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interp.py
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interp.py
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from numpy import *
import pylab
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
# http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/INT-APP/SURF-INT-global.html
# http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/INT-APP/PARA-surface.html
# http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/INT-APP/PARA-knot-generation.html
# http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/INT-APP/PARA-chord-length.html
# K. Lee, Principles of CAD/CAM/CAE Systems
def eval_curve(k,t,P,u):
# evaluate B-spline at u
# inputs
# k : B-spline order
# t : sequence of knots
# P : control points
# u : evaluation parameter value
# returns
# A : coordinates of eval point
# l : t[l] <= u < t[l+1]
# first and last k repeat, so look at t[k:-k]
l = searchsorted(t[k:-k],u,'right')+k-1
A = zeros((k,3))
for j in range(k):
A[j,:] = P[l-k+1+j,:]
for r in range(1,k):
for j in range(k-1,r-1,-1):
i = l-k+1+j
d1 = u-t[i]
d2 = t[i+k-r]-u
A[j,:] = (d1*A[j,:]+d2*A[j-1,:])/(d1+d2)
return A[k-1,:]
def eval_surf(k,l,s,t,P,u,v):
# evaluate B-spline at u
# inputs
# k,l : B-spline order
# s,t : sequence of knots
# P : control points
# u,v : evaluation parameter value
# returns
# A : coordinates of eval point
# m+1 : number of control points in u-direction
m = P.shape[0]-1
# n+1 : number of control points in v-direction
n = P.shape[1]-1
# calculate temporary control points C
C = zeros((m+1,3))
for i in range(m+1):
C[i,:] = eval_curve(l,t,P[i,:],v)
return eval_curve(k,s,C,u)
def eval_basis(k,t,i,n,u):
# n+1 : number of control points
# evaluate N_{i,k}(u)
P = zeros((n+1,3))
P[i,:] = 1.0
return eval_curve(k,t,P,u)
def interp_curve(Q,k,u=-1,t=-1):
# fit a B-spline curve to data points Q
# inputs
# Q : data points to interpolate
# k : B-spline order
# u : parameter values (optional)
# t : knot locations (optional)
# returns
# P : control points
# u : parameter values
# t : knot locations
# n+1 : number of control points
n = Q.shape[0]-1
if t == -1:
d = sqrt((Q[1:,0]-Q[0:-1,0])**2 + (Q[1:,1]-Q[0:-1,1])**2 + (Q[1:,2]-Q[0:-1,2])**2)
t = zeros((n+k+1))
t[0:k] = 0.0
t[n+1:] = 1.0
for i in range(k,n+1):
den = 0.0
for m in range(k,n+2):
den += sum(d[m-k:m-1])
t[i] = t[i-1] + sum(d[i-k:i-1]) / den
if u == -1:
u = zeros((n+1))
for j in range(n+1):
u[j] = sum(t[j+1:j+k]) / (k-1)
# form linear system
B = zeros((n+1,n+1))
for i in range(n+1):
for j in range(n+1):
B[i,j] = eval_basis(k,t,j,n,u[i])[0]
# solve to compute control points
P = linalg.solve(B,Q)
return P, u, t
def interp_surf(Q,k,l):
# fit a B-spline surface to data points Q
# inputs
# Q : data points to interpolate (m+1)x(n+1)x3
# k : B-spline order in u-direction
# l : B-spline order in v-direction
# returns
# P : control points
# u,v : parameter values
# s,t : knot values
# m+1 : number of control points in u-direction
m = Q.shape[0]-1
# n+1 : number of control points in v-direction
n = Q.shape[1]-1
# calculate parameters using the chord length method
tmp = zeros((m+1,n+1))
for j in range(0,n+1):
d = sqrt((Q[1:,j,0]-Q[0:-1,j,0])**2\
+ (Q[1:,j,1]-Q[0:-1,j,1])**2\
+ (Q[1:,j,2]-Q[0:-1,j,2])**2)
L = sum(d)
tmp[0,j] = 0.0
tmp[-1,j] = 1.0
for i in range(1,m):
tmp[i,j] = sum(d[:i]) / L
u = average(tmp,1)
for i in range(0,m+1):
d = sqrt((Q[i,1:,0]-Q[i,0:-1,0])**2\
+ (Q[i,1:,1]-Q[i,0:-1,1])**2\
+ (Q[i,1:,2]-Q[i,0:-1,2])**2)
L = sum(d)
tmp[i,0] = 0.0
tmp[i,-1] = 1.0
for j in range(1,n):
tmp[i,j] = sum(d[:j]) / L
v = average(tmp,0)
# calculate knots from parameters
s = zeros((m+k+1))
s[0:k] = 0.0
s[m+1:] = 1.0
for j in range(1,m-k+2):
s[j+k-1] = sum(u[j:j+k-1]) / (k-1)
t = zeros((n+l+1))
t[0:l] = 0.0
t[n+1:] = 1.0
for j in range(1,n-l+2):
t[j+l-1] = sum(v[j:j+l-1]) / (l-1)
# solve for temporary control points D
D = zeros((m+1,n+1,3))
for j in range(n+1):
B = zeros((m+1,m+1))
for x in range(m+1):
for y in range(m+1):
B[x,y] = eval_basis(k,s,y,m,u[x])[0]
D[:,j,:] = linalg.solve(B,Q[:,j,:])
# solve for control points P
P = zeros((m+1,n+1,3))
for i in range(m+1):
B = zeros((n+1,n+1))
for x in range(n+1):
for y in range(n+1):
B[x,y] = eval_basis(l,t,y,n,v[x])[0]
P[i,:,:] = linalg.solve(B,Q[i,:,:])
return P, u, v, s, t
def surf_test():
m = 5
n = 11
Q = zeros((m+1,n+1,3))
s = linspace(0.,1.,m+1)
t = linspace(0.,1.,n+1)
Q[:,:,0], Q[:,:,1] = meshgrid(t,s)
for i in range(m+1):
for j in range(n+1):
Q[i,j,2] = cos(6*s[i])*sin(4*t[j])*t[j]
k = l = 4
P, u, v, s, t = interp_surf(Q,k,l)
return k,l,s,t,P
'''
uu = linspace(0.,1.,20)
vv = linspace(0.,1.,20)
S = zeros((len(uu),len(vv),3))
for i in range(len(uu)):
for j in range(len(vv)):
S[i,j,:] = eval_surf(k,l,s,t,P,uu[i],vv[j])
fig = pylab.figure()
ax = Axes3D(fig)
ax.plot_surface(S[:,:,0], S[:,:,1], S[:,:,2], rstride=1, cstride=1, cmap = cm.jet)
for i in range(m+1):
ax.plot(Q[i,:,0],Q[i,:,1],Q[i,:,2],'k*')
pylab.show()
'''
def curve_test():
k = 4
np = 15
Q = zeros((np,3))
s = linspace(0.,1.,np)
for i in range(np):
Q[i,0] = cos(s[i])
Q[i,1] = sin(s[i])
Q[i,2] = s[i]
P, u, t = interp_curve(Q,k)
uu = linspace(0.,1.,100)
S = zeros((len(uu),3))
for i in range(len(uu)):
S[i,:] = eval_curve(k,t,P,uu[i])
fig = pylab.figure()
ax = Axes3D(fig)
ax.plot(Q[:,0], Q[:,1], Q[:,2],'*')
ax.plot(S[:,0], S[:,1], S[:,2])
pylab.show()