Theory

These pages explain the mathematics behind nurbspy, from curve and surface representations to geometric calculations. Equations are paired with practical interpretations and figures evaluated using the package. The representation and derivative material follows The NURBS Book Piegl and Tiller [1997].

Representations and continuity

• Bézier curves and surfaces
• B-spline curves and surfaces
• NURBS curves and surfaces
• G² continuity of NURBS curves

Curve and surface operations

The operation pages connect analytic derivatives to lengths, local frames, surface measures and inverse problems. Each page distinguishes mathematical assumptions from numerical safeguards, identifies the relevant backend APIs, and links to reproducible figure-generation code.

• Curve arc length and numerical integration
• Arc-length parameterization and coordinate inversion
• Frenet–Serret frames and torsion
• Extracting isoparametric curves from surfaces
• Surface area and surface integrals
• Surface differential geometry
• Point projection onto curves and surfaces

References

• References

Notation and assumptions

Symbol	Meaning
\(\mathbf P_i\), \(\mathbf P_{i,j}\)	Curve control points or a surface control net
\(n+1\), \(m+1\)	Numbers of control points in the two parameter directions
\(p\), \(q\)	Polynomial degrees; the corresponding orders are \(p+1\), \(q+1\)
\(U\), \(V\)	Nondecreasing knot vectors, including repeated knots
\(w_i\), \(w_{i,j}\)	Control-point weights
\(u\), \(v\)	Parameters, distinct from spatial coordinates and arc length

Unless stated otherwise, knots are normalized and clamped to \([0,1]\), weights are strictly positive, and geometry is real. Derivatives at the ends of a curve or surface are one-sided limits. Curvature requires a nonzero first derivative.

Regenerate the illustrations

From the repository root, with the development dependencies and JAX extra installed:

python docs/generate_theory_images.py
python docs/generate_theory_images.py --only frames surface_geometry

The scripts call set_plot_options() and save each numerical illustration as a 300 dpi PNG, an SVG embedded in these pages and a vector PDF for publication. They also check the illustrated calculations against exact formulas or independent numerical references. The generator and figure-code notes describe the workflow. The scripts share helper geometries and styling, so run them from the repository using the commands above.