A NURBS ambit is authentic by its order, a set of abounding ascendancy points, and a bond vector. NURBS curves and surfaces are generalizations of both B-splines and Bézier curves and surfaces, the primary aberration getting the weighting of the ascendancy credibility which makes NURBS curves rational (non-rational B-splines are a appropriate case of rational B-splines). Whereas Bézier curves advance into alone one parametric direction, usually alleged s or u, NURBS surfaces advance into two parametric directions, alleged s and t or u and v.
By evaluating a Bézier or a NURBS ambit at assorted ethics of the parameter, the ambit can be represented in Cartesian two- or three-dimensional space. Likewise, by evaluating a NURBS apparent at assorted ethics of the two parameters, the apparent can be represented in Cartesian space.
NURBS curves and surfaces are advantageous for a amount of reasons:
They are invariant beneath affine2 as able-bodied as perspective3 transformations: operations like rotations and translations can be activated to NURBS curves and surfaces by applying them to their ascendancy points.
They action one accepted algebraic anatomy for both accepted analytic shapes (e.g., conics) and free-form shapes.
They accommodate the adaptability to architecture a ample array of shapes.
They abate the anamnesis burning if autumn shapes (compared to simpler methods).
They can be evaluated analytic bound by numerically abiding and authentic algorithms.
In the next sections, NURBS is discussed in one ambit (curves). It should be acclaimed that all of it can be ambiguous to two or even added dimensions.
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