Abstract
A smooth curve on the plane can have a tangent which touches it tangentially at any number of points (Fig. 56), but this is not true for generic curves. By a small perturbation of the curve one can always achieve that no straight line will be tangential at more than two points.
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Bibliographical Comments
Y. L. Kergosien, R. Thorn: Sur les points paraboliques des surfaces. C. R. Acad. Sel., Paris, Ser. A 290 (1980), 705–710.
Y. L. Kergosien: La famille des projections orthogonales d’une surface et ses singularités. C. R. Acad. Sel., Paris, Ser. I 292 (1981), 929–932.)
O. A. Platonova: Singularities of projections of smooth surfaces. Usp. Mat. Nauk 39:1 (1984), 149–150 (English translation: Russ. Math. Surv. 39:1 (1984), 177-178).
O. A. Platonova: Singularities of the mutual disposition of a surface and a line. Usp. Mat. Nauk 36:1 (1981), 221–222 (English translation: Russ. Math. Surv. 36:1 (1981), 248-249).
O. A. Platonova: Projections of smooth surfaces (in Russian). Tr. Semin. Im. I. G. Petrovskogo 10 (1984), 135–149.
O. A. Platonova’s Thesis (in Russian). Moscow State University, 1981, 150 p.).
E. E. Landis: Tangential singularities. Funkts. Anal. Prilozh. 15:2 (1981), 36–49 (English translation: Funct. Anal. Appl. 15 (1981), 103-114).
E. E. Landis’s Thesis (in Russian). Moscow State University, 1983, 142 p.).
V. I. Arnol’d: Singularities of systems of rays. Usp. Mat. Nauk 38:2 (1983), 77–147 (English translation: Russ. Math. Surv. 38:2 (1983), 87-176).
O. P. Shcherbak: Projectively dual space curves and Legendre singularities (in Russian). Tr. Tbilis. Univ. 232-233, ser. Mat. Mekh. Astron. 13-14 (1982), 280–336.
The proofs of the theorems on the projections depend on :
V. I. Arnol’d: Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups, and singular projections of smooth surfaces. Usp. Mat. Nauk 34:2 (1979), 3–38 (English translation: Russ. Math. Surv. 34:2 (1979), 1-42).
Other approaches:
T. Banchoff, T. Gaffney, C. McCrory: Cusps of Gauss mappings. Res. Notes Math. 55, Pitman, Boston-London-Melbourne 1982.
A survey on singularities of projections:
V. V. Goryunov: Singularities of projections of full intersections, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. (Contemporary Problems of Mathematics) 22, Viniti, Moscow 1983, pp. 167–206 (English translation: J. Sov. Math. 27 (1984), 2785-2811).
See also:
V. V. Goryunov: Geometry of bifurcation diagrams of simple projections onto the line. Funkts. Anal. Prilozh. 15:2 (1981), 1–8 (English translation: Funct. Anal. Appl. 15 (1981), 77-82).
V. V. Goryunov: Projection of 0-dimensional complete intersections onto a line and the k(π,1) conjecture. Usp. Mat. Nauk 37:3 (1982), 179–180 (English translation: Russ. Math. Surv. 37:3 (1982), 206-208).
V. V. Goryunov: Bifurcation diagrams of some simple and quasihomogeneous singularities. Funkts. Anal. Prilozh. 17:2 (1983), 23–37 (English translation: Funct. Anal. Appl. 17 (1983), 97-108).
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© 1986 Springer-Verlag Berlin Heidelberg
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Arnold, V.I. (1986). Smooth Surfaces and Their Projections. In: Catastrophe Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-96937-9_12
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DOI: https://doi.org/10.1007/978-3-642-96937-9_12
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