Publication: El problema de Lagrange en cartografía
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2016-08-12
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Universidad Complutense de Madrid
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Abstract
Esta Tesis trata sobre las proyecciones de Lagrange, que son proyecciones conformes del elipsoide de revolución sobre el plano, que transforman los meridianos y paralelos en arcos circulares, y que analizamos y revisamos, con especial cuidado en las fuentes originales (Lambert, Lagrange, Bonnet, Chebyshev, etc.) Como herramienta auxiliar, introducimos en cartografía la función caracter ística de una proyección conforme: m = jf0(z)j{u100000}1, z = + iq, fundamentados en que las curvaturas de las imágenes de los meridianos y paralelos son, según J. L. Lagrange (1779): 1 = {u100000}m y 2 = mq, respectivamente (el subíndice indica derivada parcial). Parametrizamos el elipsoide mediante la longitud geodésica o geográ ca y la latitud isométrica q. Una proyección conforme es de Lagrange si y solo si m q = 0. En este trabajo resolvemos el sistema de ecuaciones: logm = 0, m q = 0. De este modo obtenemos a priori la función característica de las proyecciones de Lagrange y también realizamos una primera clasi cación: rectilíneas, formada por tres familias: Cilíndricas conformes, Cónicas y acimutales conformes y Pseudopolares, esta última es nueva en cartografía; y circulares, formada también por tres familias: De Lagrange-Lambert, Unipolares y Apolares, estas dos últimas nuevas en cartografía. En las rectilíneas todos los meridianos o todos los paralelos se transforman en rectas. En las circulares solo algunos meridianos o paralelos son rectilíneos...
This doctoral thesis is concerned with the Lagrange projections, that are conformal map projections of the oblate ellipsoid of revolution, which transform the meridians and parallels into circular arcs, and that we analyze and revise carefully from the original sources (Lambert, Lagrange, Bonnet, Chebyshev, etc.) As an auxiliar tool, we introduce, for rst time in cartography, the characteristic function of a conformal map projection: m = jf0(z)j{u100000}1, z = +iq, where is the longitude and q is the isometric latitude on the ellipsoid. We use the term characteristic owing to the curvatures of the images of the meridians and the parallels are given, according to J. L. Lagrange (1779), by: 1 = {u100000}m y 2 = mq, respectively (the subindex indicates partial derivative). A conformal map projection is of Lagrange type if and only if m q = 0. In this work we solve the system of equations: logm = 0, m q = 0. So we can a priori obtain the characteristic function of the Lagrange projections and we carry out a rst classi cation: rectilinears, which are composed by three families: conformal cylindricals, conformal conics and azimuthals, and pseudopolars projections, which is new in cartography; and circulars composed by three families as well: Lagrange-Lambert, unipolars and apolars projections, these two last new in cartography. In the rectilinear projections any of the functions i is identically zero whereas in the circulars ones these functions only vanish at some discrete points. From the characteristic function we state the fundamental property of dependence between the coordinates (X ; Y ) of the transformed points by the circular Lagrange projections and f 1 ; 2g: 2X = 1Y . Based on this property, together with the Cauchy-Riemann conditions, we get their equations...
This doctoral thesis is concerned with the Lagrange projections, that are conformal map projections of the oblate ellipsoid of revolution, which transform the meridians and parallels into circular arcs, and that we analyze and revise carefully from the original sources (Lambert, Lagrange, Bonnet, Chebyshev, etc.) As an auxiliar tool, we introduce, for rst time in cartography, the characteristic function of a conformal map projection: m = jf0(z)j{u100000}1, z = +iq, where is the longitude and q is the isometric latitude on the ellipsoid. We use the term characteristic owing to the curvatures of the images of the meridians and the parallels are given, according to J. L. Lagrange (1779), by: 1 = {u100000}m y 2 = mq, respectively (the subindex indicates partial derivative). A conformal map projection is of Lagrange type if and only if m q = 0. In this work we solve the system of equations: logm = 0, m q = 0. So we can a priori obtain the characteristic function of the Lagrange projections and we carry out a rst classi cation: rectilinears, which are composed by three families: conformal cylindricals, conformal conics and azimuthals, and pseudopolars projections, which is new in cartography; and circulars composed by three families as well: Lagrange-Lambert, unipolars and apolars projections, these two last new in cartography. In the rectilinear projections any of the functions i is identically zero whereas in the circulars ones these functions only vanish at some discrete points. From the characteristic function we state the fundamental property of dependence between the coordinates (X ; Y ) of the transformed points by the circular Lagrange projections and f 1 ; 2g: 2X = 1Y . Based on this property, together with the Cauchy-Riemann conditions, we get their equations...
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Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Sección Departamental de Astronomía y Geodesia, leída el 15-01-2016