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NURBS, an acronym for Non-Uniform Rational B-Splines, are mathematical representations used in computer graphics and 3D modeling to describe smooth curves and surfaces. NURBS provide a flexible and precise way of defining complex shapes with curves that flow smoothly between control points. These curves and surfaces are widely used in various industries, including automotive design, aerospace engineering, and industrial design.

At its core, NURBS is a mathematical algorithm that defines curves and surfaces by blending together multiple B-spline basis functions. B-splines are piecewise-defined polynomial functions that control the shape of the curve or surface. The "non-uniform" aspect of NURBS means that the spacing between control points can be varied, allowing for more control over the shape and curvature of the object being modeled.

The "rational" part of NURBS refers to the use of homogeneous coordinates. Each control point in NURBS has an associated weight that affects its influence on the shape of the curve or surface. These weights introduce a scaling factor, enabling precise control over the position and shape of the NURBS object. This weighting mechanism is particularly useful for creating smooth transitions and complex shapes.

NURBS provide several advantages over other curve and surface representations. They allow for precise control of the shape, curvature, and continuity of the object being modeled. NURBS can accurately represent both simple and highly complex shapes with a relatively small number of control points. Additionally, NURBS can be easily manipulated and modified, making them ideal for interactive design processes. Their versatility and wide adoption in computer graphics make NURBS an essential tool for creating realistic and visually appealing models and animations.

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