Non-uniform rational B-spline (NURBS) is a mathematical formula to represent the curve and surfaces in to the CAD (Computer added Design) NURBS surfaces are very accurate and smooth analytical and freeform surfaces.
These surfaces are developed in around 1950 by those engineers who were designing the object like car skins, aircraft bodies, ship hull etc. previously these design were representing simple mathematical or physical shapes. These kinds of surfaces were required when complex shapes were developed by engineers.
There were two engineers name Pierre B?zier (Who were working with Renault in France) and Paul de Casteljau (Who were working with Citro?n in France) has developed this algorithms of uniform non-rational B-splines. Pierre B?zier used splines with the help of control points to control the surface these curves are named as B?zier splines and
de Casteljau has developed algorithms? to evaluate the parametric surfaces. In 1960 it was declared that non-uniform, rational B-splines are a representation of B?zier splines. This can be also called as uniform, non-rational B-splines.

In the starting of this phenomenon it was only used by the car companies but at last it becomes a part of general computer graphics package.
We can say that editing of NURBS curves and surfaces is highly intuitive and predictable. Control points are always either connected directly to the curve/surface, or act as if they were connected by a rubber band. Depending on the type of user interface, editing can be realized via an element?s control points, which are most obvious and common for B?zier curves, or via higher level tools such as spline modeling or hierarchical editing.
When we are developing a NURBS surface of and model like Car outer body it is usually made in combination of so many other NURBS surfaces called Patches. Fitting of these patches are expressed in the form of Geometric continuity
This geometric continuities are different level :-

Positional continuity (G0)

When two curves meet at point and give a sharp edge or corner

Tangential continuity (G1)

When end vectors of both the curves are at parallel position. Both curve should be match in tangential continue manner

Curvature continuity (G2)

When end vectors of the curves are of the same length and rate of length change. If it seems there is no change or they looks like a one then it shows G2 continuity. This can be visually recognized as ?perfectly smooth


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