Non-uniform rational basis spline (NURBS)
NURBS, an acronym for Non-Uniform Rational B-Splines, is a mathematical model commonly used in Computer-Aided Design (CAD), Computer Graphics, and Engineering for generating and representing curves and surfaces. Here are some key points about NURBS:
Definition and Properties
- Non-uniform: Refers to the ability to control the spacing of the knot vector, which influences the shape of the curve or surface.
- Rational: Allows for the representation of conic sections like circles and ellipses, which cannot be represented exactly by polynomial splines.
- B-Spline: Basis Spline, which is the underlying mathematical function used to construct the NURBS curve or surface.
History and Development
The development of NURBS can be traced back to the 1950s and 1960s when researchers like Pierre Bézier and Paul de Casteljau at Renault and Citroën respectively, began working on spline functions for automobile design. Here is a brief timeline:
- 1959: Bézier introduces Bézier curves at Renault.
- 1962: de Casteljau develops the de Casteljau algorithm at Citroën.
- 1970s: Development of B-Splines by Carl de Boor and others.
- 1980s: Introduction of NURBS by Les Piegl and Wayne Tiller in their work on geometric modeling, culminating in their book "The NURBS Book" in 1995.
Applications
NURBS have become fundamental in:
- Modeling free-form shapes in CAD systems.
- Animation and visual effects in film and video games.
- Creating precise geometric shapes in engineering, architecture, and manufacturing.
- Medical imaging for accurate representation of organic shapes.
Mathematical Formulation
The mathematical definition of a NURBS curve involves:
- A set of control points that define the shape of the curve.
- A knot vector, which determines the parametric placement of the curve.
- Weights associated with each control point, which can pull the curve closer to or further from the control point.
Advantages
- Ability to represent conic sections exactly.
- Local control of the shape through the modification of weights and control points.
- Can represent both analytic (straight lines and conic sections) and free-form shapes with high precision.
Disadvantages
- Complexity in computation, especially for real-time rendering.
- Not inherently suitable for all types of geometric operations like boolean operations or offsetting.
External Links
See Also