Biharmonic Bézier surface
Encyclopedia
A biharmonic Bézier surface is a smooth polynomial
surface which conforms to the biharmonic equation
and has the same formulations as a Bézier surface
. This formulation for Bézier surfaces was developed by Juan Monterde and Hassan Ugail
. In order to generate a biharmonic Bézier surface four boundary conditions 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 can be formulated. Biharmonic Bézier surfaces are related to minimal surface
s.
i.e. surfaces that minimise the area among all the surfaces with
prescribed boundary data.
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).
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
surface which conforms to the biharmonic equation
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...
and has the same formulations as a Bézier surface
Bézier surface
Bézier surfaces are a species of mathematical spline used in computer graphics, computer-aided design, and finite element modeling.As with the Bézier curve, a Bézier surface is defined by a set of control points...
. This formulation for Bézier surfaces was developed by Juan Monterde and Hassan Ugail
Hassan Ugail
Professor Hassan Ugail, PhD is a mathematician and a computer scientist at the School of Informatics; the University of Bradford. Professor Ugail is the first Maldivian to obtain a PhD in Mathematics. He is also the first Maldivian to receive professorship in the field of Science. Prof...
. In order to generate a biharmonic Bézier surface four boundary conditions 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 can be formulated. Biharmonic Bézier surfaces are related to minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
s.
i.e. surfaces that minimise the area among all the surfaces with
prescribed boundary data.
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).
Further reading
- Related publications (Hassan UgailHassan UgailProfessor Hassan Ugail, PhD is a mathematician and a computer scientist at the School of Informatics; the University of Bradford. Professor Ugail is the first Maldivian to obtain a PhD in Mathematics. He is also the first Maldivian to receive professorship in the field of Science. Prof...
's publications). - "Biharmonic Polynomial Surfaces for Boundary-Based Smooth Shape Design"