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# B‐splines

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Abstract B‐splines are a family of smooth curves that can be constructed to interpolate or approximate a set of control points. They are used extensively for curve and surface design in engineering and media applications. Their popularity comes from the fact that they offer a simple and intuitive means of adjusting the shape of a curve or surface interactively. Any point on a B‐spline curve or surface is defined as a local blend of the control points. The most widely used blending functions are cubic. Higher order blending makes the surface smoother and consequently less detailled. The normal formulation of the B‐spline blend is in a parametric space where the control points are equally distributed. Non uniform splines use an irregular distribution of the control points to create special effects, such as discontinuities in the curve or surface. Rational splines provide a further means of user interaction by weighting each point such that the curve is pulled more strongly towards the higher weights. Copyright © 2010 John Wiley & Sons, Inc. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Density Estimation

The four parts that make up the cubic B‐spline blending curve.

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The cubic B‐spline blending curve.

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Some effects created by using nonuniform B‐splines. As the parameter µ moves from 0.33 to 0.51, only P4 and P5 appear in the blend, and the locus follows the line joining P4 to P5. From 0.51 to 0.59, only P5 appears under the blending function, and so the locus remains at P5. From 0.59 to 0.63, the locus follows the line joining P5 to P6.

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An example of blending a two‐dimensional array of knots in parameter space. The locus for the point µ = 0.62, ν = 0.45 is being calculated.

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The recursively defined family of B‐spline blending functions.

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Example of using the cubic B‐spline blending curve.

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