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Erik W Grafarend - One of the best experts on this subject based on the ideXlab platform.
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the optimal Mercator Projection and the optimal polycylindric Projection of conformal type case study indonesia
Journal of Geodesy, 1998Co-Authors: Erik W Grafarend, Rainer SyffusAbstract:As a conformal mapping of the sphere S 2R or of the ellipsoid of revolution E 2A,B the Mercator Projection maps the equator equidistantly while the transverse Mercator Projection maps the transverse metaequator, the meridian of reference, with equidistance. Accordingly, the Mercator Projection is very well suited to geographic regions which extend east-west along the equator; in contrast, the transverse Mercator Projection is appropriate for those regions which have a south-north extension. Like the optimal transverse Mercator Projection known as the Universal Transverse Mercator Projection (UTM), which maps the meridian of reference Λ0 with an optimal dilatation factor &ρcirc;=0.999 578 with respect to the World Geodetic Reference System WGS 84 and a strip [Λ0−ΛW,Λ0 + ΛE]×[ΦS,ΦN]= [−3.5∘,+3.5∘]×[−80∘,+84∘], we construct an optimal dilatation factor ρ for the optimal Mercator Projection, summarized as the Universal Mercator Projection (UM), and an optimal dilatation factor ρ0 for the optimal polycylindric Projection for various strip widths which maps parallel circles Φ0 equidistantly except for a dilatation factor ρ0, summarized as the Universal Polycylindric Projection (UPC). It turns out that the optimal dilatation factors are independent of the longitudinal extension of the strip and depend only on the latitude Φ0 of the parallel circle of reference and the southern and northern extension, namely the latitudes ΦS and ΦN, of the strip. For instance, for a strip [ΦS,ΦN]= [−1.5∘,+1.5∘] along the equator Φ0=0, the optimal Mercator Projection with respect to WGS 84 is characterized by an optimal dilatation factor &ρcirc;=0.999 887 (strip width 3∘). For other strip widths and different choices of the parallel circle of reference Φ0, precise optimal dilatation factors are given. Finally the UPC for the geographic region of Indonesia is presented as an example.
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The optimal Mercator Projection and the optimal polycylindric Projection of conformal type – case-study Indonesia
Journal of Geodesy, 1998Co-Authors: Erik W Grafarend, Rainer SyffusAbstract:As a conformal mapping of the sphere S 2R or of the ellipsoid of revolution E 2A,B the Mercator Projection maps the equator equidistantly while the transverse Mercator Projection maps the transverse metaequator, the meridian of reference, with equidistance. Accordingly, the Mercator Projection is very well suited to geographic regions which extend east-west along the equator; in contrast, the transverse Mercator Projection is appropriate for those regions which have a south-north extension. Like the optimal transverse Mercator Projection known as the Universal Transverse Mercator Projection (UTM), which maps the meridian of reference Λ0 with an optimal dilatation factor &ρcirc;=0.999 578 with respect to the World Geodetic Reference System WGS 84 and a strip [Λ0−ΛW,Λ0 + ΛE]×[ΦS,ΦN]= [−3.5∘,+3.5∘]×[−80∘,+84∘], we construct an optimal dilatation factor ρ for the optimal Mercator Projection, summarized as the Universal Mercator Projection (UM), and an optimal dilatation factor ρ0 for the optimal polycylindric Projection for various strip widths which maps parallel circles Φ0 equidistantly except for a dilatation factor ρ0, summarized as the Universal Polycylindric Projection (UPC). It turns out that the optimal dilatation factors are independent of the longitudinal extension of the strip and depend only on the latitude Φ0 of the parallel circle of reference and the southern and northern extension, namely the latitudes ΦS and ΦN, of the strip. For instance, for a strip [ΦS,ΦN]= [−1.5∘,+1.5∘] along the equator Φ0=0, the optimal Mercator Projection with respect to WGS 84 is characterized by an optimal dilatation factor &ρcirc;=0.999 887 (strip width 3∘). For other strip widths and different choices of the parallel circle of reference Φ0, precise optimal dilatation factors are given. Finally the UPC for the geographic region of Indonesia is presented as an example.
