def extract_nurbs_isocurve(n, p, uk, m, q, vk, cpw, u=None, v=None):
"""
Extract iso-curve from NURBS surface.
:param int n: Number of control points - 1 in u-direction.
:param int p: Degree in u-direction.
:param ndarray uk: Knot vector in u-direction.
:param int m: Number of control points - 1 in v-direction.
:param int q: Degree in v-direction.
:param ndarray vk: Knot vector in v-direction.
:param ndarray cpw: Control points.
:param float u: Location of *u* split (in v-direction).
:param float v: Location of *v* split (in u-direction).
:return: Knot vector and control points for NURBS iso-curve in specified
direction (uq, qw).
:rtype: tuple
"""
if u is not None:
# Split at u in v-direction.
uq = array(vk, dtype=float64)
# Special case if a >= u >= b.
if u <= uk[p]:
return uq, cpw[0, :]
if u >= uk[n + 1]:
return uq, cpw[n, :]
# Determine multiplicity.
k, s = find_span_mult(n, p, u, uk)
qw = zeros((m + 1, 4), dtype=float64)
for j in range(0, m + 1):
_, qwi = curve_knot_ins(n, p, uk, cpw[:, j], u, p - s)
qw[j] = qwi[k - s]
return uq, qw
elif v is not None:
# Split at v in u-direction.
uq = array(uk, dtype=float64)
# Special case if a >= v >= b.
if v <= vk[q]:
return uq, cpw[:, 0]
if v >= vk[m + 1]:
return uq, cpw[:, m]
# Determine multiplicity.
k, s = find_span_mult(m, q, v, vk)
qw = zeros((n + 1, 4), dtype=float64)
for i in range(0, n + 1):
_, qwi = curve_knot_ins(m, q, vk, cpw[i, :], v, q - s)
qw[i] = qwi[k - s]
return uq, qw
def curve_knot_ins(n, p, uk, cpw, u, r):
"""
Curve knot insertion.
:param int n: Number of control points - 1.
:param int p: Degree.
:param ndarray uk: Knot vector.
:param ndarray cpw: Control points.
:param float u: Knot value to insert.
:param int r: Number of times to insert knot.
:return: Knot vector and control points after knot insertion
(knots, cpw).
:rtype: tuple
*Reference:* Algorithm A5.1 from "The NURBS Book."
"""
# Find span and multiplicity.
k, s = find_span_mult(n, p, u, uk)
if r + s > p:
r = p - s
# Generate knot vector
nq = n + r
uq = zeros(nq + p + 2, dtype=float64)
for i in range(0, k + 1):
uq[i] = uk[i]
for i in range(1, r + 1):
uq[k + i] = u
for i in range(k + 1, n + p + 2):
uq[i + r] = uk[i]
# Save unchanged control points.
qw = zeros((nq + 1, 4), dtype=float64)
for i in range(0, k - p + 1):
qw[i] = cpw[i]
for i in range(k - s, n + 1):
qw[i + r] = cpw[i]
rw = zeros((p + 1, 4), dtype=float64)
for i in range(0, p - s + 1):
rw[i] = cpw[k - p + i]
# Insert knot r times.
t = k - p
for j in range(1, r + 1):
t = k - p + j
for i in range(0, p - j - s + 1):
alpha = (u - uk[t + i]) / (uk[i + k + 1] - uk[t + i])
rw[i] = alpha * rw[i + 1] + (1.0 - alpha) * rw[i]
qw[t] = rw[0]
qw[k + r - j - s] = rw[p - j - s]
# Load remaining control points.
for i in range(t + 1, k - s):
qw[i] = rw[i - t]
return uq, qw
def split_nurbs_curve(n, p, uk, cpw, u):
"""
Split NURBS curve into two segments at *u*.
:param int n: Number of control points - 1.
:param int p: Degree.
:param ndarray uk: Knot vector.
:param ndarray cpw: Control points.
:param float u: Parametric point to split curve at.
:return: New knot vector and control points for two curves
(uk1, qw1, uk2, qw2).
