Theory

These pages present the mathematical foundations of nurbspy, covering Bézier, B-spline, and NURBS curves and surfaces. The material progresses from Bézier to B-spline and then to NURBS representations, treating curves before surfaces and developing the relevant definitions, basis-function properties, and derivative formulas along the way. Most of the material follows The NURBS Book Piegl and Tiller [1997].

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