def extrude(nrb, displ, axis=None):
"""
Construct a NURBS surface/volume by
extruding a NURBS curve/surface.
Parameters
----------
nrb : NURBS
displ : array_like or float
axis : array_like or int, optional
Example
-------
>>> crv = circle()
>>> srf = extrude(crv, displ=1, axis=2)
>>> srf = bilinear()
>>> vol = extrude(srf, displ=1, axis=2)
"""
assert nrb.dim <= 2
T = transform().translate(displ, axis)
Cw = np.empty(nrb.shape+(2,4))
Cw[...,0,:] = nrb.control
Cw[...,1,:] = T(nrb.control)
UVW = nrb.knots + ([0,0,1,1],)
return NURBS(UVW, Cw)
def extrude(nrb, displ, axis=None):
"""
Construct a NURBS surface/volume by
extruding a NURBS curve/surface.
Parameters
----------
nrb : NURBS
displ : array_like or float
axis : array_like or int, optional
Example
-------
>>> crv = circle()
>>> srf = extrude(crv, displ=1, axis=2)
>>> srf = bilinear()
>>> vol = extrude(srf, displ=1, axis=2)
"""
assert nrb.dim <= 2
T = transform().translate(displ, axis)
Cw = np.empty(nrb.shape + (2, 4))
Cw[..., 0, :] = nrb.control
Cw[..., 1, :] = T(nrb.control)
UVW = nrb.knots + ([0, 0, 1, 1], )
return NURBS(UVW, Cw)
def transform(self, trans):
"""
Apply a scaling, rotation, or a translation to a NURBS object.
A NURBS object can be scaled, rotated, or translated by
applying the tranformation to the control points. To contruct
composite transformations, consult the docstrings in
:mod:`igakit.transform`.
Parameters
----------
trans : array_like
a matrix or transformation which scales, rotates,
and/or translates a NURBS object.
"""
if not isinstance(trans, transform):
trans = transform(trans)
array = self.array.copy()
array[...,:4] = trans(self.control)
self._array = np.ascontiguousarray(array)
return self
def transform(self, trans):
"""
Apply a scaling, rotation, or a translation to a NURBS object.
A NURBS object can be scaled, rotated, or translated by
applying the tranformation to the control points. To contruct
composite transformations, consult the docstrings in
:mod:`igakit.transform`.
Parameters
----------
trans : array_like
a matrix or transformation which scales, rotates,
and/or translates a NURBS object.
"""
if not isinstance(trans, transform):
trans = transform(trans)
array = self.array.copy()
array[..., :4] = trans(self.control)
self._array = np.ascontiguousarray(array)
return self
def revolve(nrb, filename, point, axis=2, angle=None):
"""
Construct a NURBS surface/volume by
revolving a NURBS curve/surface.
Parameters
----------
nrb : NURBS
point : array_like
axis : int or array_like, optional
angle : float or 2-tuple of floats, optional
Example
-------
>>> crv = line(1,2)
>>> srf = revolve(crv, point=0, axis=2, angle=[Pi/2,2*Pi])
>>> vol = revolve(srf, point=3, axis=1, angle=-Pi/2)
"""
assert nrb.dim <= 2
point = np.asarray(point, dtype='d')
assert point.ndim in (0, 1)
assert point.size <= 3
axis = np.asarray(axis)
assert axis.ndim in (0, 1)
assert 1 <= axis.size <= 3
if axis.ndim == 0:
v = np.zeros(3, dtype='d')
axis = (0, 1, 2)[int(axis)]
v[axis] = 1
else:
v = np.zeros(3, dtype='d')
v[:axis.size] = axis
norm_axis = np.linalg.norm(v)
assert norm_axis > 0
v /= norm_axis
# Transform the NURBS object to a new reference frame
# (O,X,Y,Z) centered at point and z-oriented with axis
n = [v[1], -v[0], 0] # n = cross(v, z)
gamma = np.arccos(v[2]) # cos_gamma = dot(v, z)
T = transform().translate(-point).rotate(gamma, n)
nrb = nrb.clone().transform(T)
