mainorthogonaltest.html

Delaunay三角形的网格剖分程序

<TITLE>COG 2.1: Orthogonal coordinates</TITLE>

<H1>How to Create Orthogonal Coordinates</H1>

<H2>Why orthogonal coordinates?</H2>

 <P>While arbitrary coordinates may be used successfully, for example,
for geometry description, and sometimes also for grid generation, the
Delaunay grid generation algorithm often creates non-beautiful grids
like the following:

 <P><IMG SRC="wz2dnonconform.gif">

 <P>if the coordinates are not orthogonal.  The situation becomes even
worse if we want to use anisotropic refinement.  In this case,
non-orthogonal coordinates are often unusable.

<H2>Definitions</H2>

 <P>We use <I>x,y,z</I> to denote the global coordinates (the image)
and <I>u,v,w</I> for the local coordinates.  The coordinate
transformation is defined by functions <I>x(u,v,w), y(u,v,w),
z(u,v,w)</I>.

 <UL>
 <LI> Coordinates are <B>orthogonal</B> if they preserve the angles
between the coordinate axes. So the angle between the curves
<I>u=const</I> resp. <I>v=const</I> is always 90 degree.
 <LI> Coordinates are <B>conformal</B> if they preserve other angles
too.  So the angle of the diagonal <I>u=v</I> with the axes
<I>u=const</I> resp. <I>v=const</I> should be 45 degrees too.
 <LI> Coordinates are <B>rectangular</B> if they are of the simple
type <I>x=x(u), y=y(v), z=z(w)</I>.
 </UL>

<H2>Basic rules for composition</H2>

 <P>

<H3>conformal(conformal) is conformal</H3>

 <P>That means, conformal maps form a group.

<H3>rectangular(rectangular) is rectangular</H3>

<p>That means, rectangular maps form a group too.

<H3>conformal(orthogonal) is orthogonal</H3>

<H3>orthogonal(rectangular) is orthogonal</H3>


<H3></H3>


 <P>