:rtype: tuple
"""
# Special case if a >= u >= b.
# if u <= uk[p]:
ptol = Settings.ptol
if CmpFlt.le(u, uk[p], ptol):
qw1 = zeros((n + 1, 4), dtype=float64)
qw1[:] = cpw[0]
uk1 = array(uk, dtype=float64)
uk1[:] = uk[p]
qw2 = array(cpw, dtype=float64)
uk2 = array(uk, dtype=float64)
return uk1, qw1, uk2, qw2
# elif u >= uk[n + 1]:
elif CmpFlt.ge(u, uk[n + 1], ptol):
qw1 = array(cpw, dtype=float64)
uk1 = array(uk, dtype=float64)
qw2 = zeros((n + 1, 4), dtype=float64)
qw2[:] = cpw[n]
uk2 = array(uk, dtype=float64)
uk2[:] = uk[n + 1]
return uk1, qw1, uk2, qw2
# Find multiplicity of knot.
k, s = find_span_mult(n, p, u, uk)
# Insert knot p - s times and update knot vector and control points.
if s >= p:
uq, qw = array(uk, dtype=float64), array(cpw, dtype=float64)
else:
uq, qw = curve_knot_ins(n, p, uk, cpw, u, p - s)
# Control points and knot vectors.
qw1 = qw[:k - s + 1]
qw2 = qw[k - s:]
m = n + p + 1
uk1 = zeros((k + 1) + (p - s + 1), dtype=float64)
uk2 = zeros((m - k) + p + 1, dtype=float64)
uk1[:-1] = uq[:k + (p - s) + 1]
uk1[-1] = u
uk2[1:] = uq[k - s + 1:]
uk2[0] = u
return uk1, qw1, uk2, qw2
def filter_knot_vect(n, p, uk, x):
"""
Filter the knot vector *x* so that the knot multiplicity is less than or
equal to the degree *p* when inserted into *uk*.
:param int n: Number of control points - 1.
:param int p: Degree.
:param ndarray uk: Knot vector (sorted).
:param ndarray x: Knots to add to existing knot vector (sorted).
:return: Filtered knot vector *x*.
:rtype: ndarray
"""
temp = array(uk)
xnew = []
for i in range(0, len(x)):
span, s = find_span_mult(n, p, x[i], temp)
if s <= p:
temp = insert(temp, span, x[i])
n += 1
xnew.append(x[i])
return array(xnew, dtype=float64)
def move_nurbs_surface_seam(n, p, uk, m, q, vk, cpw, uv, d):
"""
Move the seam of a closed surface to a new parameter value.
:param int n: Number of control points - 1 in u-direction.
:param int p: Degree in u-direction.
:param ndarray uk: Knot vector in u-direction.
:param int m: Number of control points - 1 in v-direction.
:param int q: Degree in v-direction.
:param ndarray vk: Knot vector in v-direction.
:param ndarray cpw: Control points.
:param float uv: New seam parameter.
:param str d: Direction to move seam in (surface must be closed in that
direction).
:return: Knot vector and control points of new surface.
:rtype: tuple
"""
ptol = Settings.ptol
if d.lower() in ['u']:
if CmpFlt.le(uv, uk[p], ptol) or CmpFlt.ge(uv, uk[n + 1], ptol):
return uk, vk, cpw
# Check multiplicity of seam parameter.
_, s = find_span_mult(n, p, uv, uk)
# Insert seam parameter s - p times.
uq, vq, qw = surface_knot_ins(n, p, uk, m, q, vk, cpw, 'u', uv, p - s)
# Find span of seam parameter in new knot vector.
nw = qw.shape[0] - 1
k = find_span(nw, p, uv, uq)
# Get index for first row of control points.
n0 = k - p
# Generate control points for new seam.
qw_seam = array(qw)
indx = 0
# For new seam to end of old seam
for i in range(n0, nw + 1):
qw_seam[indx, :, :] = qw[i, :, :]
indx += 1
# From start of old seam to new seam.
for i in range(1, n0):
qw_seam[indx, :, :] = qw[i, :, :]
