PDE surface

From Wikipedia, the free encyclopedia

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.

1 Technical details
2 Applications
3 References
4 External links

[edit] Technical details

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

$\left( \frac{\partial ^{2}}{ \partial u^{2}} + a^{2}\frac {\partial^{2}}{\partial v^{2}} \right)^{2} \underline X(u,v) = 0.$

Here $\underline{X}(u,v)$ is a function parameterised by the two parameters $u$ and $v$ such that $\underline{X}(u,v) = (x(u,v), y(u,v), z(u,v))$ where $x$ , $y$ and $z$ are the usual cartesian coordinate space. The boundary conditions on the function $\underline{X}(u,v)$ and its normal derivatives $\frac{\partial{\underline{X}}} {\partial{{n}}}$ 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.

[edit] 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.

[edit] References

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, 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).