Polygons and Polynomials

Fisher, J. Chris; Ruoff, Dieter; Shilleto, J.

doi:10.1007/978-1-4612-5648-9_23

Cited by 16 publications

(6 citation statements)

References 7 publications

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“…are suitable scaling factors, converges to an affinely regular polygon. (This theorem, attributed in [9] to Neumann, was actually discovered earlier by Darboux [5]; it has been rediscovered many times (see p. 322 of [9]). ) Moreover, apart from its aesthetic appeal, there is an important application, since the result is a central ingredient in the solution of Hammer's X-ray problem [i0].…”

Section: Introductionmentioning

confidence: 86%

Subdivision algorithms and the kernel of a polyhedron

Gardner

Kallay

1992

Discrete Comput Geom

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Abstract. The kernel K of a convex polyhedron Po, as defined by L. Fejes T6th, is the limit of the sequence (P.), where P. is the convex hull of the midpoints of the edges of P._ ~. The boundary OK of the convex body K is investigated. It is shown that ~K contains no two-dimensional faces and that ~3K need not belong to C 2. The connection with similar algorithms from CAD (computer aided design) is explained and utilized.

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Section: Introductionmentioning

confidence: 86%

Subdivision algorithms and the kernel of a polyhedron

Gardner

Kallay

1992

Discrete Comput Geom

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“…The mathematicians involved did not dwell on the history (neither Meister nor Poinsot is mentioned in any of these papers), but just used Meister's way of looking at polygons. Early papers in this vein are those by Douglas (1940), Neumann (1941), Schoenberg (1950), and Berlekamp et al (1965), while some more recent ones are Fisher et al (1981), Neumann (1982), Martini (1996), Schuster (1998), and Shephard (2003.…”

Section: Consequences Of Poinsot's Definitionmentioning

confidence: 97%

Polygons: Meister was right and Poinsot was wrong but prevailed

Grünbaum

2011

Beitr Algebra Geom

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The definitions of the term "polygon" as given and used by Meister (1724Meister ( -1788 in 1770 and by Poinsot (1777Poinsot ( -1859 in 1810 are discussed. Since it is accepted that mathematicians are free to define concepts whichever way they like, the claim that one of them is right and the other wrong may appear strange. The following pages should justify the assertion of the title by pointing out some of the errors and inconsistencies in Poinsot's work, and-more importantly-show the undesirable and harmful consequences resulting from it.

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“…The results range from Napoleon-type theorems to the elucidations of limits of iterations of various averaging operations on polygons. Detailed information about such applications, which would not be possible under the Poinsot restriction, may be found in [2], [13], [14], [19], [39], [43], [46], [47], [48], and in their references.…”

Section: Polygonsmentioning

confidence: 99%

Are Your Polyhedra the Same as My Polyhedra?

Grünbaum

2003

Algorithms and Combinatorics

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Polygons and Polynomials

Cited by 16 publications

References 7 publications

Subdivision algorithms and the kernel of a polyhedron

Subdivision algorithms and the kernel of a polyhedron

Polygons: Meister was right and Poinsot was wrong but prevailed

Are Your Polyhedra the Same as My Polyhedra?

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