21 research outputs found
Comments on generalized Heron polynomials and Robbins’ conjectures
AbstractHeron’s formula for a triangle gives a polynomial for the square of its area in terms of the lengths of its three sides. There is a very similar formula, due to Brahmagupta, for the area of a cyclic quadrilateral in terms of the lengths of its four sides. (A polygon is cyclic if its vertices lie on a circle.) In both cases if A is the area of the polygon, (4A)2 is a polynomial function of the square in the lengths of its edges. David Robbins in [D.P. Robbins, Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12 (2) (1994) 223–236. MR 95g:51027; David P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly 102 (6) (1995) 523–530. MR 96k:51024] showed that for any cyclic polygon with n edges, (4A)2 satisfies a polynomial whose coefficients are themselves polynomials in the edge lengths, and he calculated this polynomial for n=5 and n=6. He conjectured the degree of this polynomial for all n, and recently Igor Pak and Maksym Fedorchuk [Maksym Fedorchuk, Igor Pak, Rigidity and polynomial invariants of convex polytopes, Duke Math. J. 129 (2) (2005) 371–404. MR 2006f:52015] have shown that this conjecture of Robbins is true. Robbins also conjectured that his polynomial is monic, and that was shown in [V.V. Varfolomeev, Inscribed polygons and Heron polynomials (Russian. Russian summary), Mat. Sb. 194 (3) (2003) 3–24. MR 2004d:51014]. A short independent proof will be shown here
The area of cyclic polygons: Recent progress on Robbins' Conjectures
In his works [R1,R2] David Robbins proposed several interrelated conjectures
on the area of the polygons inscribed in a circle as an algebraic function of
its sides. Most recently, these conjectures have been established in the course
of several independent investigations. In this note we give an informal outline
of these developments.Comment: To appear in Advances Applied Math. (special issue in memory of David
Robbins
On the Areas of Cyclic and Semicyclic Polygons
We investigate the ``generalized Heron polynomial'' that relates the squared
area of an n-gon inscribed in a circle to the squares of its side lengths. For
a (2m+1)-gon or (2m+2)-gon, we express it as the defining polynomial of a
certain variety derived from the variety of binary (2m-1)-forms having m-1
double roots. Thus we obtain explicit formulas for the areas of cyclic
heptagons and octagons, and illuminate some mysterious features of Robbins'
formulas for the areas of cyclic pentagons and hexagons. We also introduce a
companion family of polynomials that relate the squared area of an n-gon
inscribed in a circle, one of whose sides is a diameter, to the squared lengths
of the other sides. By similar algebraic techniques we obtain explicit formulas
for these polynomials for all n <= 7.Comment: 22 page
Volumes of polytopes in spaces of constant curvature
We overview the volume calculations for polyhedra in Euclidean, spherical and
hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary
tetrahedron in and . We also present some results, which provide a
solution for Seidel problem on the volume of non-Euclidean tetrahedron.
Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle,
horocycle or one branch of equidistant curve. This is a natural hyperbolic
analog of the cyclic quadrilateral in the Euclidean plane. We find a few
versions of the Brahmagupta formula for the area of such quadrilateral. We also
present a formula for the area of a hyperbolic trapezoid.Comment: 22 pages, 9 figures, 58 reference
Critical configurations of planar robot arms
It is known that a closed polygon P is a critical point of the oriented area
function if and only if P is a cyclic polygon, that is, can be inscribed in
a circle. Moreover, there is a short formula for the Morse index. Going further
in this direction, we extend these results to the case of open polygonal
chains, or robot arms. We introduce the notion of the oriented area for an open
polygonal chain, prove that critical points are exactly the cyclic
configurations with antipodal endpoints and derive a formula for the Morse
index of a critical configuration
Errata and Addenda to Mathematical Constants
We humbly and briefly offer corrections and supplements to Mathematical
Constants (2003) and Mathematical Constants II (2019), both published by
Cambridge University Press. Comments are always welcome.Comment: 162 page