Harmonic Parameter

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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.,

[画像: h=(sum_(i=1)^(n)A_id_i)/(sum_(i=1)^(n)A_i). ]

(1)

This parameter generalizes the identity

[画像: (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

[画像: (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

[画像: (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.

lamina h

circle r

rectangle (ab)/(a+b)

square r

triangle r

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

[画像: 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)

solid h

cone (hr)/(sqrt(h^2+r^2))

cube 1/2

cuboctahedron 5/2sqrt(1/3(2-sqrt(3)))

cuboid (3abc)/(2(ab+ac+bc))

dodecahedron 1/2sqrt(1/(10)(25+11sqrt(5)))

great rhombicosidodecahedron h_1

great rhombicuboctahedron 1/2(4+3sqrt(2)-sqrt(3)-sqrt(6))