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.,

This parameter generalizes the identity

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

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

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

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_4
is root of a high-order polynomial, and

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 Resource. https://mathworld.wolfram.com/HarmonicParameter.html

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