# Map cartesian coordinates (x,y,z) to cylindrical coordinates
# (rho,theta,z) with theta in [0,2*pi] and precompute sines and
# cosines of theta angles.
Cw = nrb.control
X, Y, Z, W = (Cw[..., i] for i in range(4))
rho = np.hypot(X, Y)
theta = np.arctan2(Y, X)
theta[theta < 0] += 2 * np.pi
sines, cosines = np.sin(theta), np.cos(theta)
# Create a circular arc in the XY plane
matX = scipy.io.loadmat(filename)
knotsX = np.array(matX['knots'])
knotsX = knotsX.tolist()[0]
CX = np.transpose(np.array(matX['controlPoints']))
UX = knotsX
arc = NURBS([UX], CX)
#arc = circle(angle=angle)
Aw = arc.control
# Allocate control points and knots of the result
Qw = np.empty(nrb.shape + arc.shape + (4, ))
UVW = nrb.knots + arc.knots
# Loop over all control points of the NURBS object
# to revolve taking advantage of NumPy nd-indexing
dot = np.dot # inline numpy.dot
zeros = np.zeros # inline numpy.zeros
for idx in np.ndindex(nrb.shape):
z = Z[idx]
w = W[idx]
r = rho[idx]
r_sin_a = r * sines[idx]
r_cos_a = r * cosines[idx]
# for the sake of speed, inline
# the transformation matrix
# M = Rz(theta)*Tz(z)*Sxy(rho)
M = zeros((4, 4))
M[0, 0] = r_cos_a
M[0, 1] = -r_sin_a
M[1, 0] = r_sin_a
M[1, 1] = r_cos_a
M[2, 3] = z
M[3, 3] = 1
# Compute new 4D control points by transforming the
# arc control point and tensor-product the weights
Qi = Qw[idx]
Qi[...] = dot(Aw, M.T)
Qi[..., 3] *= w
# Create the new NURBS object and map
# back to the original reference frame
return NURBS(UVW, Qw).transform(T.invert())
def circle(radius=1, center=None, angle=None):
"""
Construct a NURBS circular arc or full circle
Parameters
----------
radius : float, optional
center : array_like, optional
angle : float or 2-tuple of floats, optional
Examples
--------
>>> crv = circle()
>>> crv.shape
(9,)
>>> P = crv([0, 0.25, 0.5, 0.75, 1])
>>> assert np.allclose(P[0], ( 1, 0, 0))
>>> assert np.allclose(P[1], ( 0, 1, 0))
>>> assert np.allclose(P[2], (-1, 0, 0))
>>> assert np.allclose(P[3], ( 0, -1, 0))
>>> assert np.allclose(P[4], ( 1, 0, 0))
>>> crv = circle(angle=3*Pi/2)
>>> crv.shape
(7,)
>>> P = crv([0, 1/3., 2/3., 1])
>>> assert np.allclose(P[0], ( 1, 0, 0))
>>> assert np.allclose(P[1], ( 0, 1, 0))
>>> assert np.allclose(P[2], (-1, 0, 0))
>>> assert np.allclose(P[3], ( 0, -1, 0))
>>> crv = circle(radius=2, center=(1,1), angle=(Pi/2,-Pi/2))
>>> crv.shape
(5,)
>>> P = crv([0, 0.5, 1])
>>> assert np.allclose(P[0], (1, 3, 0))
>>> assert np.allclose(P[1], (3, 1, 0))
>>> assert np.allclose(P[2], (1, -1, 0))
>>> crv = circle(radius=3, center=2, angle=Pi/2)
>>> crv.shape
(3,)
>>> P = crv([0, 1])
>>> assert np.allclose(P[0], ( 5, 0, 0))
>>> assert np.allclose(P[1], ( 2, 3, 0))
"""
if angle is None:
# Full circle, 4 knot spans, 9 control points
#spans = 4
#Cw = np.zeros((9,4), dtype='d')
#Cw[:,:2] = [[ 1, 0], [ 1, 1], [ 0, 1],
# [-1, 1], [-1, 0], [-1,-1],
# [ 0,-1], [ 1,-1], [ 1, 0]]
#Cw[:,:2] *= radius
#wm = np.sqrt(2)/2
#Cw[:,3] = 1; Cw[1::2,:] *= wm
#Modified to permit C^1 circle
# Full circle, 2 knot spans, 5 control points
spans = 2
Cw = np.zeros((6, 4), dtype='d')
#Cw[:,:] = [[ 1, 0, 0, 9], [ 1, 2, 0, 1], [ -1, 2, 0, 1.0/3],
