Geometry in History by S. G. Dani & Athanase Papadopoulos

Author:S. G. Dani & Athanase Papadopoulos
Language: eng
Format: epub
ISBN: 9783030136093
Publisher: Springer International Publishing

Consequently, they start from purely geometric definitions of tangent, tangent plane, osculating circle, radius of curvature etc. in the spirit of the book of Hilbert and Cohn-Vossen [69]. Their approach allows them to consider topological surfaces in 3-space with mild extra assumptions, and in particular convex surfaces, without further regularity conditions. Their starting point is a definition of tangent line, approximating circle, curvature and osculating plane, generalized from the setting of curves in space to that of sequences of points in space converging to a given point (Section 1 of [26]). At this point, the reader may stop and think about the meaning of these words, in the simplest situation, and without any differentiability assumption. Here is how Busemann and Feller proceed.

We start with a sequence of points {P n} in 3-space converging to a point P. If the lines joining P n to P converge to a line t, this limit is called the tangent line of {P n}. As in the treatise by Bonnesen and Fenchel, the pair (P, t) is said to be a line element. If the sequence {P n} has a tangent line t, then the circle in the plane (t, P n) through P n that is tangent to t is called the approximating circle of P n with respect to the line element (P, t). If ρ n denotes the radius of this circle and if the limit ρ of ρ n exists, then this limit is called the radius of curvature, and its inverse the curvature of the sequence of points {P n}. If the planes through P n and t converge to a plane E, then this plane E is called the osculating plane of {P n}.

Other metric notions associated with a sequence of points {P n} are introduced in the same paper, such as the foot of the perpendicular, normal segment and tangent segment from P n to P. Formulae for curvature radii, etc. are given in terms of limits of normal segments, tangent segments, etc. generalizing the classical formulae given in terms of derivatives.

From this, Busemann and Feller introduce as follows the notions of tangent, curvature and osculating plane for a space curve:

Let κ be a continuous space curve and P a point on κ. If for any sequence P n of points on κ converging to P the tangent and curvature exist and are independent of the choice of P n, then these tangent and curvature are defined to be the tangent and curvature of κ at P. The osculating plane of κ at P is defined similarly.

Several other metric definitions in the same spirit are given. In particular, if the tangent and the osculating plane exist but the curvature depends on the chosen sequence, then the upper limit of all the possible curvatures is called the upper curvature of κ. Choosing the sequence P n to be on the same side of P on the curve, one obtains notions of right-sided and left-sided tangent and curvature.

Busemann and

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