Timeline for Osculating conics in normal sections of a surface: how to generalize Euler's equation?
Current License: CC BY-SA 3.0
6 events
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| May 6, 2018 at 20:12 | comment | added | Robert Bryant | Ach! That was a typo. I meant to write '3-parameter family', not '6-parameter family' which is obviously wrong. Sorry about that. (I had cut out a remark about the space of quadrics that osculate to first order has dimension 6, and I garbled it.) | |
| May 6, 2018 at 20:02 | comment | added | Andreas Rüdinger | P.S. But I can be mistaken here. (Then the question would be whether the specific quadric out of the family of second order osculation quadrics which is equal to the surface built by the set of the osculating conics at a point p is wellknown/has been studied before.) P.S. By the way, I thought the family of quadrics having a contact of second order with an analytic surface at a given point is three-dimensional, cf. e.g. the beginning of projecteuclid.org/download/pdf_1/euclid.bams/1183506630. | |
| May 6, 2018 at 19:44 | comment | added | Andreas Rüdinger | @ Rober Bryant: Many thanks. I meant in the first question that the surface built by the set of the osculating conics at a point p (if we vary the normal section) does not seem to be any of the second order osculating quadrics at that point. | |
| May 6, 2018 at 18:28 | comment | added | Robert Bryant | There are some interesting questions here, but I think you need to clarify some things. For example, it's not clear what you mean by 'ocsculating quadric' at a point on a surface in $3$-space. As is well-known, while there is always a 6-parameter family of quadrics that osculate to second order to a (smooth) surface at a given point, there is generally no quadric that osculates to third order at that point, and when there is, it is not unique. (In fact, if a surface has nonvanishing Gauss curvature and at each point there is a quadric that osculates to 3rd order there, it must be a quadric.) | |
| Apr 29, 2018 at 21:18 | history | edited | Andreas Rüdinger | CC BY-SA 3.0 |
typo corrected
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| Apr 29, 2018 at 21:03 | history | asked | Andreas Rüdinger | CC BY-SA 3.0 |