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Integral formula for secantoptics and its application

Witold MozgawaMagdalena Skrzypiec — 2012

Annales UMCS, Mathematica

Some properties of secantoptics of ovals defined by Skrzypiec in 2008 were proved by Mozgawa and Skrzypiec in 2009. In this paper we generalize to this case results obtained by Cieślak, Miernowski and Mozgawa in 1996 and derive an integral formula for an annulus bounded by a given oval and its secantoptic. We describe the change of the area bounded by a secantoptic and find the differential equation for this function. We finish with some examples illustrating the above results.

Projective spaces of second order.

Andrzej MiernowskiWitold Mozgawa — 1997

Collectanea Mathematica

Grassmannians of higher order appeared for the first time in a paper of A. Szybiak in the context of the Cartan method of moving frame. In the present paper we consider a special case of higher order Grassmannian, the projective space of second order. We introduce the projective group of second order acting on this space, derive its Maurer-Cartan equations and show that our generalized projective space is a homogeneous space of this group.

Integral formula for secantoptics and its application

Witold MozgawaMagdalena Skrzypiec — 2012

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

Some properties of secantoptics of ovals defined by Skrzypiec in 2008 were proved by Mozgawa and Skrzypiec in 2009. In this paper we generalize to this case results obtained by Cieslak, Miernowski and Mozgawa in 1996 and derive an integral formula for an annulus bounded by a given oval and its secantoptic. We describe the change of the area bounded by a secantoptic and find the differential equation for this function. We finish with some examples illustrating the above results.

Rotation indices related to Poncelet’s closure theorem

Waldemar CieślakHorst MartiniWitold Mozgawa — 2014

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

Let CRCr denote an annulus formed by two non-concentric circles CR, Cr in the Euclidean plane. We prove that if Poncelet’s closure theorem holds for k-gons circuminscribed to CRCr, then there exist circles inside this annulus which satisfy Poncelet’s closure theorem together with Cr, with n- gons for any n > k.

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