indx += 1
# Set last row.
qw_seam[-1, :, :] = qw_seam[0, :, :]
# Generate new knot vector.
uq_seam = array(uq)
# From seam to end
indx = p + 1
for i in range(k + 1, nw + p + 1):
uq_seam[indx] = uq[i] - uv
indx += 1
# From start to seam
for i in range(p + 1, k + 1 - p):
uq_seam[indx] = uq_seam[indx - 1] + (uq[i] - uq[i - 1])
indx += 1
uq_seam[-p - 1:] = uq[-1]
return uq_seam, vk, qw_seam
if d.lower() in ['v']:
if CmpFlt.le(uv, vk[q], ptol) or CmpFlt.ge(uv, vk[m + 1], ptol):
return uk, vk, cpw
# Check multiplicity of seam parameter.
_, s = find_span_mult(m, q, uv, vk)
# Insert seam parameter s - q times.
uq, vq, qw = surface_knot_ins(n, p, uk, m, q, vk, cpw, 'v', uv, q - s)
# Find span of seam parameter in new knot vector.
mw = qw.shape[1] - 1
k = find_span(mw, q, uv, vq)
# Get index for first row of control points.
m0 = k - q
# Generate control points for new seam.
qw_seam = array(qw)
indx = 0
# For new seam to end of old seam
for i in range(m0, mw + 1):
qw_seam[:, indx, :] = qw[:, i, :]
indx += 1
# From start of old seam to new seam.
for i in range(1, m0):
qw_seam[:, indx, :] = qw[:, i, :]
indx += 1
# Set last row.
qw_seam[:, -1, :] = qw_seam[:, 0, :]
# Generate new knot vector.
vq_seam = array(vq)
# From seam to end
indx = q + 1
for i in range(k + 1, mw + q + 1):
vq_seam[indx] = vq[i] - uv
indx += 1
# From start to seam
for i in range(q + 1, k + 1 - q):
vq_seam[indx] = vq_seam[indx - 1] + (vq[i] - vq[i - 1])
indx += 1
vq_seam[-q - 1:] = vq[-1]
return uk, vq_seam, qw_seam
def surface_knot_ins(n, p, uk, m, q, vk, cpw, d, uv, r):
"""
Surface knot insertion.
:param int n: Number of control points - 1 in u-direction.
:param int p: Degree in u-direction.
:param ndarray uk: Knot vector in u-direction.
:param int m: Number of control points - 1 in v-direction.
:param int q: Degree in v-direction.
:param ndarray vk: Knot vector in v-direction.
:param ndarray cpw: Control points.
:param str d: Direction to insert knot *uv* ("u": u-direction,
"v": v-direction.
:param float uv: Knot to insert.
:param int r: Number of times to insert knot.
:return: Knot vector and control points after knot insertion (uq, vq, cpw).
:rtype: tuple
*Reference:* Algorithm A5.3 from "The NURBS Book."
"""
if d.lower() in ['u']:
# Generate u-direction knot vector.
k, s = find_span_mult(n, p, uv, uk)
if r + s > p:
r = p - s
nq = n + r
uq = zeros(nq + p + 2, dtype=float64)
for i in range(0, k + 1):
uq[i] = uk[i]
for i in range(1, r + 1):
uq[k + i] = uv
for i in range(k + 1, n + p + 2):
uq[i + r] = uk[i]
# Copy v-direction knot vector.
vq = array(vk)
# Save alphas.
alpha = zeros((p + 1, r + 1), dtype=float64)
for j in range(1, r + 1):
t = k - p + j
for i in range(0, p - j - s + 1):
alpha[i, j] = (uv - uk[t + i]) / (uk[i + k + 1] - uk[t + i])
# Insert knot for each row at each column.
qw = zeros((nq + 1, m + 1, 4), dtype=float64)
for col in range(0, m + 1):
# Save unchanged control points.
for i in range(0, k - p + 1):
qw[i, col] = cpw[i, col]
for i in range(k - s, n + 1):
qw[i + r, col] = cpw[i, col]
