Difference between revisions of "Catalan surface"
From Encyclopedia of Mathematics
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| − | A [[Ruled surface|ruled surface]] whose rectilinear generators are all parallel to the same plane. Its line of restriction (cf. [[Ruled surface|Ruled surface]]) is planar. The position vector of a Catalan surface is | + | {{TEX|done}} |
| + | A [[Ruled surface|ruled surface]] whose rectilinear generators are all parallel to the same plane. Its line of restriction (cf. [[Ruled surface|Ruled surface]]) is planar. The position vector of a Catalan surface is $r=\rho(u)+vl(u)$, where $l''(u)\neq0$, $(l,l',l'')=0$. If all the generators of a Catalan surface intersect the same straight line, then the surface is a [[Conoid|conoid]]. | ||
====References==== | ====References==== | ||
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| − | + | * {{Ref|1}} E. Catalan, "Mémoire sur les surfaces gauches à plan directeur" , Paris (1843) | |
| − | + | * {{Ref|a1}} W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German) {{ZBL|0366.53001}} | |
| − | + | * {{Ref|a2}} R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 31–35 {{ZBL|0425.53001}} | |
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Latest revision as of 07:34, 17 March 2023
A ruled surface whose rectilinear generators are all parallel to the same plane. Its line of restriction (cf. Ruled surface) is planar. The position vector of a Catalan surface is $r=\rho(u)+vl(u)$, where $l''(u)\neq0$, $(l,l',l'')=0$. If all the generators of a Catalan surface intersect the same straight line, then the surface is a conoid.
References
- [1] E. Catalan, "Mémoire sur les surfaces gauches à plan directeur" , Paris (1843)
- [a1] W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German) Zbl 0366.53001
- [a2] R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 31–35 Zbl 0425.53001
How to Cite This Entry:
Catalan surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Catalan_surface&oldid=18468
Catalan surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Catalan_surface&oldid=18468
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article