Skip to main content
SpringerLink
Log in

Theory of Map Projection: From Riemann Manifolds to Riemann Manifolds

Handbook of Geomathematics

  • 366 Accesses

Abstract

The Theory of Map Projections is based here on the transformation of Riemann manifolds to Riemann manifolds. Section 2 offers some orientation based on simultaneous diagonalization of two symmetric matrices. We separate simply connected regions. In detail, we review the pullback versus pushforward operations.Section 3 introduces the first multiplicative measure of deformation: the Cauchy–Green deformation tensor, its polar decomposition, as well as singular value decomposition. An example is the Hammer retroazimuthal projection.A second multiplicative measure of deformation is presented in Sect. 4: stretch, or length distortion, and Tissot portrait as well. The Euler–Lagrange deformation tensor presented in Sect. 5 is the first additive measure of deformation based on the difference of the metrics {ds2, dS2}. Section 6 introduces a review of 25 different measures of deformation. First, angular shear is the second additive measure, also called angular distorsion, left and right.Section 8 introduces a third multiplicative measure of deformation, called relative angular shear. In contrast, the equivalence theorem of conformal mapping in Sect. 9 is based on Korn–Lichtenstein equations. Areal distortion in Sect. 10 offers a popular alternative based on the fourth multiplicative and additive measure of deformation, namely, dual deformation called areomorphism. Section 11 offers an equivalence theorem of equiareal mapping. The highlight is our review of canonical criteria in Sect. 12: (i) isometry; (ii) equidistant mapping of submanifolds; (iii) in particular, canonical conformism, areomorphism, isometry, and equidistance; and finally, (iv) optimal map projections. Please study Sect. 13, the exercise: the Armadillo double projection.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • Airy GB (1861) Explanation of a projection by balance of errors for maps applying to a very large extent of the Earth’s surface: comparison of this projection with other projections. Philos Mag 22:409–442

    Google Scholar 

  • Almansi E (1911) Sulle deformazioni finite di solidi elastici isotropi, Note I. Atti Accad naz Lincei Re Serie Quinta 201:705–714

    Google Scholar 

  • Amalvict M, Livieratos E (1988) Surface mapping of a rotational reference ellipsoid onto a triaxial counterpart through strain parameters. Manuscripta Geodaetica 13:133–138

    Google Scholar 

  • Blaschke W, Leichtweiß K (1973) Elementare Differentialgeometrie. Springer, Berlin/Heidelberg/New York

    Book  MATH  Google Scholar 

  • Boulware DG, Brown LS, Peccei RD (1970) Deep-inelastic electroproduction and conformal symmetry. Phys Rev D 2:293–298

    Article  Google Scholar 

  • Bourguignon JP (1970) Transformation infinitesimal conformes fermées des variétés riemanniennes connexes complètes. C R Acad Sci Ser A 270:1593–1596

    MathSciNet  MATH  Google Scholar 

  • Cardoso JF, Souloumiac A (1996) Jacobi angles for simultaneous diagonalization. SIAM J Matrix Anal Appl 17:161–164

    Article  MathSciNet  MATH  Google Scholar 

  • Cauchy A (1828) Sur les équations qui expriment les conditions d’équilibre ou les lois du mouvement intérieur d’un corps solide, élastique, ou non élastique. Ex De Math 3:160–187

    Google Scholar 

  • Cauchy A (1889) Oeuvres complétes, Iie série, tome VII. Gauthier–Villars et Fils, pp 82–93

    Google Scholar 

  • Cauchy A (1890) Sur l’équilibre et le mouvement d’un systéme de points matériels sollicités par des forces d’attraction ou de répulsion mutuelle, Oeuvres Complétes, Iie série, tome VIII. Gauthier–Villars et Fils, pp 227–252

    Google Scholar 

  • Chen BY (1973) Geometry of submanifolds. Marcel Dekker, New York

    MATH  Google Scholar 

  • Chen BY, Yano K (1973) Special conformally flat spaces and canal hypersurfaces. Tohoku Math J 25:77–184

    Article  MathSciNet  Google Scholar 

  • Chern SS, Hartman P, Wintner A (1954) On isothermic coordinates. Commentarii Mathematici Helvetici 28:301–309

    Article  MathSciNet  MATH  Google Scholar 

  • Chu MT (1991a) A continuous Jacobi-like approach to the simultaneous reduction of real matrices. Linear Algebra Appl 147:75–96

    Article  MathSciNet  MATH  Google Scholar 

  • Chu MT (1991b) Least squares approximation by real normal matrices with specified spectrum. SIAM J Matrix Anal Appl 12:115–127

    Article  MathSciNet  MATH  Google Scholar 

  • Crumeyrolle A (1990) Orthogonal and sympletic Clifford algebras. Kluwer Academic, Dordrecht/Boston/London