# Load auxiliary control points.
rw = zeros((p + 1, 4), dtype=float64)
for i in range(0, p - s + 1):
rw[i] = cpw[k - p + i, col]
# Insert the knot r times.
t = k - p
for j in range(1, r + 1):
t = k - p + j
for i in range(0, p - j - s + 1):
rw[i] = (alpha[i, j] * rw[i + 1] +
(1.0 - alpha[i, j]) * rw[i])
qw[t, col] = rw[0]
qw[k + r - j - s, col] = rw[p - j - s]
# Load remaining control points.
for i in range(t + 1, k - s):
qw[i, col] = rw[i - t]
return uq, vq, qw
if d.lower() in ['v']:
# Generate v-direction knot vector.
k, s = find_span_mult(m, q, uv, vk)
if r + s > q:
r = q - s
mq = m + r
vq = zeros(mq + q + 2, dtype=float64)
for i in range(0, k + 1):
vq[i] = vk[i]
for i in range(1, r + 1):
vq[k + i] = uv
for i in range(k + 1, m + q + 2):
vq[i + r] = vk[i]
# Copy u-direction knot vector.
uq = array(uk)
# Save alphas.
alpha = zeros((q + 1, r + 1), dtype=float64)
for j in range(1, r + 1):
t = k - q + j
for i in range(0, q - j - s + 1):
alpha[i, j] = (uv - vk[t + i]) / (vk[i + k + 1] - vk[t + i])
# Insert knot for each column at each row.
qw = zeros((n + 1, mq + 1, 4), dtype=float64)
for row in range(0, n + 1):
# Save unchanged control points.
for i in range(0, k - q + 1):
qw[row, i] = cpw[row, i]
for i in range(k - s, m + 1):
qw[row, i + r] = cpw[row, i]
# Load auxiliary control points.
rw = zeros((q + 1, 4), dtype=float64)
for i in range(0, q - s + 1):
rw[i] = cpw[row, k - q + i]
# Insert the knot r times.
t = k - q
for j in range(1, r + 1):
t = k - q + j
for i in range(0, q - j - s + 1):
rw[i] = (alpha[i, j] * rw[i + 1] +
(1.0 - alpha[i, j]) * rw[i])
qw[row, t] = rw[0]
qw[row, k + r - j - s] = rw[q - j - s]
# Load remaining control points.
for i in range(t + 1, k - s):
qw[row, i] = rw[i - t]
return uq, vq, qw
def split_nurbs_surface(n, p, uk, m, q, vk, cpw, u=None, v=None):
"""
Split NURBS surface into two segments in specified direction.
:param int n: Number of control points - 1 in u-direction.
:param int p: Degree in u-direction.
:param ndarray uk: Knot vector in u-direction.
:param int m: Number of control points - 1 in v-direction.
:param int q: Degree in v-direction.
:param ndarray vk: Knot vector in v-direction.
:param ndarray cpw: Control points.
:param u: Location of *u* split (in v-direction).
:type u: float or None
:param v: Location of *v* split (in u-direction).
:type v: float or None
:return: New knot vectors and control points for two surfaces
(uk1, vk1, qw1, uk2, vk2, qw2).
:rtype: tuple
"""
if u is not None:
# Split at u in v-direction.
vk12 = array(vk, dtype=float64)
if u <= uk[p]:
qw1 = zeros((n + 1, m + 1, 4), dtype=float64)
for i in range(0, n + 1):
qw1[i, :] = cpw[0, :]
qw2 = array(cpw, dtype=float64)
uk1 = array(uk, dtype=float64)
uk1[:] = uk[p]
uk2 = array(uk, dtype=float64)
return uk1, vk12, qw1, uk2, vk12, qw2
if u >= uk[n + 1]:
qw1 = array(cpw, dtype=float64)
qw2 = zeros((n + 1, m + 1, 4), dtype=float64)
for i in range(0, n + 1):
qw2[i, :] = cpw[n, :]
uk1 = array(uk, dtype=float64)
uk2 = array(uk, dtype=float64)
uk2[:] = uk[n + 1]
