The Elementary Differential Geometry of Plane Curves
by Fowler
Format: Paperback
ISBN13: 9781152645196
Paperback|9781152645196
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Overview
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1920 Excerpt: ... has contact of the highest possible order with the given curve at P. This osculating curve will, in general, have contact of order n and, if (t)=g fi (), 2(0. -i. -2-.. -n-x, the conditions for contact of order n which determine the a'a are (4-432) +rt0) = 0, (r=l, 2, --X). It will be observed that (0)=M(, )=0, (r=l,2, ..., X), for all values of the parameters, since all members of the family have contact at least of order X. We can now argue exactly as before in 4-30 and arrive at the following theorem. Theorem 4-433. In general, the member of the family 4-431, which has osculating contact with a given curve at P, has contact of order n with the given curve, and is the limit of curves of the family which have contact of order X at P and pass through n--points on the given curve in the neighbourhood of P. As an example, the foregoing theorem may be used to identify the Newtonian circle of curvature with the osculating circle. The Newtonian circle of curvature ( 3'40) is the limit as Q-- P of the circle touching a given curve at P and cutting it at Q. The foregoing discussion shows that this limit is a circle with second order contact with the given curve at P, that is, as such a circle is unique, the osculating circle. 4 50. Similar problems in three dimensions. The treatment of the distance of a point Q from a surface /O. y, -) = 0, when Q is near an ordinary point P of the surface, is substantially the same as the treatment for plane curves. We obtain the following theorem. Theorem 4 51. If P(a, b, c) is an ordinary points on the surface f((r), yj z)=0, and a is the distance from a neighbouring point Q (x, y, z) to the surface, Or, curves. t That is, a point such that / ', fb', fc' are not simul.
| ISBN-13 | 9781152645196 |
|---|---|
| ISBN-10 | 1152645196 |
| Weight | 0.25 Pounds |
| Dimensions | 9.00 x 6.00 x 0.16 In |
| List Price | $14.14 |
| Format | Paperback |
|---|---|
| Pages | 66 pages |
| Publisher | |
| Published On | 2010-01-01 |
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