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the solution of the korn lichtenstein equations of conformal mapping the direct generation of ellipsoidal gaus kruger conformal coordinates or the transverse Mercator Projection
Journal of Geodesy, 1998Co-Authors: Erik W Grafarend, Rainer SyffusAbstract:The differential equations which generate a general conformal mapping of a two-dimensional Riemann manifold found by Korn and Lichtenstein are reviewed. The Korn–Lichtenstein equations subject to the integrability conditions of type vectorial Laplace–Beltrami equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and ellipsoidal latitude. The related coefficient constraints are collected in two corollaries. We present the constraints to the general solution of the Korn–Lichtenstein equations which directly generates Gaus–Kruger conformal coordinates as well as the Universal Transverse Mercator Projection (UTM) avoiding any intermediate isometric coordinate representation. Namely, the equidistant mapping of a meridian of reference generates the constraints in question. Finally, the detailed computation of the solution is given in terms of bivariate polynomials up to degree five with coefficients listed in closed form.
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The solution of the Korn–Lichtenstein equations of conformal mapping: the direct generation of ellipsoidal Gauß–Krüger conformal coordinates or the Transverse Mercator Projection
Journal of Geodesy, 1998Co-Authors: Erik W Grafarend, Rainer SyffusAbstract:The differential equations which generate a general conformal mapping of a two-dimensional Riemann manifold found by Korn and Lichtenstein are reviewed. The Korn–Lichtenstein equations subject to the integrability conditions of type vectorial Laplace–Beltrami equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and ellipsoidal latitude. The related coefficient constraints are collected in two corollaries. We present the constraints to the general solution of the Korn–Lichtenstein equations which directly generates Gaus–Kruger conformal coordinates as well as the Universal Transverse Mercator Projection (UTM) avoiding any intermediate isometric coordinate representation. Namely, the equidistant mapping of a meridian of reference generates the constraints in question. Finally, the detailed computation of the solution is given in terms of bivariate polynomials up to degree five with coefficients listed in closed form.
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the oblique Mercator Projection of the ellipsoid of revolution ie a 2 b
Journal of Geodesy, 1995Co-Authors: J. Engels, Erik W GrafarendAbstract:While the standardMercator Projection / transverse Mercator projecton maps the equator / the transverse metaequator equivalent to the meridian of referenceequidistantly, theoblique Mercator Projection aims at aconformal mapping of the ellipsoid of revolution constraint to anequidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive themapping equations of theoblique Mercator Projection being characterized to beconformal andequidistant on the oblique metaequator extending results ofM. Hotine (1946, 1947).
J. Engels - One of the best experts on this subject based on the ideXlab platform.
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the hotine rectified skew orthomorphic Projection oblique Mercator Projection revisited
2001Co-Authors: E W Grafarend, J. EngelsAbstract:Based on the contribution ”The oblique Mercator Projection on the ellipsoid of revolution \( _{n,m}^{(0)} \) ” , Journal of Geodesy 70 (1995) 38-50, also known as the Hotine rectified skew orthomorphic Projection, an alternative representation of the conformai mapping of Hotine type is presented.
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The oblique Mercator Projection of the ellipsoid of revolution IE _a ^2 ,_b
Journal of Geodesy, 1995Co-Authors: J. Engels, E. GrafarendAbstract:While the standard Mercator Projection / transverse Mercator projecton maps the equator / the transverse metaequator equivalent to the meridian of reference equidistantly , the oblique Mercator Projection aims at a conformal mapping of the ellipsoid of revolution constraint to an equidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive the mapping equations of the oblique Mercator Projection being characterized to be conformal and equidistant on the oblique metaequator extending results of M. Hotine (1946, 1947).
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the oblique Mercator Projection of the ellipsoid of revolution ie a 2 b
Journal of Geodesy, 1995Co-Authors: J. Engels, Erik W GrafarendAbstract:While the standardMercator Projection / transverse Mercator projecton maps the equator / the transverse metaequator equivalent to the meridian of referenceequidistantly, theoblique Mercator Projection aims at aconformal mapping of the ellipsoid of revolution constraint to anequidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive themapping equations of theoblique Mercator Projection being characterized to beconformal andequidistant on the oblique metaequator extending results ofM. Hotine (1946, 1947).
