PDE surfaces are used in geometric modelling and computer graphics

Computer graphics

Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

 for creating smooth surfaces conforming to a given boundary configuration. PDE surfaces utilise partial differential equations to generate a surface which usually satisfy a mathematical boundary value problem

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 restraints, called the boundary conditions...

.

PDE surfaces were first introduced into the area of geometric modelling and computer graphics

Computer graphics

Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

 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

Elliptic operator

In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is...

 of the form


Here is a function parameterised by the two parameter

Parameter

Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

s and such that where , and are the usual cartesian coordinate

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

 space. The boundary conditions on the function and its
normal derivatives
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 conditions. The parameter is a special design parameter which controls the relative smoothing of the surface in the and directions.

Applications

PDE surfaces can be utilised in many application areas. These include computer-aided design

Computer-aided design

Computer-aided design , also known as computer-aided design and drafting , is the use of computer technology for the process of design and design-documentation. Computer Aided Drafting describes the process of drafting with a computer...

, interactive design, parametric design, computer animation

Computer animation

Computer animation is the process used for generating animated images by using computer graphics. The more general term computer generated imagery encompasses both static scenes and dynamic images, while computer animation only refers to moving images....

, computer-aided physical analysis and design optimisation.

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.