __init__(self, coords=None, center=None, radius=None,
angles=None, angle_spec=at.DEG):
"""Create a circular arc."""
def fix_angles(angles):
while angles[1] < angles[0]:
angles[1] += 2*np.pi
while angles[1] > angles[0] + 2*np.pi:
angles[1] -= 2*np.pi
return angles
# Internally, we store the coords
Curve.__init__(self)
self.nparts = 1
self.closed = False
if coords is not None:
self.coords = Coords(coords)
if self.coords.shape != (3, 3):
raise ValueError("Expected 3 points")
self._center = self.coords[1]
v = self.coords-self._center
self.radius = at.length(self.coords[0]-self.coords[1])
try:
self.normal = at.unitVector(np.cross(v[0], v[2]))
except Exception:
pf.warning("The three points defining the Arc seem to be "
"colinear: I will use a random orientation.")
self.normal = gt.anyPerpendicularVector(v[0])
angles = gt.rotationAngle(
Coords([1., 0., 0.]).rotate(
at.rotMatrix2([0., 0., 1.], self.normal)),
v[[0, 2]], self.normal, at.RAD)
self._angles = fix_angles(angles)
else:
if center is None:
center = [0., 0., 0.]
if radius is None:
radius = 1.0
if angles is None:
angles = (0., 360.)
try:
self._center = center
self.radius = radius
self.normal = [0., 0., 1.]
angles = [a * angle_spec for a in angles]
self._angles = fix_angles(angles)
begin, end = self.sub_points(np.array([0.0, 1.0]), 0)
self.coords = Coords([begin, self._center, end])
except Exception:
print("Invalid data for Arc")
raise
def getCenter(self):
return self._center
def getAngles(self, angle_spec=at.DEG):
return (self._angles[0]/angle_spec, self._angles[1]/angle_spec)
def getAngleRange(self, angle_spec=at.DEG):
return ((self._angles[1]-self._angles[0])/angle_spec)
def _set_coords(self, coords):
if isinstance(coords, Coords) and coords.shape == self.coords.shape:
return self.__class__(coords)
else:
raise ValueError(
f"Invalid reinitialization of {self.__class__} coords")
def __str__(self):
return f"""ARC
Center {self._center}, Radius {self.radius}, Normal {self.normal}
Angles={self.getAngles()}
Pt0={self.coords[0]}; Pt1={self.coords[1]}; Pt2={self.coords[2]}
"""
def dxftext(self):
return "Arc(%s,%s,%s,%s,%s,%s)" % tuple(
list(self._center) + [self.radius]+list(self.getAngles()))
[docs] def sub_points(self, t, j):
a = t*(self._angles[-1]-self._angles[0])
X = Coords(np.column_stack([np.cos(a), np.sin(a), np.zeros_like(a)]))
X = X.scale(self.radius).rotate(self._angles[0]/at.DEG).rotate(
at.rotMatrix2([0., 0., 1.], self.normal)).translate(self._center)
return X
[docs] def sub_directions(self, t, j):
a = t*(self._angles[-1]-self._angles[0])
X = Coords(np.column_stack([-np.sin(a), np.cos(a), np.zeros_like(a)]))
X = X.rotate(self._angles[0]/at.DEG).rotate(
at.rotMatrix2([0., 0., 1.], self.normal))
return X
[docs] def approx(self, ndiv=None, chordal=0.001):
"""Return a PolyLine approximation of the Arc.
Approximates the Arc by a sequence of inscribed straight line
segments.
If `ndiv` is specified, the arc is divided in precisely `ndiv`
segments.
If `ndiv` is not given, the number of segments is determined
from the chordal distance tolerance. It will guarantee that the
distance of any point of the arc to the chordal approximation
is less or equal than `chordal` times the radius of the arc.
"""
if ndiv is None:
phi = 2. * np.arccos(1.-chordal)
rng = abs(self._angles[1] - self._angles[0])
ndiv = int(np.ceil(rng/phi))
return Curve.approx(self, ndiv=ndiv)
[docs]def arc2points(x0, x1, R, pos='-'):
"""Create an arc between two points
Given two points x0 and x1, this constructs an arc with radius R
through these points. The two points should have the same z-value.
The arc will be in a plane parallel with the x-y plane and wind
positively around the z-axis when moving along the arc from x0 to x1.
If pos == '-', the center of the arc will be at the left when going
along the chord from x0 to x1, creating an arc smaller than a half-circle.
If pos == '+', the center of the arc will be at the right when going
along the chord from x0 to x1, creating an arc larger than a half-circle.
If R is too small, an exception is raised.