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The Hotine oblique Mercator Projection of \mathbb{E}_{{a,b}}^{2}
International Association of Geodesy Symposia, 1995Co-Authors: J. Engels, E. GrafarendAbstract:While the standard Mercator Projection/transverse Mercator Projection maps the equator/the transverse metaequator equivalent to the meridian of reference equidistantly, the oblique Mercator Projection aims at a conformal mapping of the ellipsoid of revolution constraint to an equidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive the mapping equations of the oblique Mercator Projection being characterized to be conformal and equidistant on the oblique metaequator extending results of M. Hotine (1946, 1947). The paper is submitted for publication in manuscripta geodaetica.
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the hotine oblique Mercator Projection of mathbb e _ a b 2
1995Co-Authors: J. Engels, E. GrafarendAbstract:While the standard Mercator Projection/transverse Mercator Projection maps the equator/the transverse metaequator equivalent to the meridian of reference equidistantly, the oblique Mercator Projection aims at a conformal mapping of the ellipsoid of revolution constraint to an equidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive the mapping equations of the oblique Mercator Projection being characterized to be conformal and equidistant on the oblique metaequator extending results of M. Hotine (1946, 1947). The paper is submitted for publication in manuscripta geodaetica.
E. Grafarend - One of the best experts on this subject based on the ideXlab platform.
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The oblique Mercator Projection of the ellipsoid of revolution IE _a ^2 ,_b
Journal of Geodesy, 1995Co-Authors: J. Engels, E. GrafarendAbstract:While the standard Mercator Projection / transverse Mercator projecton maps the equator / the transverse metaequator equivalent to the meridian of reference equidistantly , the oblique Mercator Projection aims at a conformal mapping of the ellipsoid of revolution constraint to an equidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive the mapping equations of the oblique Mercator Projection being characterized to be conformal and equidistant on the oblique metaequator extending results of M. Hotine (1946, 1947).
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The Hotine oblique Mercator Projection of \mathbb{E}_{{a,b}}^{2}
International Association of Geodesy Symposia, 1995Co-Authors: J. Engels, E. GrafarendAbstract:While the standard Mercator Projection/transverse Mercator Projection maps the equator/the transverse metaequator equivalent to the meridian of reference equidistantly, the oblique Mercator Projection aims at a conformal mapping of the ellipsoid of revolution constraint to an equidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive the mapping equations of the oblique Mercator Projection being characterized to be conformal and equidistant on the oblique metaequator extending results of M. Hotine (1946, 1947). The paper is submitted for publication in manuscripta geodaetica.
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the hotine oblique Mercator Projection of mathbb e _ a b 2
1995Co-Authors: J. Engels, E. GrafarendAbstract:While the standard Mercator Projection/transverse Mercator Projection maps the equator/the transverse metaequator equivalent to the meridian of reference equidistantly, the oblique Mercator Projection aims at a conformal mapping of the ellipsoid of revolution constraint to an equidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive the mapping equations of the oblique Mercator Projection being characterized to be conformal and equidistant on the oblique metaequator extending results of M. Hotine (1946, 1947). The paper is submitted for publication in manuscripta geodaetica.
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the optimal universal transverse Mercator Projection
1995Co-Authors: E. GrafarendAbstract:The Korn-Lichtenstein partial differential equations subject to an integrability condition of Laplace-Beltrami type which govern conformal mapping are reviewed. They are completed by an extensive review of deformation measures (Cauchy-Green deformation tensor, Euler-Lagrange deformation tensor, simultaneous diagonalization of a pair of symmetric matrices) extending the Tissot deformation portrait. W.r.t. one system of isometric parameters which cover a surface (oriented two-dimensional Riemann manifold) the d’Alembert-Euler equations (Cauchy-Riemann equations) subject to an integrability condition of Laplace-Beltrami type are solved in real analysis by various systems of functions (fundamental solution: 2d-polynomial, separation of variables) plus a properly chosen boundary value problem, namely the equidistant mapping of one parameter line. Finally the optimal transverse Mercator Projection is outlined by solving a boundary value problem of the d’Alembert-Euler equations (Cauchy-Riemann equations) of a biaxial ellipsoid (ellipsoid of revolution) where a dilatation factor of a central meridian is to be determined. It is proven that for a non-symmetric and a symmetric UTM strip the total areal distortion approaches zero once the total departure from an isometry is minimized. According to the “Geodetic Reference System 1980” for a strip [-l E ,+l E ] × [B S ,B N ] = [-3.5°,+3.5°] × [80°S,84°N] - the standard UTM strip - an optimal dilatation factor is p = 0.999,578, while for a strip [-2°, +2°] × [80°S, 84°N] - the standard Gauβ-Kruger strip - an optimal dilatation factor is p = 0.999,864. The paper is being published in manuscripta geodaetica.
