AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |
Back to Blog
![]() ![]() 30, 1944 MA THEMA TICS: KASNER AND DE CICCO 163 event, it is possible to find a function X = X(f) such that X satisfies the partial differential equation: Xxx + X1y = +(X). A family of curvesf(x, y) = const., is of the type X if and only iff satisfies a certain partial differential equation of the fourth order. In the present paper, we shall say that a family of curves is of the type X if it represents the scale curves of a conformal map of a surface z upon a plane 7r such that the gaussian curvature G is constant along each of the scale curves. Any family of 0o1 curves can represent the scale curves of a con- formal map upon a plane ir of any one of a certain class of surfaces 2. If 2 contains more than one such isothermal system (M) of geodesics, then it must contain o 2 such systems (M) and the surface z must be of constant gaussian curvature G.3 3. If, in addition, it be desired that the base family (M) consist wholly of geodesics upon I, then 2 must be applicable to a surface of revolution and the family (M) must correspond to meridians. The above result states that this can be accomplished if and only if (M) is an isothermal family of curves. Suppose it is desired to traverse a surface 2 always cutting a given base family (M) upon 2 at a constant angle (thus defining general loxodromes with respect to (M) on 2) in such a way that when 2 is mapped upon a plane 7r, the path described is represented in 7r by a straight line (for any angle). Kasner has, proved that the complete system of co2 isogonal trajec- tories of a given base family (M) of o 1curves upon a surface 2 is linear (in the analytic sense, as distinguished from the algebraic sense) if and only if (M) is an isothermal family.2 This theorem is important in cartography for the following reasons. In the non-conformal case, there is always a double infinity of scale curves. We shall be chiefly concerned with these scale curves. Therefore in the conformal case, there are ao 1 scale curves, defined by the finite equation u(x, y) = const. A scale curve on 2 or ir is the locus of a point along which the scale function a does not vary. The classic projections of Ptolemy, Mercator and Lambert appear in new light. We wish to present some theorems concerning the scale function a in any general conformal mapping of a surface 2 upon a plane ir, and to apply the results to the cartography of the sphere. It is independent of the direction if and only if the mapping is conformal. The scale function o- = ds/dS, which is the ratio of the corresponding elements of arc length in ir and in 2, depends, in general, not only upon the point but also upon the direction. Let a surface 2 be mapped in point-to-point fashion upon a plane 7r with cartesian coordinates (x, y). ![]() SCALE CUR VES IN CONFORMAL MAPS' BY EDWARD KASNER AND JoHN DE CiCco DEPARTMENTS OF MATHEMATICS, COLUMBIA UNIVERSITY, ILLINOIS INSTITUTE OF TECHNOLOGY Communicated 1. 162 MA THEMA TICS: KASNER AND DE CICCO PROC. ![]()
0 Comments
Read More
Leave a Reply. |