sphopt: Volumen eines sphärischen Dreiecks

$$\displaystyle \begin{array}{rl} V & = \frac{1}{6} \cdot A,B,C \\& = \frac{1}{6} \cdot \sin a \cdot \sin h_a \\& = \frac{1}{6} \cdot \sin a \cdot \sin b \cdot \sin \gamma \\& = \frac{1}{6} \cdot \sqrt{1-(\cos ^2 a + \cos ^2 b + \cos ^2 c )+ 2 \cdot \cos a \cdot \cos b \cdot \cos c } \\& = \frac{1}{6} \cdot \sqrt{ \sin s \cdot \sin (s-a) \cdot \sin (s-b) \cdot \sin (s-c) } \\& = \frac{ 1 - (\cos ^2 \alpha + \cos ^2 \beta + \cos ^2 \gamma ) - 2 \cdot \cos \alpha \cdot \cos \beta \cdot \cos \gamma } { 6 \cdot \sin \alpha \cdot \sin \beta \cdot \sin \gamma } \end{array}$

Conjecture 1: The function

$\begin{displaymath} V = V(x,y,z) = \frac{ 1 - (\cos ^2 x + \cos ^2 y + \cos ^2 z ) - 2 \cdot \cos x \cdot \cos y \cdot \cos z } { 6 \cdot \sin x \cdot \sin y \cdot \sin z } \end{displaymath}$

is strict concave for 0° < x,y,z < 180° and 0° < x+y+z < 540° .

Remarks:

The function $\sqrt{ 1-(x^2+y^2+z^2)+2 x y z }$ is strict concave for 0 ≤ x,y,z ≤ 1
Let γ = 2 β and α = β, i.e. let the triangle be isoscele. Then:

V = sin (2 α) · ( 1 - cot ² α · cot ² β ) / 6
It is well known that this function is strict concave for 0° < α,β < 180° and 90° < α + β .

The volume of the isolateral triangle
$\begin{displaymath} U(\epsilon) = \frac{1}{12} \cdot ( 3 - \cot^2 \frac{\epsilon+180^\circ}{6} ) \cdot \cot \frac{\epsilon+180^\circ}{6} \end{displaymath}$
is strict concave as function of its area ε, 0° < ε < 360°

Remarks:

If the conjecture is right, for each combinatorical type the polyhedron with largest volume is unique.

The number of possible combinatorical type of the best polyhedron had been reduced to two for N=10,11,13; the combinatorcal type of the best polyhedron is known for N=4...8,9,12,14.

To solve the problem for N=10 and N=14 it would be sufficient to prove conjecture 2:
Let E be a vertex of degree k. Let the triangles around E and the triangles edge-neighboured to them be given. Let R be a regular vertex of degree k and let the edge-neighboured triangles to the vertex's triangles be isoscele. If both structures have same area, the volume of the triangles around E is allways smaller or equal the volume of the triangles around R.

Literatur:

[1] Sigl, Rudolph
Ebene und Sphärische Trigonometrie
H.Wichmann-Verlag, Karlsruhe, 1977

[2] Wimmer, Lienhard:
Über das maximale Volumen der konvexen Hülle von Punkten auf der Einheitskugel
Dissertation, Salzburg, 1997

Lienhard Wimmer
2004-01-09

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