Rainer Syffus - One of the best experts on this subject based on the ideXlab platform.
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the optimal Mercator Projection and the optimal polycylindric Projection of conformal type case study indonesia
Journal of Geodesy, 1998Co-Authors: Erik W Grafarend, Rainer SyffusAbstract:As a conformal mapping of the sphere S 2R or of the ellipsoid of revolution E 2A,B the Mercator Projection maps the equator equidistantly while the transverse Mercator Projection maps the transverse metaequator, the meridian of reference, with equidistance. Accordingly, the Mercator Projection is very well suited to geographic regions which extend east-west along the equator; in contrast, the transverse Mercator Projection is appropriate for those regions which have a south-north extension. Like the optimal transverse Mercator Projection known as the Universal Transverse Mercator Projection (UTM), which maps the meridian of reference Λ0 with an optimal dilatation factor &ρcirc;=0.999 578 with respect to the World Geodetic Reference System WGS 84 and a strip [Λ0−ΛW,Λ0 + ΛE]×[ΦS,ΦN]= [−3.5∘,+3.5∘]×[−80∘,+84∘], we construct an optimal dilatation factor ρ for the optimal Mercator Projection, summarized as the Universal Mercator Projection (UM), and an optimal dilatation factor ρ0 for the optimal polycylindric Projection for various strip widths which maps parallel circles Φ0 equidistantly except for a dilatation factor ρ0, summarized as the Universal Polycylindric Projection (UPC). It turns out that the optimal dilatation factors are independent of the longitudinal extension of the strip and depend only on the latitude Φ0 of the parallel circle of reference and the southern and northern extension, namely the latitudes ΦS and ΦN, of the strip. For instance, for a strip [ΦS,ΦN]= [−1.5∘,+1.5∘] along the equator Φ0=0, the optimal Mercator Projection with respect to WGS 84 is characterized by an optimal dilatation factor &ρcirc;=0.999 887 (strip width 3∘). For other strip widths and different choices of the parallel circle of reference Φ0, precise optimal dilatation factors are given. Finally the UPC for the geographic region of Indonesia is presented as an example.
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The optimal Mercator Projection and the optimal polycylindric Projection of conformal type – case-study Indonesia
Journal of Geodesy, 1998Co-Authors: Erik W Grafarend, Rainer SyffusAbstract:As a conformal mapping of the sphere S 2R or of the ellipsoid of revolution E 2A,B the Mercator Projection maps the equator equidistantly while the transverse Mercator Projection maps the transverse metaequator, the meridian of reference, with equidistance. Accordingly, the Mercator Projection is very well suited to geographic regions which extend east-west along the equator; in contrast, the transverse Mercator Projection is appropriate for those regions which have a south-north extension. Like the optimal transverse Mercator Projection known as the Universal Transverse Mercator Projection (UTM), which maps the meridian of reference Λ0 with an optimal dilatation factor &ρcirc;=0.999 578 with respect to the World Geodetic Reference System WGS 84 and a strip [Λ0−ΛW,Λ0 + ΛE]×[ΦS,ΦN]= [−3.5∘,+3.5∘]×[−80∘,+84∘], we construct an optimal dilatation factor ρ for the optimal Mercator Projection, summarized as the Universal Mercator Projection (UM), and an optimal dilatation factor ρ0 for the optimal polycylindric Projection for various strip widths which maps parallel circles Φ0 equidistantly except for a dilatation factor ρ0, summarized as the Universal Polycylindric Projection (UPC). It turns out that the optimal dilatation factors are independent of the longitudinal extension of the strip and depend only on the latitude Φ0 of the parallel circle of reference and the southern and northern extension, namely the latitudes ΦS and ΦN, of the strip. For instance, for a strip [ΦS,ΦN]= [−1.5∘,+1.5∘] along the equator Φ0=0, the optimal Mercator Projection with respect to WGS 84 is characterized by an optimal dilatation factor &ρcirc;=0.999 887 (strip width 3∘). For other strip widths and different choices of the parallel circle of reference Φ0, precise optimal dilatation factors are given. Finally the UPC for the geographic region of Indonesia is presented as an example.