    Book  Google Scholar 

  • De Azcárraga JA, Izquierdo JM (1995) Lie groups, Lie algebras, cohomology and some applications in physics. Cambridge monographs on mathematical physics. Cambridge University Press, Cambridge/New York

    Book  Google Scholar 

  • De Moor B, Zha H (1991) A tree of generalizations of the ordinary singular value decomposition. Linear Algebra Appl 147:469–500

    Article  MathSciNet  MATH  Google Scholar 

  • Dermanis A, Livieratos E (1993) Dilatation, shear, rotation and energy: analysis of map projections. Bollettino di Geodesia e Science Affini 42:53–68

    Google Scholar 

  • Do Carmo MP (1994) Differential forms and applications. Springer, Berlin/New York

    Book  MATH  Google Scholar 

  • Do Carmo M, Dajczer M, Mercuri F (1985) Compact conformally flat hypersurfaces. Trans Am Math Soc 288:189–203

    Article  MathSciNet  MATH  Google Scholar 

  • Duda FP, Martins LC (1995) Compatibility conditions for the Cauchy–Green strain fields: solutions for the plane case. J Elast 39:247–264

    Article  MathSciNet  MATH  Google Scholar 

  • Eisenhart LP (1949) Riemannian geometry. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Euler L (1755) Principes généraux des mouvement des fluides, Memoirs de l’Accad. des Sciences de Berlin 11:274–315

    Google Scholar 

  • Euler L (1777) Über die Abbildung einer Kugelfläche in die Ebene. Acta Academiae Scientiarum Petropolitanae, Petersburg

    Google Scholar 

  • Ferrara S, Grillo AF, Gatto R (1972) Conformal algebra in two space-time dimensions and the Thirring model. Il Nuovo Cinmento 12A:959–968

    Article  MathSciNet  Google Scholar 

  • Finger J (1894) Über die allgemeinsten Beziehungen zwischen den Deformationen und den zugehörigen Spannungen in aerotropen und isotropen Substanzen. Sitzber Akad Wiss Wien (2a) 103:1073–1100

    Google Scholar 

  • Finzi A (1922) Sulle varieta in rappresentazione conforme con la varieta euclidea a piu di tre dimensioni. Rend Acc Lincei Classe Sci Ser 5–31:8–12

    Google Scholar 

  • Flanders H (1970) Differential forms with applications to the physical sciences, 4th printing. Academic, London

    Google Scholar 

  • Gauss CF (1822) Allgemeine Auflösung der Aufgabe, die Teile einer gegebenen Fläche auf einer anderen gegebenen Fläche so abzubilden, daß die Abbildung dem Abgebildeten in den kleinsten Teilen ähnlich wird, 1822, Abhandlungen Königl. Gesellschaft der Wissenschaften zu Göttingen, Bd. IV, pp 189–216, Göttingen 1838

    Google Scholar 

  • Gauss CF (1844) Untersuchungen über Gegenstände der höheren Geodäsie, erste Abhandlung, Abhandlungen der Königl. Gesellschaft der Wissenschaften zu Göttingen, Bd. 2 (1844), Ges. Werke IV, pp 259–334, Göttingen 1880

    Google Scholar 

  • Gere JM, Weaver W (1965) Matrix algebra for engineers. D. Van Nostrand, New York

    Google Scholar 

  • Goenner H, Grafarend EW, You RJ (1994) Newton mechanics as geodesic flow on Maupertuis’ manifolds: the local isometric embedding into flat spaces. Manuscripta Geodaetica 19:339–345

    Google Scholar 

  • Grafarend EW (1995) The optimal universal Mercator projection. Manuscripta Geodaetica 20:421–468

    Google Scholar 

  • Grafarend EW, Syffus R (1998a) The optimal mercator projection and the optimal polycylindric projection of conformal type – case study Indonesia. J Geod 72:251–258

    Article  MATH  Google Scholar 

  • Grafarend EW, Syffus R (1998b) The solution of the Korn-Lichtenstein equations of conformal mapping: the direct generation of ellipsoidal Gauss-Krueger conformal coordinates or the transverse mercator projection. J Geod 72:282–293

    Article  MATH  Google Scholar 

  • Grafarend EW, Krumm F (2006) Map projections, cartographic informationsystems, pp 713, Springer, Berlin/New York

    Google Scholar 

  • Grafarend EW, You RJ, Syffus R (2014) Map projections, cartographic information systems, second ed, vol 1: pp 413, vol 2: pp 510, Berlin/New York

    Google Scholar 

  • Green G (1839) On the laws of reflection and refraction of light at the common surface of two non-crystallized media. Trans Camb Philos Soc 7:1–24

    Google Scholar 

  • Hedrick ER, Ingold L (1925a) Analytic functions in three dimensions. Trans Am Math Soc 27: 551–555

