IGA. Robin Bouclier

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and sixth control points have been moved from 1 to 0.5 , 2 and image respectively)

      1.2.2.4. Surfaces and volumes

      B-splines and NURBS are not restricted to parametric curves. Obviously, multivariate entities such as surfaces and volumes can also be described in the parametric form by using these technologies. The standard construction principle is identical to that of curves. In particular, a B-spline surface is a bivariate function of the form:

      [1.17] images

      where the bivariate B-spline basis functions image take degree p in the ξ1-direction and degree q in the ξ2-direction. They are obtained by the tensor product of their univariate counterparts:

      As for the univariate case, the construction of a NURBS surface requires the additional definition of one weight per control point. More specifically, a NURBS surface reads:

      [1.19] images

      where the bivariate piecewise rational basis functions image are computed from the bivariate B-spline functions image as:

      [1.20] images

      Let us eventually introduce the case of parametric volumes. These are trivariate functions obtained with an additional tensor product at the B-spline level. A NURBS volume can be expressed as follows:

      [1.21] images

      [1.22] images

Schematic illustration of example of a hollow cylinder described with a quadratic NURBS volume.

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