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the solution of the korn lichtenstein equations of conformal mapping the direct generation of ellipsoidal gaus kruger conformal coordinates or the transverse Mercator Projection
Journal of Geodesy, 1998Co-Authors: Erik W Grafarend, Rainer SyffusAbstract:The differential equations which generate a general conformal mapping of a two-dimensional Riemann manifold found by Korn and Lichtenstein are reviewed. The Korn–Lichtenstein equations subject to the integrability conditions of type vectorial Laplace–Beltrami equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and ellipsoidal latitude. The related coefficient constraints are collected in two corollaries. We present the constraints to the general solution of the Korn–Lichtenstein equations which directly generates Gaus–Kruger conformal coordinates as well as the Universal Transverse Mercator Projection (UTM) avoiding any intermediate isometric coordinate representation. Namely, the equidistant mapping of a meridian of reference generates the constraints in question. Finally, the detailed computation of the solution is given in terms of bivariate polynomials up to degree five with coefficients listed in closed form.
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The solution of the Korn–Lichtenstein equations of conformal mapping: the direct generation of ellipsoidal Gauß–Krüger conformal coordinates or the Transverse Mercator Projection
Journal of Geodesy, 1998Co-Authors: Erik W Grafarend, Rainer SyffusAbstract:The differential equations which generate a general conformal mapping of a two-dimensional Riemann manifold found by Korn and Lichtenstein are reviewed. The Korn–Lichtenstein equations subject to the integrability conditions of type vectorial Laplace–Beltrami equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and ellipsoidal latitude. The related coefficient constraints are collected in two corollaries. We present the constraints to the general solution of the Korn–Lichtenstein equations which directly generates Gaus–Kruger conformal coordinates as well as the Universal Transverse Mercator Projection (UTM) avoiding any intermediate isometric coordinate representation. Namely, the equidistant mapping of a meridian of reference generates the constraints in question. Finally, the detailed computation of the solution is given in terms of bivariate polynomials up to degree five with coefficients listed in closed form.
M Nazim - One of the best experts on this subject based on the ideXlab platform.
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some equations for the space oblique Mercator Projection
Photogrammetric Engineering and Remote Sensing, 1993Co-Authors: M NazimAbstract:The Space Oblique Mercator (SOM) Projection is found most suitable for continuous mapping of satellite imagery. Simple working equations with modification in the solution of λ″ and constants for the Clarke and Everest spheroids are produced here for facilitating calculations and improving working accessibility to the general user.
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Application of space oblique Mercator Projection in remote sensing
Journal of the Indian Society of Remote Sensing, 1991Co-Authors: M Nazim, R P KalaAbstract:The concept of SOM was originated by Colvocoresses, which was mathematically developed and later modified by Snyder. It is most suitable for continuous mapping of satellite imagery, true to scale along ground track. Simplified mathematical formulation without complicated integral equations and constants of SOM Projection for Landsat, SPOT and IRS satellites on Clarke and Everest spheroids are presented in this paper for ease of calculations.
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application of space oblique Mercator Projection in remote sensing
Journal of The Indian Society of Remote Sensing, 1991Co-Authors: M Nazim, R P KalaAbstract:The concept of SOM was originated by Colvocoresses, which was mathematically developed and later modified by Snyder. It is most suitable for continuous mapping of satellite imagery, true to scale along ground track.