Harmonic Parameter

Author

Eric W. Weisstein
February 27, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SolidGeometry/HarmonicParameter.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HarmonicParameter.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

HarmonicParameter[Polygon[m_List?MatrixQ],p_:{0,0}]:=Module[
{sides=Partition[m,2,1,1]},
Plus@@Abs[Norm[Subtract@@#]Distance[Point[p],Line[#]]&/@sides]/Plus@@(Norm/@(Subtract@@@sides))]/; Dimensions[m][[2]]==2

HarmonicParameter[poly_]:=Module[
{
a=PolygonArea[#[[1]]]&/@Flatten[poly],
d=Norm[Centroid[#]]&/@Flatten[poly]
},
a.d/Plus@@a
]

Laminas

Rectangle

Simplify[HarmonicParameter[Polygon[{{-a,-b},{-a,b},{a,b},{-a,b}}s /2]],s>0&&a>0&&b>0]

((2(a/2)b+2(b/2)a)/(2(a+b)))

Square

Simplify[HarmonicParameter[Polygon[{{-1,-1},{-1,1},{1,1},{-1,1}}s a/2]],s>0&&a>0]

((a b)/(a+b))/.b->a

((4a/2a)/(4a))

Triangle

p=FullSimplify[Incenter[Triangle[a,b,c]],TriangleQ[]][[1]]

FullSimplify[HarmonicParameter[Polygon@@Triangle[a,b,c],p],TriangleQ[]]

Simplify[%42/.Abs[x_]:>x]

FullSimplify[CircleRadius[Incircle[Triangle[a,b,c]]],TriangleQ[]]

h=FullSimplify[ExactTrilinears[Orthocenter].{a,b,c}] /(a+b+c)

Inradius[]

FullSimplify[h/Inradius[],TriangleQ[]]

Solids

Sort[{PolyhedronName[#],{v,s,h}=ToExact[#,MaxOrder->16,WorkingPrecision->500,CheckPrecision->600,Weight->10^4]&/@Through[{PolyhedronVolume,PolyhedronSurfaceArea,HarmonicParameter}[Polyhedron[#]]],ToExact[3v/s-h]}&/@Join[Archimedeans,Platonics]
]//ColumnForm