"""
if x0[2] != x1[2]:
raise ValueError("The points should b in a plane // xy plane")
xm = (x0+x1) / 2 # center of points
xd = (x0-x1) / 2 # half length vector
do = at.normalize([xd[1], -xd[0], 0]) # direction of center
xl = np.dot(xd, xd) # square length
if R**2 < xl:
raise ValueError("The radius should at least be %s" % np.sqrt(xl))
dd = np.sqrt(R**2 - xl) # distance of center to xm
if pos == '+':
xc = xm - dd * do # Center is to the left when going from x0 to x1
else:
xc = xm + dd * do
xx = [1., 0., 0.]
xz = [0., 0., 1.]
angles = gt.rotationAngle([xx, xx], [x0-xc, x1-xc], m=[xz, xz])
if dir == '-':
angles = reversed(angles)
xc[2] = x0[2]
return Arc(center=xc, radius=R, angles=angles)
# TODO: implement this
# class Spiral(Curve):
# """A class representing a spiral curve."""
# def __init__(self, turns=2.0, nparts=100, rfunc=None):
# Curve.__init__(self)
# # TODO: implement the rfunc
# if rfunc is None:
# rfunc = lambda x: x
# self.coords = Coords([0., 0., 0.]).replic(npoints+1).hypercylindrical()
# self.nparts = nparts
# self.closed = False
##############################################################################
# Other functions
[docs]def binomial(n, k):
"""Compute the binomial coefficient Cn,k.
This computes the binomial coefficient Cn,k = fac(n) // fac(k) // fac(n-k).
Example:
>>> print([ binomial(3,i) for i in range(4) ])
[1, 3, 3, 1]
"""
import math
f = math.factorial
return f(n) // f(k) // f(n-k)
_binomial_coeffs = {}
_bezier_power_matrix = {}
[docs]def binomialCoeffs(p):
"""Compute all binomial coefficients for a given degree p.
Returns an array of p+1 values.
For efficiency reasons, the computed values are stored in the module,
in a dict with p as key. This allows easy and fast lookup of already
computed values.
Example:
>>> print(binomialCoeffs(4))
[1 4 6 4 1]
"""
if p not in _binomial_coeffs:
_binomial_coeffs[p] = np.array([binomial(p, i) for i in range(p+1)])
return _binomial_coeffs[p]
[docs]def bezierPowerMatrix(p):
"""Compute the Bezier to power curve transformation matrix for degree p.
Bezier curve representations can be converted to power curve representations
using the coefficients in this matrix.
Returns a (p+1,p+1) shaped array with a zero upper triangular part.
For efficiency reasons, the computed values are stored in the module,
in a dict with p as key. This allows easy and fast lookup of already
computed values.
Example:
>>> print(bezierPowerMatrix(4))
[[ 1. 0. 0. 0. 0.]
[ -4. 4. 0. 0. 0.]
[ 6. -12. 6. 0. 0.]
[ -4. 12. -12. 4. 0.]
[ 1. -4. 6. -4. 1.]]
>>> 4 in _bezier_power_matrix
True
"""
if p in _bezier_power_matrix:
return _bezier_power_matrix[p]
M = np.zeros((p+1, p+1))
# Set corner elements
M[0, 0] = M[p, p] = 1.0
M[p, 0] = -1.0 if p % 2 else 1.0
# Compute first column, last row and diagonal
sign = -1.0
for i in range(1, p):
M[i, i] = binomialCoeffs(p)[i]
M[i, 0] = M[p, p-i] = sign*M[i, i]
sign = -sign
# Compute remaining elements
k1 = (p+1)//2
pk = p-1
for k in range(1, k1):
sign = -1.0
for j in range(k+1, pk+1):
M[j, k] = M[pk, p-j] = sign * binomialCoeffs(p)[k] * binomialCoeffs(p-k)[j-k]
sign = -sign
pk -= 1
_bezier_power_matrix[p] = M
return M
Formex.toCurve = convertFormexToCurve
[docs]def positionCoordsObj(objects, path, normal=0, upvector=2,
avgdir=False, enddir=None):
""" Position a sequence of Coords objects along a path.
At each point of the curve, a copy of the Coords object is created, with
its origin in the curve's point, and its normal along the curve's direction.
In case of a PolyLine, directions are pointing to the next point by default.
If avgdir==True, average directions are taken at the intermediate points
avgdir can also be an array like sequence of shape (N,3) to explicitely set
the directions for ALL the points of the path
Missing end directions can explicitely be set by enddir, and are by default
taken along the last segment.
enddir is a list of 2 array like values of shape (3). one of the two can
also be an empty list.
If the curve is closed, endpoints are treated as any intermediate point,
and the user should normally not specify enddir.
The return value is a sequence of the repositioned Coords objects.
"""
points = path.coords
if isinstance(avgdir, bool):
if avgdir:
directions = path.avgDirections()
else:
directions = path.directions()
else:
directions = np.asarray(avgdir).reshape(len(avgdir), -1)
missing = points.shape[0] - directions.shape[0]
if missing == 1:
lastdir = (points[-1] - points[-2]).reshape(1, 3)
directions = np.concatenate([directions, lastdir], axis=0)
elif missing == 2:
lastdir = (points[-1] - points[-2]).reshape(1, 3)
firstdir = (points[1] - points[0]).reshape(1, 3)
directions = np.concatenate([firstdir, directions, lastdir], axis=0)
if enddir:
for i, j in enumerate([0, -1]):
if enddir[i]:
directions[j] = Coords(enddir[i])
directions = at.normalize(directions)
if at.isInt(normal):
normal = Coords(at.unitVector(normal))
if at.isInt(upvector):
upvector = Coords(at.unitVector(upvector))
sequence = [object.rotate(at.rotMatrix2(normal, d, upvector)).translate(p)
for object, d, p in zip(objects, directions, points)]
return sequence
# End