return uk1, vk12, qw1, uk2, vk12, qw2
# Find multiplicity of knot.
k, s = find_span_mult(n, p, u, uk)
# Insert knot p - s times and update knot vector and control points.
if s >= p:
uq, qw = array(uk, dtype=float64), array(cpw, dtype=float64)
else:
uq, _, qw = surface_knot_ins(n, p, uk, m, q, vk, cpw, 'u', u,
p - s)
qw1 = qw[:k - s + 1, :]
qw2 = qw[k - s:, :]
r = n + p + 1
uk1 = zeros((k + 1) + (p - s + 1), dtype=float64)
uk2 = zeros((r - k) + p + 1, dtype=float64)
uk1[:-1] = uq[:k + (p - s) + 1]
uk1[-1] = u
uk2[1:] = uq[k - s + 1:]
uk2[0] = u
return uk1, vk12, qw1, uk2, vk12, qw2
elif v is not None:
# Split at v in u-direction.
uk12 = array(uk, dtype=float64)
if v <= vk[q]:
qw1 = zeros((n + 1, m + 1, 4), dtype=float64)
for j in range(0, m + 1):
qw1[:, j] = cpw[:, 0]
qw2 = array(cpw, dtype=float64)
vk1 = array(vk, dtype=float64)
vk1[:] = vk[q]
vk2 = array(vk, dtype=float64)
return uk12, vk1, qw1, uk12, vk2, qw2
if v >= vk[m + 1]:
qw1 = array(cpw, dtype=float64)
qw2 = zeros((n + 1, m + 1, 4), dtype=float64)
for j in range(0, m + 1):
qw2[:, j] = cpw[:, m]
vk1 = array(vk, dtype=float64)
vk2 = array(vk, dtype=float64)
vk2[:] = vk[m + 1]
return uk12, vk1, qw1, uk12, vk2, qw2
# Find multiplicity of knot.
k, s = find_span_mult(m, q, v, vk)
# Insert knot q - s times and update knot vector and control points.
if s >= q:
vq, qw = array(vk, dtype=float64), array(cpw, dtype=float64)
else:
_, vq, qw = surface_knot_ins(n, p, uk, m, q, vk, cpw, 'v', v,
q - s)
qw1 = qw[:, :k - s + 1]
qw2 = qw[:, k - s:]
r = m + q + 1
vk1 = zeros((k + 1) + (q - s + 1), dtype=float64)
vk2 = zeros((r - k) + q + 1, dtype=float64)
vk1[:-1] = vq[:k + (q - s) + 1]
vk1[-1] = v
vk2[1:] = vq[k - s + 1:]
vk2[0] = v
return uk12, vk1, qw1, uk12, vk2, qw2
def interpolate_curves(curves,
q=3,
method='chord',
inplace=False,
auto_reverse=True):
"""
Create a surface by interpolating the list of section curves.
:param list curves: List of NURBS curves to skin. Order of the curves in
the list will determine the longitudinal (*v*) direction.
:param int q: Degree in v-direction. Will be adjusted if not enough
skinning curves are provided (q = ncurves - 1).
:param str method: Option to specify method for selecting parameters
("uniform", "chord", or "centripetal").
:param bool inplace: Option to modify the curves in the list in-place. If
this option is *False*, then new curves will be created. If this option
is *True*, then the curves in the list will be modified in-place.
:param bool auto_reverse: Option to check direction of curves and
reverse if necessary.
:return: NURBS surface data (cpw, uk, vk, p, q).
:rtype: tuple
*Reference:* "The NURBS Book" Section 10.3.
