By B. Guenin

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A tour of subriemannian geometries, their geodesics and applications

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Extra info for Combinatorial Optimization [Lecture notes]

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72; Locher [1940], p. 207f. 11a A polarity is uniquely determined by a polar triangle and a determined by a three-plane and a pair pair of corresponding elements such of corresponding elements such that that they are not contained in the trian- they are not contained in the threegle. plane. 10. Polar triangles are classified according to the number of elliptic and hyperbolic involutions of conjugate polar points on their sides (or conjugate polar lines in their vertices) induced by the polarity.

3: (13) (Bb) There is exactly one invariant plane, say ε1. 4 that if there is exactly one invariant plane then there is exactly one invariant point. Therefore, the invariant plane ε1 can only be of class (Ib) or (IV). Ib) Let ε1 be of class (Ib) and let A2 be the invariant point and A3A4 the invariant line in ε1. 7) must pass through A2 without lying in ε1. This yields class (3b). IV) Let ε1 be of class (IV) and let A4 be the invariant point in ε1 and A3A4 the invariant line in ε1. There can be no additional invariant line without producing additional invariant points or planes, hence one has finally class (14).

223, p. 35–48. Reprint from: Mathematisch-Physikalische Korrepondenz 2005, 224: 35–48. 1 Curves and developables This section contains some remarks concerning topological properties of curves and developables which will be useful later on. Definition A point-arc is a set of points satisfying the same properties of order as a segment of a range of points: There are two senses in which a point can move through the arc; every countable subset of points of an arc can be put in natural order; the set of all points of an arc is continuous.