# Define the order of the basis polynomials
# Linear (p = 1), Quadratic (p = 2), Cubic (p = 3), etc.
# Set p = n (number of control points minus one) to obtain a Bezier
p = 3
q = 3
# Define the knot vectors (clamped spline)
# p+1 zeros, n-p equispaced points between 0 and 1, and p+1 ones. In total r+1 points where r=n+p+1
# q+1 zeros, m-p equispaced points between 0 and 1, and q+1 ones. In total s+1 points where s=m+q+1
U = np.concatenate((np.zeros(p), np.linspace(0, 1, n - p + 2), np.ones(p)))
V = np.concatenate((np.zeros(q), np.linspace(0, 1, m - q + 2), np.ones(q)))
# Create and plot the NURBS surface
nurbsSurface = nrb.NurbsSurface(control_points=P,
weights=W,
u_degree=p,
v_degree=q,
u_knots=U,
v_knots=V)
nurbsSurface.plot(surface=True, control_points=True)
S_func = nurbsSurface.get_value
# -------------------------------------------------------------------------------------------------------------------- #
# Check the analytic derivatives against a finite difference aproximation
# -------------------------------------------------------------------------------------------------------------------- #
# Define a (u,v) parametrization suitable for finite differences. 1D arrays of (u,v) query points
h = 1e-4
hh = h + h**2
Nu, Nv = 50, 50
u = np.linspace(0.00 + h, 1.00 - h, Nu)
v = np.linspace(0.00 + h, 1.00 - h, Nv)
[u, v] = np.meshgrid(u, v, indexing='xy')
# Third row P[:, 0, 2] = [0.00, 3.00, 2.00] P[:, 1, 2] = [1.00, 2.00, 2.00] P[:, 2, 2] = [2.00, 1.50, 2.00] P[:, 3, 2] = [3.00, 2.00, 2.00] P[:, 4, 2] = [4.00, 3.00, 2.00] # Fourth row P[:, 0, 3] = [0.50, 3.00, 3.00] P[:, 1, 3] = [1.00, 2.50, 3.00] P[:, 2, 3] = [2.00, 2.00, 3.00] P[:, 3, 3] = [3.00, 2.50, 3.00] P[:, 4, 3] = [3.50, 3.00, 3.00] # Create and the NURBS surface nurbsSurface = nrb.NurbsSurface(control_points=P) # Define point to be projected P0 = np.asarray([0.50, 0.50, 0.50]).reshape((3,1)) # Compute projected time u0, v0 = nurbsSurface.project_point_to_surface(P0) S0 = nurbsSurface.get_value(u0, v0) # Plot the NURBS surface and the projected point fig, ax = nurbsSurface.plot() Px, Py, Pz = P0.flatten() Sx, Sy, Sz = S0.flatten() ax.plot([Px, Sx], [Py, Sy], [Pz, Sz], color='black', linestyle='--', marker='o', markeredgecolor='r', markerfacecolor='w') plt.show()
P[:, 1, 1] = [1.00, 2.00, 1.00] P[:, 2, 1] = [2.00, 1.50, 1.00] P[:, 3, 1] = [3.00, 2.00, 1.00] P[:, 4, 1] = [4.00, 3.00, 1.00] # Third row P[:, 0, 2] = [0.00, 3.00, 2.00] P[:, 1, 2] = [1.00, 2.00, 2.00] P[:, 2, 2] = [2.00, 1.50, 2.00] P[:, 3, 2] = [3.00, 2.00, 2.00] P[:, 4, 2] = [4.00, 3.00, 2.00] # Fourth row P[:, 0, 3] = [0.50, 3.00, 3.00] P[:, 1, 3] = [1.00, 2.50, 3.00] P[:, 2, 3] = [2.00, 2.00, 3.00] P[:, 3, 3] = [3.00, 2.50, 3.00] P[:, 4, 3] = [3.50, 3.00, 3.00] # Define the array of control point weights W = np.zeros((n, m)) W[:, 0] = np.asarray([1, 1, 5, 1, 1]) W[:, 1] = np.asarray([1, 1, 5, 1, 1]) W[:, 2] = np.asarray([1, 1, 5, 1, 1]) W[:, 3] = np.asarray([1, 1, 0.5, 1, 1]) # Create and plot the Bezier surface bezierSurface = nrb.NurbsSurface(control_points=P, weights=W) bezierSurface.plot(control_points=True) plt.show()
m = np.shape(P)[2] - 1 # Define the order of the basis polynomials # Linear (p = 1), Quadratic (p = 2), Cubic (p = 3), etc. # Set p = n (number of control points minus one) to obtain a Bezier p = 3 q = 3 # Define the knot vectors (clamped spline) # p+1 zeros, n-p equispaced points between 0 and 1, and p+1 ones. In total r+1 points where r=n+p+1 # q+1 zeros, m-p equispaced points between 0 and 1, and q+1 ones. In total s+1 points where s=m+q+1 U = np.concatenate((np.zeros(p), np.linspace(0, 1, n - p + 2), np.ones(p))) V = np.concatenate((np.zeros(q), np.linspace(0, 1, m - q + 2), np.ones(q))) # Create and plot the B-Spline surface bsplineSurface = nrb.NurbsSurface(control_points=P, u_degree=p, v_degree=q, u_knots=U, v_knots=V) fig, ax = bsplineSurface.plot(surface=False, control_points=False) # Plot the blade surface using a colormap based on curvature bsplineSurface.plot_surface(fig, ax, color='mean_curvature', colorbar=True) # Plot the vectors normal to the surface bsplineSurface.plot_normals(fig, ax, scale=0.30) # Plot isoparametric curves bsplineSurface.plot_isocurve_u(fig, ax, np.linspace(0, 1, 5)) bsplineSurface.plot_isocurve_v(fig, ax, np.linspace(0, 1, 5)) # Show the figure plt.show()
# Define the array of control points n_dim, n, m = 3, 2, 2 P = np.zeros((n_dim, n, m)) # First row P[:, 0, 0] = [0.00, 0.00, 0.00] P[:, 1, 0] = [0.00, 1.00, 0.00] P[:, 1, 0] = [0.00, 1.00, 0.00] # Second row P[:, 0, 1] = [1.00, 0.00, 0.00] P[:, 1, 1] = [1.00, 1.00, 0.00] # Create and plot the NURBS surface nurbsSurface1 = nrb.NurbsSurface(control_points=P) # -------------------------------------------------------------------------------------------------------------------- # # Create the second NURBS surface # -------------------------------------------------------------------------------------------------------------------- # # Define the array of control points n_dim, n, m = 3, 2, 2 P = np.zeros((n_dim, n, m)) # First row P[:, 0, 0] = [0.00, 1.00, 0.00] P[:, 1, 0] = [0.00, 2.00, 0.00] # Second row P[:, 0, 1] = [1.00, 1.00, 0.00]