NURBS (Non-Uniform Rational B-Splines) is a mathematical model commonly used in Computer-Aided Design (CAD) to represent complex curves and surfaces. NURBS provides a precise and flexible way to define and control the shape of freeform geometric entities, making it an essential tool for designing complex and organic shapes.

NURBS curves and surfaces are defined by a set of control points, weights, and knot vectors. The control points determine the general shape of the curve or surface, while the weights and knot vectors control the degree of influence of each control point and the smoothness of the resulting geometry.

One of the key advantages of NURBS is its ability to represent both analytical shapes (such as circles, ellipses, and cylinders) and freeform shapes (such as car bodies, aircraft wings, and organic forms) with a single, unified mathematical model. This makes NURBS a versatile and powerful tool for designing and modeling complex shapes in various industries, including automotive, aerospace, and industrial design.

NURBS curves and surfaces have several important properties that make them well-suited for CAD applications:

1. Precision: NURBS can represent complex shapes with high accuracy and precision, making them ideal for designing and manufacturing parts with tight tolerances.

2. Smoothness: NURBS curves and surfaces are inherently smooth, with continuity and differentiability properties that ensure a high degree of fairness and aesthetic quality.

3. Editability: NURBS provides a wide range of tools and techniques for editing and modifying curves and surfaces, such as control point manipulation, weight adjustment, and knot insertion/removal.

4. Interoperability: NURBS is a standard format supported by most CAD software packages, making it easy to exchange and share NURBS models between different applications and platforms.

In CAD software, NURBS is typically used in conjunction with other modeling tools and techniques, such as sketches, extrusions, and boolean operations. Designers can create NURBS curves and surfaces from scratch, or can fit NURBS entities to existing geometry using various curve and surface fitting algorithms.

Most CAD software packages provide a range of tools and options for creating and editing NURBS geometry, including control point editing, weight manipulation, and curve and surface analysis tools. These tools allow designers to quickly and easily create and modify complex shapes, and to ensure that the resulting geometry meets the desired form, fit, and function requirements.