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Harmonic Parameter


The harmonic parameter of a polyhedron is the weighted mean of the distances d_i from a fixed interior point to the faces, where the weights are the areas A_i of the faces, i.e.,

 h=(sum_(i=1)^(n)A_id_i)/(sum_(i=1)^(n)A_i).
(1)

This parameter generalizes the identity

 (dV)/(dr)=S,
(2)

where V is the volume, r is the inradius, and S is the surface area, which is valid only for symmetrical solids, to

 (dV)/(dh)=S.
(3)

The harmonic parameter is independent of the choice of interior point (Fjelstad and Ginchev 2003). In addition, it can be defined not only for polyhedron, but any n-dimensional solids that have n-dimensional content V and (n-1)-dimensional content S.

Expressing the area A and perimeter p of a lamina in terms of h gives the identity

 (dA)/(dh)=p.
(4)

The following table summarizes the harmonic parameter for a few common laminas. Here, r is the inradius of a given lamina, and a and b are the side lengths of a rectangle.

Expressing V and S for a solid in terms of h then gives the identity

 h=(3V)/S.
(5)

The following table summarizes the harmonic parameter for a few common solids, where some of the more complicated values are given by the polynomial roots

 h_1=(256x^8-64512x^7-4257024x^6+34098944x^5+167319904x^4-806004288x^3-327993296x^2+816428176x+373301041)_6 
h_2=(31622400x^8-6045062400x^7+65176660800x^6-187266038400x^5+85961856960x^4+136958389920x^3+42447187200x^2+5102095680x+214358881)_5 
h_3=(3603193611264x^(12)-38078720649216x^(10)+49184509540608x^8-3562375387968x^6+308526620112x^4-3065029992x^2+38950081)_3
(6)

h_4 is root of a high-order polynomial, and

 h_5=1/(1202)[1/(10)(121461425+53168861sqrt(5)-4sqrt(30(28343974350325+12675597513679sqrt(5)))]^(1/2).
(7)

See also

Surface Area, Volume

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References

Fjelstad, P. and Ginchev, I. "Volume, Surface Area, and the Harmonic Mean." Math. Mag. 76, 126-129, 2003.

Referenced on Wolfram|Alpha

Harmonic Parameter

Cite this as:

Weisstein, Eric W. "Harmonic Parameter." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HarmonicParameter.html

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