    Article  MathSciNet  MATH  Google Scholar 

  • Hedrick ER, Ingold L (1925b) The Beltrami equations in three dimensions. Trans Am Math Soc 27:556–562

    MathSciNet  MATH  Google Scholar 

  • Hencky H (1928) Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Zeitschrift f techn Physik 9:215–220, 457

    Google Scholar 

  • Heitz S (1988) Coordinates in geodesy. Springer, Berlin/Heidelberg/New York

    Book  Google Scholar 

  • Higham NJ (1986) Computing the polar decomposition. SIAM J Sci Stat Comput 7:1160–1174

    Article  MathSciNet  MATH  Google Scholar 

  • Jacobi CGJ (1839) Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwrdigen analytischen Substitution. Crelles J 19:309–313

    Article  MathSciNet  MATH  Google Scholar 

  • Karni Z, Reiner M (1960) Measures of deformation in the strained and in the unstrained state. Bull Res Counc Isr 8c 89. Jerusalem

    Google Scholar 

  • Kavrajski VV (1958) Ausgewählte Werke, Mathematische Kartographie, Allgemeine Theorie der kartographischen Abbildungen, Kegel-und Zylinderabbildungen, ihre Anwendungen (russ.) GS VMP, Moskau

    Google Scholar 

  • Kenney C, Laub AJ (1991) Polar decomposition and matrix sign function condition estimates. SIAM J Sci Stat Comput 12:488–504

    Article  MathSciNet  MATH  Google Scholar 

  • Klingenberg W (1982) Riemannian geometry. de Gruyter, Berlin/New York

    MATH  Google Scholar 

  • Koenig R, Weise KH (1951) Mathematische Grundlagen der höheren Geodäsie und Kartographie, Bd. I. Springer, Berlin/Heidelberg/New York

    Book  MATH  Google Scholar 

  • Korn A (1914) Zwei Anwendungen der Methode der sukzessiven Annäherungen. In: Mathematischen Abhandlungen Hermann Amandus Schwarz zu seinem fünfzigjährigen Doktorjubiläum. Springer, Berlin/Heidelberg/New York, pp 215–229

    Google Scholar 

  • Krueger L (1903) Bemerkungen zu C.F. Gauss: Conforme Abbildungen des Sphäroids in der Ebene, C.F. Gauss, Werke, Königl. Ges. Wiss. Göttingen Bd. IX (1903), pp 195–204

    Google Scholar 

  • Krueger L (1922) Zur stereographischen Projektion. P. Stankiewicz, Berlin

    Google Scholar 

  • Kuiper NH (1949) On conformally-flat spaces in the large. Ann Math 50:916–924

    Article  MathSciNet  MATH  Google Scholar 

  • Kuiper NH (1950) On compact conformally Euclidean spaces of dimension ¿ 2. Ann Math 52: 478–490

    Article  MathSciNet  MATH  Google Scholar 

  • Kulkarni RS (1969) Curvature structures and conformal transformations. J Differ Geom 4:425–451

    Google Scholar 

  • Kulkarni RS (1972) Conformally flat manifolds. Proc Natl Acad Sci USA 69:2675–2676

    Article  MATH  Google Scholar 

  • Kulkarni RS, Pinkall U (eds) (1988) Conformal geometry. Vieweg, Braunschweig/Wiesbaden

    MATH  Google Scholar 

  • Lagrange de JL (1781) Sur la construction des cartes geographiques. Nouveaux Mémoires de l’Academie Royale des Sciences et Belles Lettres de Berlin 161–210. Berlin

    Google Scholar 

  • Lancaster GM (1969) A characterization of certain conformally Euclidean spaces of class one. Proc Am Math Soc 21:623–628

    Article  MathSciNet  MATH  Google Scholar 

  • Lancaster GM (1973) Canonical metrics for certain conformally Euclidean spaces of dimension three and codimension one. Duke Math J 40:1–8

    Article  MathSciNet  MATH  Google Scholar 

  • Lichtenstein L (1911) Beweis des Satzes, daß jedes hinreichend kleine, im wesentlichen stetig gekrümmte, singularitätenfreie Flächenstück auf einem Teil einer Ebene zusammenhängend und in den kleinsten Teilen ähnlich abgebildet werden kann. Preußische Akademie der Wissenschaften, Berlin

    Google Scholar 

  • Lichtenstein L (1916) Zur Theorie der konformen Abbildung. Konforme Abbildung nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebiete. Anzeiger der Akademie der Wissenschaften in Krakau 2–4:192–217

    Google Scholar 

  • Liouville J (1850) Extension au cas de trois dimensions de la question du tracé géographique, Note VI, by G. Monge: application de l’analyse à la géométrie, 5ème èdition revue corrigèe par M. Liouville, Bachelier, Paris

