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**A general 4th-order PDE method to generate Bézier surfaces from the boundary.**
*(English)*
Zbl 1083.65014

Summary: We present a method for generating Bézier surfaces from the boundary information based on a general 4th-order partial differential equations (PDE). This is a generalisation of our previous work on harmonic and biharmonic Bézier surfaces whereby we studied the Bézier solutions for Laplace and the standard biharmonic equation, respectively.

Here we study the Bézier solutions of the Euler-Lagrange equation associated with the most general quadratic functional. We show that there is a large class of fourth-order operators for which Bézier solutions exist and hence we show that such operators can be utilised to generate Bézier surfaces from the boundary information. As part of this work we present a general method for generating these Bézier surfaces. Furthermore, we show that some of the existing techniques for boundary based surface design, such as Coons patches and Bloor-Wilson PDE method, are indeed particular cases of the generalised framework we present here.

Here we study the Bézier solutions of the Euler-Lagrange equation associated with the most general quadratic functional. We show that there is a large class of fourth-order operators for which Bézier solutions exist and hence we show that such operators can be utilised to generate Bézier surfaces from the boundary information. As part of this work we present a general method for generating these Bézier surfaces. Furthermore, we show that some of the existing techniques for boundary based surface design, such as Coons patches and Bloor-Wilson PDE method, are indeed particular cases of the generalised framework we present here.

### MSC:

65D17 | Computer-aided design (modeling of curves and surfaces) |

### Keywords:

general biharmonic operator; PDE freeform surfaces; quadratic functionals; permanence principle; Coons patches; Laplace equation; biharmonic equation; graphical examples; Bézier surfaces; Euler-Lagrange equation
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\textit{J. Monterde} and \textit{H. Ugail}, Comput. Aided Geom. Des. 23, No. 2, 208--225 (2006; Zbl 1083.65014)

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### References:

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