[vtkusers] insphere radius of a tetrahedron

[Philipp.Batchelor]

I just did a google search, I found this site/code for different shape measures
of a tetrahedron, in particular ratio in/out
gives:
(obviously, Randy was talking about the regular tetrahedron)

/*--------------------------------------------------------------------------- 
 * FUNCTION: TetShapeMeasure_RadiusRatio 
 * 
 * Return the tetrahedron shape measure rho given by three times rho_in 
 * divided by rho_out where rho_in is the inradius and rho_out is the 
 * circumradius of the tetrahedron.  This tetrahedron shape measure is 
 * independant of the sign of the volume of the tetrahedron because the 
 * volume is squared. 
 * 
 *---------------------------------------------------------------------------*/ 
....
cf http://www.erc.msstate.edu/~rksingh/angle_cal.html
....
// The formula comes from: 
   // A. Liu and B. Joe 
   // ``Relation Between Tetraherdon Shape Measure'' 
   // BIT 34 (1994), 268--287. 
   return   ( 216 * volume * volume ) 
          / ( ( areaBCD + areaACD + areaABD + areaABC ) * 
              sqrt(   ( a + b + c ) * ( a + b - c ) 
                    * ( a + c - b ) * ( b + c - a ) ) ); 

Ph. Batchelor

Will Schroeder wrote:
> 
> The inscribed sphere radius is not directly related to the circumsphere. In
> the limiting case (where the tetra approaches flat) the insphere r ~ 0,
> while the circumsphere R -> infinity. Go to
> http://mathforum.org/dr.math/faq/formulas/faq.polyhedron.html#tetrahedron
> for the equations. If you code it up, please let's add it into vtkTetra as
> a companion to   "static double Circumsphere(double  p1[3], double p2[3],
> double p3[3], double p4[3], double center[3]);" The ratio of r/R is a good
> measure of the quality of the tetrahedron, and is used by finite element
> analysts.
> Will
> 
> At 06:50 AM 4/25/2002 -0500, Randy Heiland wrote:
> >If you have the outer, isn't the inner just 1/3 of it?
> >
> >Randy
> >
> >------------------------
> >Hi everybody,
> >
> >I would like to calculate the insphere radius of a tetrahedron...I used
> >Circumsphere() function for the outer radius...
> >
> >How can I compute easily the inner sphere radius of a tetrahedron in VTK or
> >mathematically using VTK....
> >
> >I would be greatful if you help me.....
> >
> >Thanks....
> >
> >Rupi
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-- 
----------------------------------
Dr Philippe Batchelor
King's College London  	
London	SE1 9RT

phone: 	+44 207 955 4223
fax:	+44 207 955 4532
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