Some Global Properties of Plane Curves
The rosettes.
On Poncelet's porism
We consider circular annuli with Poncelet's porism property. We prove two identities which imply Chapple's, Steiner's and other formulas. All porisms can be expressed in the form in which elliptic functions are not used.
Affine invariants of annuli
A family of regular annuli is considered. Affine invariants of annuli are introduced.
Circuminscribed polygons in a plane annulus
Each oval and a natural number n ≥ 3 generate an annulus which possesses the Poncelet's porism property. A necessary and sufficient condition of existence of circuminscribed n-gons in an annulus is given.
On rosettes and antipodal rosettes.
Affine invariants of annuli
A family of regular annuli is considered. Affine invariants of annuli are introduced.
On Poncelet’s porism
We consider circular annuli with Poncelet’s porism property. We prove two identities which imply Chapple’s, Steiner’s and other formulas. All porisms can be expressed in the form in which elliptic functions are not used.
Rotation indices related to Poncelet’s closure theorem
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 ngons for any n > k.
Rotation indices related to Poncelet’s closure theorem
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.
A measure of symmetry for ovals
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