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| '''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]]. | | Hi there, I am Andrew Berryhill. For a while I've been in Mississippi but now I'm contemplating other choices. Distributing manufacturing has been his profession for some time. To climb is something she would never give up.<br><br>Also visit my site tarot readings; [http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going www.zavodpm.ru], |
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| PDE surfaces were first introduced into the area of [[geometric modelling]] and [[computer graphics]] by two British mathematicians, Malcolm Bloor and Michael Wilson.
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| ==Technical details==
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| The PDE method involves generating a surface for some boundary by means of solving an [[Elliptic operator|elliptic partial differential equation]] of the form
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| :<math>
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| \left( \frac{\partial ^{2}}{ \partial
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| u^{2}} + a^{2}\frac {\partial^{2}}{\partial v^{2}} \right)^{2}
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| X(u,v) = 0.
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| </math>
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| Here <math>X(u,v) </math> is a function parameterised by the two [[parameter]]s <math> u </math> and <math> v </math> such that <math>X(u,v) = (x(u,v), y(u,v), z(u,v)) </math> where <math> x </math>, <math> y </math> and <math> z </math> are the usual [[Cartesian coordinate system|cartesian coordinate]] space. The boundary conditions on the function <math>X(u,v) </math> and its
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| normal derivatives <math> \partial{X}/\partial{{n}} </math>
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| are imposed at the edges of the surface patch.
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| 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
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| values. In this way a surface is obtained as a smooth transition between
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| the chosen set of [[boundary condition]]s. The parameter <math> a </math> is a special design parameter which controls the relative smoothing of the surface in the <math> u </math> and <math> v </math> directions.
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| ==Applications==
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| 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.
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| ==References==
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| #M.I.G. Bloor and M.J. Wilson, ''Generating Blend Surfaces using Partial Differential Equations'', Computer Aided Design, 21(3), 165-171, (1989).
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| #[[Hassan Ugail|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).
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| #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).
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| #H. Du and H. Qin, ''Direct Manipulation and Interactive Sculpting of PDE surfaces'', Computer Graphics Forum, 19(3), C261-C270, (2000).
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| #H. Ugail, ''Spine Based Shape Parameterisations for PDE surfaces'', Computing, 72, 195--204, (2004).
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| #L. You, P. Comninos, J.J. Zhang, ''PDE Blending Surfaces with C2 Continuity'', Computers and Graphics, 28(6), 895-906, (2004).
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| ==External links==
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| * [http://www.inf.brad.ac.uk/research/dve-sbd.php Simulation based design, DVE research (University of Bradford, UK)]. (A java applet demonstrating the properties of PDE surfaces)
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| * [http://www.maths.leeds.ac.uk/Applied/ Dept Applied Mathematics, University of Leeds] details on Bloor and Wilsons work.
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| [[Category:Surfaces]]
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| [[Category:Computer graphics]]
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| [[Category:Elliptic partial differential equations]]
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| [[Category:Computer-aided design]]
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| [[Category:Multivariate interpolation]]
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Hi there, I am Andrew Berryhill. For a while I've been in Mississippi but now I'm contemplating other choices. Distributing manufacturing has been his profession for some time. To climb is something she would never give up.
Also visit my site tarot readings; www.zavodpm.ru,