Ruletták a Minkowski síkon G.Horváth Ákos Department of Geometry, Mathematical Institute, Budapest University of Technology and Economics (BME) (közös munka Vitor Balestro-val és Horst Martini-vel) Geometria szeminárium, BME, Geometria tanszék Budapest, 2016 Október 25

Motion of rigid systems in the Euclidean plane (roulettes) -The plane S’ moving on the fixed plane S -Fixed frame with coordinates (x,y) -Moving frame with coordinates (u,v) -The parameter  is non-vanishing (the motion is nontranslative) -For every value of  there is a point of the moving plane for which the velocity vector vanishes. This point (the so-called instantaneous centre) defines in the moving plane the moving polode (centroid) and in the fixed plane the fixed polode (centroid), respectively. -The velocity vector at a point is orthogonal to the position vector from the instantaneous centre to the point, implying that the move is an instantaneous rotation about the instantaneous centre.

Main theorem of planar kinematics and the definition of roulettes The moving polode rolls without slipping (or without friction) on the fixed polode , and this is the only rolling process which corresponds to the given motion of the planes. Every non-translatory planar motion of a rigid mechanical system in the plane can be considered as the rolling process of a curve (rigidly connected with the system) on a fixed curve in the plane.

The second Euler –Savary equation Let r and r’ be the curvature radiuses of the fixed and moving polodes at its common point K, and denote by a, the length of the common velocity vector at K. It can be proved with some calculation the second Euler-Savary equation:

The first Euler –Savary equation (geometric form) The acceleration vector at P can be get as the derivate of the velocity vector at P. The calculation leads to a sum of two vectors, one of them is parallel to the normal PK with a fixed length dependent only on the angular acceleration of the plane around K. The getting other component goes through a fixed point L of the common normal of the polodes, the length of the vector LK is equal to the length of the common velocity vector at K (denoted by a). The Thalesian circle above the segment KL is the circle of inflection. If O and I are the curvature centre of the roulette at P and the second point of intersection of the line KP with the circle of inflection we can prove the equality:

A kerék mozgása I (az ívtávolságok rögzitettek)

Kerék mozgása II (az ívhossz a forgatás függvénye)

Konklúzió szög ívhossz kapcsolat nem működik jól, ha a mozgás paramétere az ívhossz a fix görbén, akkor nem periodikus görbét kapunk, ha a mozgó görbe elfordulásával paraméterezünk, akkor a görbe két pontja közötti ív hossza függ az ív mozgó görbén elfoglalt helyzetétől. Ötlet: Kapcsoljuk a szögmérést az ívhossz számításhoz.

Flexible motions in Minkowski plane Generalized angle measure:

Generalized rotation

Area based general rotation (Kepler’s law)

Nephroid (R=2r, epicyclois)

Ruletta definíció:

Busemann Fenchel, Alexandrov, Busemann, Jensen (1954)

Busemann görbület kifejezése az Euklideszi görbülettel V: r-dimenziós altér, U(V): V metszete a Minkowski egységgömbbel, s(V): az r-dimenziós egységgömb térfogatának az aránya az U(V) Euklideszi térfogatával.

Second Euler- Savary equation

First Euler-Savary equation

Bizonyítás:

A bizonyítás folytatása

Az inflexiós görbe

Thank you for your attention!