"""
# Step 0: Adjust desired degree in case not enough curves are provided.
if len(curves) - 1 < q:
q = len(curves) - 1
# Step 1: Make sure each curve in a NURBS curve.
nurbs_curves = []
for c in curves:
if inplace:
nurbs_curves.append(c)
else:
nurbs_curves.append(c.copy())
if not nurbs_curves:
return None
# Check direction of curves by comparing the derivatives at the midpoint.
if auto_reverse:
cu_0 = nurbs_curves[0].deriv(0.5, 1, rtype='ndarray')
for c in nurbs_curves[1:]:
cu_i = c.deriv(0.5, 1, rtype='ndarray')
if dot(cu_0, cu_i) < 0.:
c.reverse()
cu_0 = cu_i
# Step 2: Make sure each curve has a common degree.
pmax = 0
for c in nurbs_curves:
if c.p > pmax:
pmax = c.p
for c in nurbs_curves:
if c.p < pmax:
c.elevate_degree(pmax, inplace=True)
# Step 3: Make sure each curve has a similar knot vector.
# Find each unique interior knot value in every curve.
a, b = [], []
for c in nurbs_curves:
a.append(c.a)
b.append(c.b)
amin = min(a)
bmax = max(b)
for c in nurbs_curves:
c.set_domain(amin, bmax)
all_knots = []
for c in nurbs_curves:
nu, _, knots = c.get_mult_and_knots()
for uk in knots[:nu]:
all_knots.append(uk)
all_knots.sort()
unique_knots = []
m = len(all_knots) - 1
i = 0
while i <= m:
unique_knots.append(all_knots[i])
# while i <= m and all_knots[i] == unique_knots[-1]:
while i <= m and CmpFlt.eq(all_knots[i], unique_knots[-1],
Settings.ptol):
i += 1
# For each curve, find the multiplicity and then the max multiplicity of
# each unique knot value.
all_mult = []
for c in nurbs_curves:
row = []
for ui in unique_knots:
_, mult = find_span_mult(c.n, c.p, ui, c.uk)
row.append(mult)
all_mult.append(row)
mult_arr = array(all_mult, dtype=int32)
max_mult = amax(mult_arr, axis=0)
# For each curve, insert the knot the required number of times, if any.
for mult, ui in zip(max_mult, unique_knots):
for c in nurbs_curves:
_, c_mult = find_span_mult(c.n, c.p, ui, c.uk)
if c_mult < mult:
c.insert_knot(ui, mult - c_mult, inplace=True, domain='global')
# Step 4: Compute v-direction parameters between [0, 1].
# Find parameters between each curve by averaging each segment.
temp = array([c.cpw for c in nurbs_curves], dtype=float64)
pnts_matrix = temp.transpose((1, 0, 2))
c0 = nurbs_curves[0]
n = c0.n
m = len(nurbs_curves) - 1
v_matrix = zeros((n + 1, m + 1), dtype=float64)
for i in range(0, n + 1):
if method.lower() in ['u', 'uniform']:
v = uniform(pnts_matrix[i, :], 0., 1.)
elif method.lower() in ['ch', 'chord']:
v = chord_length(pnts_matrix[i, :], 0., 1.)
else:
v = centripetal(pnts_matrix[i, :], 0., 1.)
v_matrix[i] = v
# Average each column.
v = mean(v_matrix, axis=0, dtype=float64)
v[0] = 0.0
v[-1] = 1.0
# Step 5: Compute knot vector by averaging.
uk = c0.uk
s = m + q + 1
vk = zeros(s + 1, dtype=float64)
vk[s - q:] = 1.0
for j in range(1, m - q + 1):
temp = 0.
for i in range(j, j + q):
temp += v[i]
vk[j + q] = 1.0 / q * temp
# Step 6: Perform n + 1 interpolations for v-direction.
cpw = zeros((n + 1, m + 1, 4), dtype=float64)
for i in range(0, n + 1):
qp = pnts_matrix[i]
# Set up system of linear equations.
a = zeros((m + 1, m + 1), dtype=float64)
for j in range(0, m + 1):
span = find_span(m, q, v[j], vk)
a[j, span - q:span + 1] = basis_funs(span, v[j], q, vk)
# Solve for [a][cp] = [qp] using LU decomposition.
lu, piv = lu_factor(a, overwrite_a=True, check_finite=False)
cpw[i, :] = lu_solve((lu, piv),
qp,
trans=0,
overwrite_b=True,
check_finite=True)
return cpw, uk, vk, pmax, q