27.32.1.177: /* Hopf monads and bimonads */ Fixed typo, Fixed grammar

2014-12-15T13:07:58Z

Hopf monads and bimonads: Fixed typo, Fixed grammar

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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],

	~~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 operator\|elliptic partial differential equation]] of the form~~

	~~:<math>~~
	~~\left( \frac{\partial ^{2}}{ \partial~~
	~~u^{2}} + a^{2}\frac {\partial^{2}}{\partial v^{2}} \right)^{2}~~
	~~X(u,v) = 0~~.
	~~</math>~~

	~~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~~
	~~normal derivatives <math> \partial{X}~~/~~\partial{{n}} <~~/~~math>~~
	~~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]]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.~~

	~~==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==~~
	~~#M.I.G. Bloor and M.J. Wilson, ''Generating Blend Surfaces using Partial Differential Equations'', Computer Aided Design, 21(3), 165~~-~~171, (1989).~~
	~~#[[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).~~
	~~#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==~~
	* [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)~~
	* [http://www.maths.leeds.ac.uk/Applied/ Dept Applied Mathematics, ~~University of Leeds] details on Bloor and Wilsons work.~~

	~~[[Category:Surfaces]]~~
	~~[[Category:Computer graphics]]~~
	~~[[Category:Elliptic partial differential equations]]~~
	~~[[Category:Computer-aided design]]~~
	~~[[Category:Multivariate interpolation]]~~

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes

2009-12-16T22:57:18Z

remove Erik9bot category,outdated, tag and general fixes

New page

'''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 operator|elliptic partial differential equation]] of the form

:<math>
\left( \frac{\partial ^{2}}{ \partial
u^{2}} + a^{2}\frac {\partial^{2}}{\partial v^{2}} \right)^{2}
X(u,v) = 0.
</math>

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
normal derivatives <math> \partial{X}/\partial{{n}} </math>
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]]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.

==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==
#M.I.G. Bloor and M.J. Wilson, ''Generating Blend Surfaces using Partial Differential Equations'', Computer Aided Design, 21(3), 165-171, (1989).
#[[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).
#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==
* [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)
* [http://www.maths.leeds.ac.uk/Applied/ Dept Applied Mathematics, University of Leeds] details on Bloor and Wilsons work.

[[Category:Surfaces]]
[[Category:Computer graphics]]
[[Category:Elliptic partial differential equations]]
[[Category:Computer-aided design]]
[[Category:Multivariate interpolation]]

Monoidal monad - Revision history

27.32.1.177: /* Hopf monads and bimonads */ Fixed typo, Fixed grammar

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes