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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:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
File history
Click on a date/time to view the file as it appeared at that time.
| 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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