# [-1, -2, 0, 1.0/3], [1, -2, 0, 1], [1 ,0 ,0 ,9]]
Cw[:, :] = [[1 * 9, 0, 0, 9], [1, 2, 0, 1],
[-1 * 1.0 / 3, 2 * 1.0 / 3, 0, 1.0 / 3],
[-1 * 1.0 / 3, -2 * 1.0 / 3, 0, 1.0 / 3], [1, -2, 0, 1],
[1 * 9, 0, 0, 9]]
Cw[:, :2] *= radius
else:
raise Exception('For angle<360 new C^1 knots not yet implemented',
'For angle<360 new C^1 knots not yet implemented')
Pi = np.pi # inline numpy.pi
# Determine start and end angles
if isinstance(angle, (tuple, list)):
start, end = angle
if start is None: start = 0
if end is None: end = 2 * Pi
else:
start, end = 0, angle
# Compute sweep and number knot spans
sweep = end - start
quadrants = (0.0, Pi / 2, Pi, 3 * Pi / 2)
spans = np.searchsorted(quadrants, abs(sweep))
# Construct a single-segment NURBS circular arc
# centered at the origin and bisected by +X axis
alpha = sweep / (2 * spans)
sin_a = np.sin(alpha)
cos_a = np.cos(alpha)
tan_a = np.tan(alpha)
x = radius * cos_a
y = radius * sin_a
wm = cos_a
xm = x + y * tan_a
Ca = [[x, -y, 0, 1], [wm * xm, 0, 0, wm], [x, y, 0, 1]]
# Compute control points by successive rotation
# of the controls points in the first segment
Cw = np.empty((2 * spans + 1, 4), dtype='d')
R = transform().rotate(alpha + start, 2)
Cw[0:3, :] = R(Ca)
if spans > 1:
R = transform().rotate(2 * alpha, 2)
for i in range(1, spans):
n = 2 * i + 1
Cw[n:n + 2, :] = R(Cw[n - 2:n, :])
# Translate control points to center
if center is not None:
T = transform().translate(center)
Cw = T(Cw)
# Compute knot vector in the range [0,1]
#a, b = 0, 1
#U = np.empty(2*(spans+1)+2, dtype='d')
#U[0], U[-1] = a, b
#U[1:-1] = np.linspace(a,b,spans+1).repeat(2)
U = np.array([0.0, 0.0, 0.0, 0.0, 0.5, 0.5, 1.0, 1.0, 1.0, 1.0])
# Return the new NURBS object
return NURBS([U], Cw)
def rotate(self, angle, axis=2):
t = transform().rotate(angle, axis)
return self.transform(t)
def scale(self, scale, axis=None):
t = transform().scale(scale, axis)
return self.transform(t)
def move(self, displ, axis=None):
t = transform().move(displ, axis)
return self.transform(t)
def translate(self, displ, axis=None):
t = transform().translate(displ, axis)
return self.transform(t)
def revolve(nrb, point, axis=2, angle=None):
"""
Construct a NURBS surface/volume by
revolving a NURBS curve/surface.
Parameters
----------
nrb : NURBS
point : array_like
axis : int or array_like, optional
angle : float or 2-tuple of floats, optional
Example
-------
>>> crv = line(1,2)
>>> srf = revolve(crv, point=0, axis=2, angle=[Pi/2,2*Pi])
>>> vol = revolve(srf, point=3, axis=1, angle=-Pi/2)
"""
assert nrb.dim <= 2
point = np.asarray(point, dtype='d')
assert point.ndim in (0, 1)
assert point.size <= 3
axis = np.asarray(axis)
assert axis.ndim in (0, 1)
assert 1 <= axis.size <= 3
if axis.ndim == 0:
v = np.zeros(3, dtype='d')
axis = (0,1,2)[int(axis)]
v[axis] = 1
else:
v = np.zeros(3, dtype='d')
v[:axis.size] = axis
norm_axis = np.linalg.norm(v)
assert norm_axis > 0
v /= norm_axis
# Transform the NURBS object to a new reference frame
# (O,X,Y,Z) centered at point and z-oriented with axis
n = [v[1], -v[0], 0] # n = cross(v, z)
gamma = np.arccos(v[2]) # cos_gamma = dot(v, z)
T = transform().translate(-point).rotate(gamma, n)
nrb = nrb.clone().transform(T)
