PDE surfaces are used in

geometric modelling __NOTOC__ Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes. The shapes studied in geometric modeling are mostly two- or three-dimensi ...

and

computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...

for creating smooth surfaces conforming to a given boundary configuration. PDE surfaces use

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

to generate a surface which usually satisfy a mathematical

boundary value problem In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...

. PDE surfaces were first introduced into the area of

and

by two British mathematicians, Malcolm Bloor and Michael Wilson.

Technical details

The PDE method involves generating a surface for some boundary by means of solving an elliptic partial differential equation of the form :

\left( \frac + a^\frac  \right)^
X(u,v) = 0.

Here

X(u,v)

is a function parameterised by the two

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

u

and

v

such that

X(u,v) = (x(u,v), y(u,v), z(u,v))

where

x

y

and

z

are the usual

cartesian coordinate A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

space. The boundary conditions on the function

X(u,v)

and its normal derivatives

\partial/\partial

are imposed at the edges of the surface patch. With the above formulation it is notable that the elliptic partial differential operator in the above PDE represents a smoothing process in which the value of the function at any point on the surface is, in some sense, a weighted average of the surrounding values. In this way, a surface is obtained as a smooth transition between the chosen set of

boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...

s. The parameter

a

is a special design parameter which controls the relative smoothing of the surface in the

u

and

v

directions. When

a=1

, the PDE is 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 ...

X_ + 2X_ + X_=0

. The biharmonic equation is the equation produced by applying the Euler-Lagrange equation to the simplified

thin plate energy functional The exact thin plate energy functional (TPEF) for a function f(x,y) is :\int_^ \int_^ (\kappa_1^2 + \kappa_2^2) \sqrt \,dx \,dy where \kappa_1 and \kappa_2 are the principal curvatures of the surface mapping f at the point (x,y). This is the sur ...

X_^2 + 2X_^2 + X_{vv}^2

. So solving the PDE with

a=1

is equivalent to minimizing the thin plate energy functional subject to the same boundary conditions.

Applications

PDE surfaces can be used in many application areas. These include computer-aided design, interactive design, parametric design,

computer animation Computer animation is the process used for digitally generating animations. The more general term computer-generated imagery (CGI) encompasses both static scenes ( still images) and dynamic images ( moving images), while computer animation re ...

, computer-aided physical analysis and design optimisation.

Related publications

#M.I.G. Bloor and M.J. Wilson, ''Generating Blend Surfaces using Partial Differential Equations'', Computer Aided Design, 21(3), 165-171, (1989). # H. Ugail, M.I.G. Bloor, and M.J. Wilson, ''Techniques for Interactive Design Using the PDE Method'',

ACM Transactions on Graphics ''ACM Transactions on Graphics'' (TOG) is a bimonthly peer-reviewed scientific journal that covers the field of computer graphics. It was established in 1982 and is published by the Association for Computing Machinery. TOG publishes two special iss ...

, 18(2), 195-212, (1999). #J. Huband, W. Li and R. Smith, ''An Explicit Representation of Bloor-Wilson PDE Surface Model by using Canonical Basis for Hermite Interpolation'', Mathematical Engineering in Industry, 7(4), 421-33 (1999). #H. Du and H. Qin, ''Direct Manipulation and Interactive Sculpting of PDE surfaces'', Computer Graphics Forum, 19(3), C261-C270, (2000). #H. Ugail, ''Spine Based Shape Parameterisations for PDE surfaces'', Computing, 72, 195--204, (2004). #L. You, P. Comninos, J.J. Zhang, ''PDE Blending Surfaces with C2 Continuity'', Computers and Graphics, 28(6), 895-906, (2004).

External links

Simulation based design, DVE research (University of Bradford, UK)
(A java applet demonstrating the properties of PDE surfaces)
Dept Applied Mathematics, University of Leeds
details on Bloor and Wilsons work. Surfaces Computer graphics Elliptic partial differential equations Computer-aided design Multivariate interpolation