Geodesic
Generating spirals with predefined boundary conditions
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 11, Tome 372 (2009), pp. 62-81 
A. I. Kurnosenko. Generating spirals with predefined boundary conditions. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 11, Tome 372 (2009), pp. 62-81. http://geodesic.mathdoc.fr/item/ZNSL_2009_372_a5/
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     author = {A. I. Kurnosenko},
     title = {Generating spirals with predefined boundary conditions},
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     year = {2009},
     volume = {372},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_372_a5/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Spirality, considered as monotonicity of curvature, is preserved under inversions. This property is used to construct a spiral transition curve with predefined curvature elements at the end points. These boundary conditions define two invariant values: Coxeter's inversive distance and the width of the lense. To solve the problem, it is sufficient to realize corresponding values on two curvature elements of any known spiral. The rest is achieved by inversion. In particular, any boundary conditions, compatible with spirality, can be satisfied by inverting an arc of logarithmic spiral. Bibl. – 9 titles.

[1] A. I. Kurnosenko, “Obschie svoistva ploskikh spiralnykh krivykh”, Zap. nauchn. semin. POMI, 353, 2008, 93–115 | MR

[2] D. S. Meek, D. J. Walton, “Planar spirals that match $G^2$ Hermite data”, Comp. Aided Geom. Design, 15 (1998), 103–126 | DOI | MR | Zbl

[3] J. M. Ali, R. M. Tookey, J. V. Ball, A. A. Ball, “The generalised Cornu spiral and its application to span generation”, J. Comp. Appl. Math., 102 (1999), 37–47 | DOI | MR | Zbl

[4] D. A. Dietz, B. Piper, “Interpolation with cubic spirals”, Comp. Aided Geom. Design, 21 (2004), 165–180 | DOI | MR | Zbl

[5] A. I. Kurnosenko, “Korotkie spirali”, Zap. nauch. semin. POMI, 372, 2009, 34–43 | MR

[6] A. I. Kurnosenko, “Dlinnye spirali”, Zap. nauch. semin. POMI, 372, 2009, 44–52 | MR

[7] A. Ruinskii, “Inversii ravnostoronnei giperboly”, Matem. prosv. Cer. 3, 2000, no. 4, 120–126

[8] A. I. Kurnosenko, “Inversnyi invariant pary okruzhnostei”, Zap. nauchn. semin. POMI, 261, 1999, 167–186 | MR | Zbl

[9] A. I. Markushevich, L. A. Markushevich, Vvedenie v teoriyu analiticheskikh funktsii, Prosveschenie, M., 1977 | Zbl