Dynamical Systems of Simplices in Dimension Two or Three
Identifieur interne : 000690 ( France/Analysis ); précédent : 000689; suivant : 000691Dynamical Systems of Simplices in Dimension Two or Three
Auteurs : Gérald Bourgeois ; Sébastien Orange [France]Source :
- Lecture Notes in Computer Science [ 0302-9743 ] ; 2011.
Abstract
Abstract: Let T 0 = (A 0 B 0 C 0 D 0) be a tetrahedron, G 0 be its centroid and S be its circumsphere. Let (A 1,B 1,C 1,D 1) be the points where S intersects the lines (G 0 A 0,G 0 B 0,G 0 C 0,G 0 D 0) and T 1 be the tetrahedron (A 1 B 1 C 1 D 1). By iterating this construction, a discrete dynamical system of tetrahedra (T i ) is built. The even and odd subsequences of (T i ) converge to two isosceles tetrahedra with at least a geometric speed. Moreover, we give an explicit expression of the lengths of the edges of the limit. We study the similar problem where T 0 is a planar cyclic quadrilateral. Then (T i ) converges to a rectangle with at least geometric speed. Finally, we consider the case where T 0 is a triangle. Then the even and odd subsequences of (T i ) converge to two equilateral triangles with at least a quadratic speed. The proofs are largely algebraic and use Gröbner bases computations.
Url:
DOI: 10.1007/978-3-642-21046-4_1
Affiliations:
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