# Map cartesian coordinates (x,y,z) to cylindrical coordinates
# (rho,theta,z) with theta in [0,2*pi] and precompute sines and
# cosines of theta angles.
Cw = nrb.control
X, Y, Z, W = (Cw[...,i] for i in range(4))
rho = np.hypot(X, Y)
theta = np.arctan2(Y, X); theta[theta<0] += 2*np.pi
sines, cosines = np.sin(theta), np.cos(theta)
# Create a circular arc in the XY plane
arc = circle(angle=angle)
Aw = arc.control
# Allocate control points and knots of the result
Qw = np.empty(nrb.shape + arc.shape + (4,))
UVW = nrb.knots + arc.knots
# Loop over all control points of the NURBS object
# to revolve taking advantage of NumPy nd-indexing
dot = np.dot # inline numpy.dot
zeros = np.zeros # inline numpy.zeros
for idx in np.ndindex(nrb.shape):
z = Z[idx]
w = W[idx]
r = rho[idx]
r_sin_a = r*sines[idx]
r_cos_a = r*cosines[idx]
# for the sake of speed, inline
# the transformation matrix
# M = Rz(theta)*Tz(z)*Sxy(rho)
M = zeros((4,4))
M[0,0] = r_cos_a; M[0,1] = -r_sin_a
M[1,0] = r_sin_a; M[1,1] = r_cos_a
M[2,3] = z
M[3,3] = 1
# Compute new 4D control points by transforming the
# arc control point and tensor-product the weights
Qi = Qw[idx]
Qi[...] = dot(Aw, M.T)
Qi[...,3] *= w
# Create the new NURBS object and map
# back to the original reference frame
return NURBS(UVW, Qw).transform(T.invert())
def circle(radius=1, center=None, angle=None):
"""
Construct a NURBS circular arc or full circle
Parameters
----------
radius : float, optional
center : array_like, optional
angle : float or 2-tuple of floats, optional
Examples
--------
>>> crv = circle()
>>> crv.shape
(9,)
>>> P = crv([0, 0.25, 0.5, 0.75, 1])
>>> assert np.allclose(P[0], ( 1, 0, 0))
>>> assert np.allclose(P[1], ( 0, 1, 0))
>>> assert np.allclose(P[2], (-1, 0, 0))
>>> assert np.allclose(P[3], ( 0, -1, 0))
>>> assert np.allclose(P[4], ( 1, 0, 0))
>>> crv = circle(angle=3*Pi/2)
>>> crv.shape
(7,)
>>> P = crv([0, 1/3., 2/3., 1])
>>> assert np.allclose(P[0], ( 1, 0, 0))
>>> assert np.allclose(P[1], ( 0, 1, 0))
>>> assert np.allclose(P[2], (-1, 0, 0))
>>> assert np.allclose(P[3], ( 0, -1, 0))
>>> crv = circle(radius=2, center=(1,1), angle=(Pi/2,-Pi/2))
>>> crv.shape
(5,)
>>> P = crv([0, 0.5, 1])
>>> assert np.allclose(P[0], (1, 3, 0))
>>> assert np.allclose(P[1], (3, 1, 0))
>>> assert np.allclose(P[2], (1, -1, 0))
>>> crv = circle(radius=3, center=2, angle=Pi/2)
>>> crv.shape
(3,)
>>> P = crv([0, 1])
>>> assert np.allclose(P[0], ( 5, 0, 0))
>>> assert np.allclose(P[1], ( 2, 3, 0))
"""
if angle is None:
# Full circle, 4 knot spans, 9 control points
spans = 4
Cw = np.zeros((9,4), dtype='d')
Cw[:,:2] = [[ 1, 0], [ 1, 1], [ 0, 1],
[-1, 1], [-1, 0], [-1,-1],
[ 0,-1], [ 1,-1], [ 1, 0]]
Cw[:,:2] *= radius
wm = np.sqrt(2)/2
Cw[:,3] = 1; Cw[1::2,:] *= wm
else:
Pi = np.pi # inline numpy.pi
# Determine start and end angles
if isinstance(angle, (tuple, list)):
start, end = angle
if start is None: start = 0
if end is None: end = 2*Pi
else:
start, end = 0, angle
# Compute sweep and number knot spans
sweep = end - start
quadrants = (0.0, Pi/2, Pi, 3*Pi/2)
spans = np.searchsorted(quadrants, abs(sweep))
# Construct a single-segment NURBS circular arc
# centered at the origin and bisected by +X axis
alpha = sweep/(2*spans)
sin_a = np.sin(alpha)
cos_a = np.cos(alpha)
tan_a = np.tan(alpha)
x = radius*cos_a
y = radius*sin_a
wm = cos_a
xm = x + y*tan_a
Ca = [[ x, -y, 0, 1],
[wm*xm, 0, 0, wm],
[ x, y, 0, 1]]
# Compute control points by successive rotation
# of the controls points in the first segment
Cw = np.empty((2*spans+1,4), dtype='d')
R = transform().rotate(alpha+start, 2)
Cw[0:3,:] = R(Ca)
if spans > 1:
R = transform().rotate(2*alpha, 2)
for i in range(1, spans):
n = 2*i+1
Cw[n:n+2,:] = R(Cw[n-2:n,:])
# Translate control points to center
if center is not None:
T = transform().translate(center)
Cw = T(Cw)
# Compute knot vector in the range [0,1]
a, b = 0, 1
U = np.empty(2*(spans+1)+2, dtype='d')
U[0], U[-1] = a, b
U[1:-1] = np.linspace(a,b,spans+1).repeat(2)
# Return the new NURBS object
return NURBS([U], Cw)
def rotate(self, angle, axis=2):
t = transform().rotate(angle, axis)
return self.transform(t)
def scale(self, scale, axis=None):
t = transform().scale(scale, axis)
return self.transform(t)
def move(self, displ, axis=None):
t = transform().move(displ, axis)
return self.transform(t)
def translate(self, displ, axis=None):
t = transform().translate(displ, axis)
return self.transform(t)