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Radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds

Shukla S. S., Yadav A.

Demonstratio Mathematica

2014

Vol. 47, nr 4

994--1011

In this paper, we study radical transversal lightlike submanifolds and screen slant radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds giving some non-trivial examples of these submanifolds. Integrability conditions of distributions D and RadTM on radical transversal lightlike submanifolds and screen slant radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds, have been obtained. We also study totally contact umbilical radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds.

Od Merkatora do Space Oblique Mercator

Wieczorek M., Zalewski W.

Polski Przegląd Kartograficzny

2005

T. 37, nr 3

196-212

Tematem artykułu jest historia teorii wiernokątnych odwzorowań walcowych. Zaprezentowano w nim znane odwzorowanie Merkatora oraz jego liczne modyfikacje, których autorami byli kolejno J.H. Lambert, J.L. Lagrange, J.C.F. Gauss, J.H. Kruger, M. Hotine oraz J.P. Snyder. W podsumowaniu dano tabelaryczne zestawienie nazw dotyczących tego typu odwzorowań spotykanych w polskiej i anglojęzycznej literaturze, będące próbą uporządkowania tej mocno zróżnicowanej terminologii.

Encountered problems with the naming of projections of Mercator, Gauss, Gauss-Kruger and UTM led the authors to this attempt to systematize the terminology in the field. Gerard Kremer (Mercator) - the founder of modern cartography, who is most famous for his 1569 map of the world, was the first to apply the conformal cylindrical projection in normal aspect. Now it is referred to as Mercator projection, although Ch'ien Lo-Chih (940), Erhard Etzlaub (1511) and Edward Wright (1599) are also sometimes considered to be its authors. Henry Bond (1645) and James Gregory (1668) worked on the mathematical formula of this projection. The theory of conformal projections of a sphere onto a plane intrigued not only cartographers, but also scientists of other disciplines. In 1772 German mathematician J.H. Lambert invented differential equation, from which he developed the formula for Mercator projection in transverse aspect. Further research on conformity were conducted by French mathematician and astronomer J.L. Lagrange. In 1779 he generally solved the problem of conformal projection of an oblate surface onto a plane. He analysed a specific case of a projection of an oblate ellipsoid flattened at the poles onto a sphere, which later became known as Lagrange conformal projection. In literature it is also referred to as Mollweide's conformal projection of an ellipsoid onto a sphere. Also C. Gauss researched this field. In 1825 he elaborated a differential equation for a conformal projection of any two surfaces onto one another. He was looking for a solution, which would make it possible to relate spherical image to the projected area better 4han in the case of Lagrange's projection. The results he presented in detail for oblate ellipsoid. And it is this particular case, when an oblate ellipsoid is projected onto a sphere, which is referred to as Gauss projection. It is also referred to as a conformal transformation of a sphere on a side of a transverse cylinder, or a conformal transformation of a sphere onto a tube in transverse aspect, known as Lambert-Gauss or Mercator-Gauss projection. C. Gauss also authored a two stage projection used for triangulation, first applied in the area of Hannover (Hannover coordinates). L. Krtiger retrieved it from manuscripts and developed further; so that now it is known as Gauss-Kruger projection. It is a transverse tangent conformal cylindrical projection of an ellipsoid with the scale reduction factor along the central meridian being 1.0. In the late 1940s a variant of this projection (mo=0.9996) called UTM (Universal Transverse Mercator) was widely applied. Earlier research had been conducted on possible applications of two diagonal variants of conformal projections (M. Rosemund, M. Hotine). An ellipsoid version of diagonal Mercator projection which is scale conformal along the central meridian found wider use. It was authored by Hotine, and known as modified diagonal orthomorphic projection. It was used for presentation of images from the first series of Landsat satellites. However a tangent conformal cylindrical diagonal transverse ellipsoid projection, scale conformal along the satellite's path (mo=1.0) proved more practical for such applications. Its mathematical formula was developed by J.P.Snyder in 1981. It is usually referred to as SOM (Space Oblique Mercator) projection.