A biharmonic Bézier surface is a smooth

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

surface which conforms to the

biharmonic equation In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of t ...

and has the same formulations as a

Bézier surface Bézier surfaces are a type of mathematical spline used in computer graphics, computer-aided design, and finite element modeling. As with Bézier curves, a Bézier surface is defined by a set of control points. Similar to interpolation in many ...

. This formulation for Bézier surfaces was developed by Juan Monterde and Hassan Ugail. In order to generate a biharmonic Bézier surface four

boundary conditions In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...

defined by Bézier control points are usually required. It has been shown that given four boundary conditions a unique solution to the chosen general fourth order

elliptic partial differential equation In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which gene ...

can be formulated. Biharmonic Bézier surfaces are related to

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

s. i.e. surfaces that minimise the area among all the surfaces with prescribed boundary data.

External links

Related publications

1. J. Monterde and H. Ugail, On Harmonic and Biharmonic Bézier Surfaces, Computer Aided Geometric Design, 21(7), 697–715, (2004). 2. J. Monterde and H. Ugail, A general 4th-order PDE method to generate Bézier surfaces from the boundary, Computer Aided Geometric Design, 23(2), 208–225, (2006).

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