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Summary
| Description |
English: This shows how the ability to create piecewise parabolic B-splines in 3D allows NURBS to follow a perfect circle:
The black triangle shows the 2D NURBS control points without weights (w=1). The Blue dotted line shows the corresponding 3D B-spline in en:homogeneous coordinates. The blue parabolas are the corresponding B-spline, consisting of three parabolas. Each is parallel to the opposite face of the gray cone, so projecting it onto the w=1 plane results in a circular arc (red; see en:conic section). The knots are: [0,0,0,1/3,1/3,2/3,2/3,1,1,1]. The NURBS control points are: [[1.0, 0.0, 1.0], [1, sqrt(3), 0.5], [-0.5, sqrt(3)/2, 1.0], [-2.0, 0.0, 0.5], [-0.5, -sqrt(3)/2, 1.0], [1, -sqrt(3), 0.5], [1.0, 0.0, 1.0]]. and so the corresponding 3D B-spline control points are: [[1.0, 0.0, 1.0], [0.5, sqrt(3)/2, 0.5], [-0.5, sqrt(3)/2, 1.0], [-1.0, 0.0, 0.5], [-0.5, -sqrt(3)/2, 1.0], [0.5, -sqrt(3)/2, 0.5], [1.0, 0.0, 1.0]].The source code is shown below. |
| Date | |
| Source | en:File:NURBS-circle-3D.png |
| Author | Nicoguaro |
| SVG development |  This W3C-invalid plot was created with Matplotlib. |
| Source code | Python codefrom __future__ import division
from scipy.interpolate import splev
from numpy import (sqrt, linspace, sin, cos, array, hstack, transpose, pi,
ones_like, zeros_like, where, r_)
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.tri import Triangulation
def splevn(t, tks):
"""Like splev but control points are vectors."""
return zip(*[splev(t, [tks[0], p, tks[-1]]) for p in zip(*tks[1])])
# Knots, weighted control points, degree:
arcNURBS = [[0,0,0,0.25,0.25,0.5,0.5,0.75,0.75,1,1,1],
[[1,0,1],
[sqrt(0.5), sqrt(0.5), sqrt(0.5)],
[0,1,1],
[-sqrt(0.5), sqrt(0.5), sqrt(0.5)],
[-1,0,1],
[-sqrt(0.5), -sqrt(0.5), sqrt(0.5)],
[0,-1,1],
[sqrt(0.5), -sqrt(0.5), sqrt(0.5)],
[1,0,1]], 2]
theta = linspace(0, 2*pi, 7)
pts = transpose([cos(theta), sin(theta)])
w = 1.0 / array([1.,2,1,2,1,2,1])
arcNURBS = [array([0.,0,0,1,1,2,2,3,3,3]) / 3.0,
hstack([pts, w[:,None]]),
2]
Pw = array(arcNURBS[1])
t = linspace(arcNURBS[0][0], arcNURBS[0][-1], 6*15+1)
x, y, w = transpose(splevn(t, arcNURBS))
fig = plt.figure(figsize=(10, 10))
ax = fig.add_subplot(111, projection='3d')
ax.view_init(azim=-50, elev=20)
for setRange in (ax.set_xlim, ax.set_ylim):
setRange(-1.5, 1.5)
ax.set_zlim(-0.5, 1.5)
# Shade in the cone:
xs = hstack([[0], x/w])
ys = hstack([[0], y/w])
ws = hstack([[0], ones_like(w)])
tris = []
for i in range(len(x)):
tris.append([0, (i + 1) % len(xs), (i + 2) % len(xs)])
color = (0.9,)*3
cone = ax.plot_trisurf(xs, ys, tris, ws, color=color, edgecolor=color,
alpha=1.0, lw=0)
Pw = transpose(arcNURBS[1])
ax.plot(Pw[0]/Pw[2], Pw[1]/Pw[2], Pw[2]/Pw[2], 'k.-',
label='NURBS control w/o weights')
p = ax.plot(Pw[0], Pw[1], Pw[2], '.-', color='b',
label='3D control polygon')[0]
p.set_zorder(-1000)
ax.plot(x, y, w, 'b', label='3D parabolas')
ax.plot(x/w, y/w, ones_like(w), color=(0.85,0.1,0.1), linewidth=2,
label='NURBS circle at w = 1')
if True:
ax.plot(Pw[0], Pw[1], ones_like(Pw[2]), '.-', color='b', alpha=0.2)
ax.plot(x, y, ones_like(w), 'b', alpha=0.2)
if not True:
plots = []
plots += ax.plot(Pw[0]/Pw[2], Pw[1]/Pw[2], 0*Pw[2]/Pw[2], 'k.-', alpha=0.2)
plots += ax.plot(Pw[0], Pw[1], 0*Pw[2], '.-', color='b', alpha=0.2)
plots += ax.plot(x, y, 0*w, 'b', alpha=0.2)
plots += ax.plot(x/w, y/w, zeros_like(w), color=(0.85,0.1,0.1),
linewidth=2, alpha=0.2)
for i, p in enumerate(plots):
p.set_zorder(-1000-i)
DRAW_RULES_ON_CONE = False
if DRAW_RULES_ON_CONE:
for i in r_[:len(t):10]:
ax.plot([0, x[i], x[i]/w[i]], [0, y[i], y[i]/w[i]], [0, w[i], 1],
'k.--')
for i in where(Pw[-1] != 1)[0]:
ax.plot([0, Pw[0,i]/Pw[2,i]], [0, Pw[1,i]/Pw[2,i]], [0, 1], 'k.--',
alpha=0.5)
plt.xlabel(r"$x$", fontsize=18)
plt.ylabel(r"$y$", fontsize=18)
ax.set_zlabel(r"$z$", fontsize=18)
plt.legend()
plt.savefig("NURBS-circle-3D.svg", transparent=True, bbox_inches="tight")
plt.show()
|
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 23:51, 15 February 2016 | 715 × 738 (75 KB) | Nicoguaro | Crop image | |
| 23:49, 15 February 2016 | 900 × 900 (75 KB) | Nicoguaro | User created page with UploadWizard |
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