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Biplanar Surfaces of Order Three

Published online by Cambridge University Press:  20 November 2018

Tibor Bisztriczky
Tibor Bisztriczky*
Affiliation:
University of Calgary, Calgary, Alberta
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0. Introduction. A surface of order three, F, in the real projective threespace P3 is met by every line, not in F, in at most three points. F is biplanar if it contains exactly one non-differentiable point v and the set of tangents of F at v is the union of two distinct planes, say τ1 and τ2. In the present paper, we classify and describe those biplanar F which contain the line τ1 ∩ τ2.

We describe a surface by determining the tangent plane sections of the surface at the differentiable points. This approach was introduced in [1] and it is based upon A. Marchaud's definition of “surfaces of order three” in [4].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 396 - 418
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Bisztriczky, T., Surfaces of order three with a peak, I. J. of Geometry, 11/1 (1978), 55–83.Google Scholar
2. Bisztriczky, T., Uniplanar surfaces of order three, to appear.Google Scholar
3. Haupt, O. and Runneth, H., Geometrische ordnungen (Springer-Verlag, Berlin, 1967).Google Scholar
4. Marchaud, A., Sur les surfaces du troisième ordre de la Géométrie finie, J. Math. Pur. Appl. 18 (1939), 323–362.Google Scholar
5. Marchaud, A., Sur les propriétés différentielles du premier ordre des surfaces simples de Jordan et quelques applications, Ann. Ec. Norm. Sup. 63 (1947), 81–108.Google Scholar
6. Scherk, P., tîber differenzierbare kurven und bôgen I. zum begriff der charakteristik, Casopis Pëst. Mat. 66 (1937), 165–171.Google Scholar