This project aims to implement the NURBS curve and surface computation algorithms in native Python with minimum possible dependencies.
Currently, the Curve and Surface classes can be used for data storage and evaluation of B-Spline and NURBS curves and surfaces. Additionally, Grid class can be used to generate simple 2D control point grids for use with the Surface class.
NURBS-Python currently implements the following algorithms from The NURBS Book by Piegl & Tiller:
- Algorithm A2.1: FindSpan
- Algorithm A2.2: BasisFuns
- Algorithm A2.3: DersBasisFuns
- Algorithm A3.1: CurvePoint
- Algorithm A3.2: CurveDerivsAlg1
- Algorithm A3.3: CurveDerivCpts
- Algorithm A3.4: CurveDerivsAlg2
- Algorithm A3.5: SurfacePoint
- Algorithm A3.6: SurfaceDerivsAlg1
- Algorithm A4.1: CurvePoint (from weighted control points)
- Algorithm A4.3: SurfacePoint (from weighted control points)
The data structure in Curve and Surface classes is implemented using Python properties. The following table shows the properties defined in these classes:
| Curve Properties | Surface Properties | Notes |
|---|---|---|
| degree | degree_u | Degree of the curve/surface |
| degree_v | ||
| knotvector | knotvector_u | Knot vectors |
| knotvector_v | ||
| ctrlpts | ctrlpts | 1D array of control points |
| ctrlpts2D | 2D array of control points in [u][v] format | |
| ctrlptsw | ctrlptsw | 1D array of weighted control points |
| weights | weights | Weights vector |
| delta | delta | Evaluation delta for knots |
| curvepts | surfpts | Evaluated points |
After setting the required parameters, the curve or the surface can be evaluated using evaluate() or evaluate_rational() methods. Then, the evaluated curve points can be obtained from curvepts property and the evaluated surface points can be obtained from surfpts property. The curve and surface derivatives can be evaluated using derivatives() method. An easy way to get 1st derivatives using tangent() method is available in both classes.
Surface class has methods for transposing the surface by swapping U and V directions, tranpose(), and finding surface normals, normal().
Both classes have read_ctrlpts() and read_ctrlptsw() methods for reading control points and weighted control points, respectively, from a text file. The details on the file format are explained in FORMATS.md file.
utilities module has some extra features for several mathematical operations:
autogen_knotvector()generates a uniform knot vector according to the input degree and number of control pointsnormalize_knotvector()normalizes the knot vector between 0 and 1cross_vector()computes the cross production of the input vectorsnormalize_vector()generates a unit vector from the input vector
Other functions in the utilities module are used as helper functions in evaluation methods of Curve and Surface classes.
Grid module is capable of generating simple 2D control point grids for use with the Surface class. Please check ex_grid01.py file and the documentation for details on how to use the Grid class and its features.
One of the major goals of this project is implementing all these algorithms with minimum dependencies. Currently, the NURBS package can run with plain Python and therefore, it has no extra dependencies, like NumPy or similar. The code was tested with Python versions 2.7.12 and 3.5.3.
On the other hand, the plotting part of the examples requires Matplotlib installed in your Python distribution. If you don't need any plotting, you basically won't need Matplotlib at all.
nurbs\directory includes theCurveandSurfaceclasses along with theutilitiesmoduleex*.pyfiles are testing scripts for demonstrating curve and surface evaluationsdata\directory contains sample control points for the testing scripts
Curves:
- ex_curve01.py
- ex_curve02.py
- ex_curve03.py
- ex_curve04.py (A full circle)
Surfaces:
Grid Generator
The following plots are generated using Matplotlib.
Displays the control points polygon and the evaluated curve using the an auto-generated uniform knot vector.
Displays the evaluated curve using an auto-generated uniform knot vector and the tangent vector at u = 0.6.
Displays the control points polygon and the evaluated curve using the an auto-generated uniform knot vector. Tangent vectors are shown in quiver plots.
Displays the control points polygon and the evaluated NURBS curve for a full circle.
Thanks to @jedufour for ex_surface03.py example.
If you have any questions related to the NURBS-Python package, please don't hesitate to contact the author.
- Onur Rauf Bingol (@orbingol)
- John-Eric Dufour (@jedufour), bug fixing and contribution of surface example 3
- Jan Heczko (@heczis), bug fixing
I would like to thank my PhD adviser, Dr. Adarsh Krishnamurthy, for his guidance and supervision throughout the course of this project. If you are interested in this Python package, please have a look at our research group's web page for more projects and contact information.