Definitions

# PDE surface

PDE surfaces are used in geometric modelling and computer graphics 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.

PDE surfaces were first introduced into the area of geometric modelling and computer graphics 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{partial ^{2}}{ partial u^{2}} + a^{2}frac {partial^{2}}{partial v^{2}} right)^{2} X(u,v) = 0.

Here $X\left(u,v\right)$ is a function parameterised by the two parameters $u$ and $v$ such that $X\left(u,v\right) = \left(x\left(u,v\right), y\left(u,v\right), z\left(u,v\right)\right)$ where $x$, $y$ and $z$ are the usual cartesian coordinate space. The boundary conditions on the function $X\left(u,v\right)$ and its normal derivatives $partial\left\{X\right\}/partial$ 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 conditions. The parameter $a$ is a special design parameter which controls the relative smoothing of the surface in the $u$ and $v$ directions.

## Applications

PDE surfaces can be utilised in many application areas. These include computer-aided design, interactive design, parametric design, computer animation, computer-aided physical analysis and design optimisation.

## References

1. M.I.G. Bloor and M.J. Wilson, Generating Blend Surfaces using Partial Differential Equations, Computer Aided Design, 21(3), 165-171, (1989).
2. H. Ugail, M.I.G. Bloor, and M.J. Wilson, Techniques for Interactive Design Using the PDE Method, ACM Transactions on Graphics, 18(2), 195-212, (1999).
3. 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).
4. H. Du and H. Qin, Direct Manipulation and Interactive Sculpting of PDE surfaces, Computer Graphics Forum, 19(3), C261-C270, (2000).
5. H. Ugail, Spine Based Shape Parameterisations for PDE surfaces, Computing, 72, 195--204, (2004).
6. L. You, P. Comninos, J.J. Zhang, PDE Blending Surfaces with C2 Continuity, Computers and Graphics, 28(6), 895-906, (2004).