Parametric Curves and Surfaces

Parametric curves and surfaces represent smooth free-form shapes as functions of one or two parameters, giving designers compact, controllable descriptions of geometry.

Definition

A parametric curve or surface maps an interval or rectangle of parameter values to points in space, typically as a weighted combination of control points using polynomial or rational basis functions.

Scope

This topic covers Bezier curves and the de Casteljau algorithm, B-spline and NURBS representations with their knot vectors and local control, continuity conditions between segments, and the tensor-product construction that extends these curves to surfaces.

Core questions

• How can a smooth curve be specified and edited through a few control points?
• What continuity holds where curve or surface pieces join?
• Why are rational forms such as NURBS needed?
• How do curve constructions generalize to surfaces?

Key concepts

• Bezier curves
• de Casteljau algorithm
• B-splines and knot vectors
• NURBS
• Geometric and parametric continuity
• Tensor-product surfaces

Key theories

Bezier curves and de Casteljau evaluation

A Bezier curve is a Bernstein-polynomial blend of its control points, evaluated stably by repeated linear interpolation, with the curve lying inside the convex hull of and tangent to its control polygon.

B-splines and NURBS

B-splines provide local control and adjustable smoothness through a knot vector, and their rational generalization, NURBS, can represent conic sections exactly, making it the standard in computer-aided design.

Clinical relevance

Parametric curves and surfaces are the geometric backbone of computer-aided design, font and vector-graphics outlines, animation paths, and industrial surface design in automotive and aerospace engineering.

History

Developed independently by Bezier at Renault and de Casteljau at Citroen in the early 1960s, the methods were unified and extended by de Boor's B-spline theory and standardized as NURBS in CAD systems.

Key figures

• Pierre Bezier
• Paul de Casteljau
• Carl de Boor

Seminal works

• farin2002
• piegl1997

Frequently asked questions

Why are Bezier curves so widely used?

They are defined by a small set of control points that intuitively shape the curve, are easy and numerically stable to evaluate, and stay within the convex hull of their controls, which makes them predictable to edit.

What does the N in NURBS add over plain B-splines?

Non-uniform rational B-splines use weights and rational basis functions, which lets them represent circles, ellipses, and other conic sections exactly, something polynomial B-splines cannot do.