<TITLE>COG 2.1: Cogeometry</TITLE> <P>The aim of this file is to give a description of the abstract,theoretical concept I have named <B>cogeometry</B>.  This may beuseful for a better understanding of the idea behind the softwarepackage <A HREF="cog.html">COG 1.0</A>.<H1>Cogeometry</H1><H2>Contravariant Geometry Description</H2> <P>A geometry description describes the subdivision of the space intodifferent regions, boundary faces between these regions, similarboundary edges and boundary vertices in 3D.  This can be easilygeneralized for arbitrary dimension where the geometry descriptiondescribes the subdivision of the space into boundary faces of eachcodimension between 0 (regions) and n (boundary vertices). <P>A <B>Cogeometry</B> is a new type of geometry description which ismuch easier to create and modify compared with a usual geometrydescription with boundary grids or boundary mappings.  The notion<B>cogeometry</B> for this type of geometry description was createdsimilar to the notion <B>cohomology</B> in algebraic topology. <P>The main difference to the usual geometry description is that itis <A HREF="cogdef.html#contravariance"> contravariant</A>.  Thisseems much more natural, because the object we want to describe - thegeometry - has also contravariant behaviour.  That means, we have anatural <B>pre-image</B> of a geometry, but there is no such natural<B>image</B> operation: Once we have a map X-&gt;Y and a geometry onY, we can define a geometry on X (the pre-image) by a simple rule: theregions in X are the (set-theoretical) pre-images of the regions in Y. <P>In the other direction, we have no such natural operation. Ageometry on is something meaningful, but there is no way to define animage of a geometry for maps which are not reversible. <P>There are a lot of <A HREF="cogoperations.html">interestingoperations</A> which allow to define and modify geometries which arede-facto only special cases of this <B>preimage-operation</B>.  Ageometry description which is contravariant can handle such operationsmuch easier compared with a non-contravariant geometry description.<H2>Definition</H2> <P>It is possible to give a nice, abstract definition of a cogeometryon the space <B>R<SUP>n</SUP></B> by n+1 functions. Each of thesefunctions <B>F<SUB>k</SUB></B> allows to define intersections ofk-segments with simplices of dimension k. <P>It works even for infinite-dimensional spaces, in this case weneed an infinite number of such functions. But there is an interestingsubclass of cogeometries - with <B>finite codimension</B> - which iscompletely defined also by a finite number of them. <P>In this sense, a cogeometry is a variant of a dualconstruction. Instead of defining a k-segment using a map from ak-simplex into the space, we use a function <B>F<SUB>k</SUB></B>defined on such maps. <P>In the n-dimensional case, we need only the first (n+1) ofthem.<H2>Implementation of the concept in COG 1.0</H2> <P>In <A HREF="cog.html">COG 1.0</A>, we plan to implement only thefirst four of these functions. For simplicity of implementation, eachof the functions <B>F<SUB>k</SUB></B> for k&gt;0 is splitted into twowith slightly different parameter list. <P>This leads to the C++ class definition you can find in <AHREF="cog.hxx"> cog.hxx</A>