    Google Scholar 

  • Macvean DB (1968) Die Elementarbeit in einem Kontinuum und die Zuordnung von Spannungs-und Verzerrungstensoren. Zeitschrift für Angewandte Mathematik und Physik 19:137–185

    Google Scholar 

  • Markuschewitsch AI (1955) Skizzen zur Geschichte der analytischen Funktion. Deutscher Verlag der Wissenschaften, Belin

    Google Scholar 

  • Mirsky L (1960) Symmetric gauge functions and unitary invariant norms. Q J Math 11:50–59

    Article  MathSciNet  MATH  Google Scholar 

  • Misner CW (1978) Harmonic maps as models for physical theores. Phys Rev D 18:4510–4524

    Article  MathSciNet  Google Scholar 

  • Mitra SK, Rao CR (1968) Simultaneous reduction of a pair of quadratic forms. Sankya A 30: 312–322

    Google Scholar 

  • Morman KN Jr (1986) The generalized strain measure with application to nonhomogeneous deformations in rubber-like solids. J Appl Mech 53:726–728

    Article  Google Scholar 

  • Moore JD (1977) Conformally flat submanifolds of Euclidean space. Math Ann 225:89–97

    Article  MathSciNet  MATH  Google Scholar 

  • Mueller B (1991) Kartenprojektionen des dreiachsigen Ellipsoids, Diplomarbeit Geodätisches Institut Universität, Stuttgart

    Google Scholar 

  • Newcomb RW (1960) On the simultaneous diagonalization of two semidefinite matrices. Q J Appl Math 19:144–146

    MathSciNet  Google Scholar 

  • Nishikawa S (1974) Conformally flat hypersurfaces in a Euclidean space. Tohoku Math J 26:563–572

    Article  MathSciNet  MATH  Google Scholar 

  • Piola G (1836) Nuova analisi per tutte le questioni della meccanica molecolare. Memorie Mat Fis Soc Ital Sci Modena 21:155–321

    Google Scholar 

  • Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York

    MATH  Google Scholar 

  • Ricci G (1918) Sulla determinazione di varieta dotate di proprietà intrinseche date a priori – note I. Rend Acc Lincei Classe Sci Ser 5, vol 19, Rome 1918

    Google Scholar 

  • Riemann B (1851) Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Größe, Inauguraldissertation, Göttingen

    Google Scholar 

  • Samelson H (1969) Orientability of hypersurfaces in Rn. Proc Am Math Soc 22:301–302

    MathSciNet  MATH  Google Scholar 

  • Schering E (1857) Über die conforme Abbildung des Ellipsoides auf der Ebene, Göttingen 1857, Nachdruck in: Ges. Math. Werke von E. Schering, eds. R. Haussner und K. Schering, 1.Bd. Mayer und Müller Verlag, Berlin 1902

    Google Scholar 

  • Schmehl H (1927) Untersuchungen über ein allgemeines Erdellipsoid, Veröffentlichungen des Preußischen Geodätischen Institutes, Neue Folge Nr. 98, Potsdam

    Google Scholar 

  • Schoen R (1984) Conformal deformation of a Riemannian metric to constant scalar curvature. J Differ Geom 20:479–495

    MathSciNet  MATH  Google Scholar 

  • Schouten JA (1921) Über die konforme Abbildung n-dimensionaler Mannigfaltigkeiten mit quadratischer Maßbestimmung auf eine Mannigfaltigkeit mit Euklidischer Maßbestimmung. Math Z 11:58–88

    Article  MathSciNet  MATH  Google Scholar 

  • Searle SR (1982) Matrix algebra useful for statistics. Wiley, New York

    MATH  Google Scholar 

  • Seth BR (1964a) Generalized strain measure with applications to physical problems. In: Reiner M, Abir D (eds) Second order effects in elasticity, plasticity and fluid dynamics. Pergamon Press, Oxford, pp 162–172

    Google Scholar 

  • Seth BR (1964b) Generalized strain and transition concepts for elastic-plastic deformation-creep and relaxation. IUTAM symposium. Pergamon Press, München, pp 383–389

    Google Scholar 

  • Shougen W, Shuqin Z (1991) An algorithm for Ax = (λ)Bx with symmetric and positive-definite A and B. SIAM J Matrix Anal Appl 12:654–660

    Article  MathSciNet  MATH  Google Scholar 

  • Spivak M (1979) A comprehensive introduction to differential geometry, vol 4, 2nd edn. Publish or Perish, Boston

    Google Scholar 

  • Stein EM, Weiss G (1968) Generalization of the Cauchy-Riemann equations and representations of the rotation group. Am J Math 90:163–196

    Article  MathSciNet  MATH  Google Scholar 

  • Ting TCT (1985) Determination of C 1 ∕ 2, \(C_{-1/2}\) and more general isotropic tensor functions of C. J Elast 15:319–323

    Article  MATH  Google Scholar 

  • Tissot NA (1881) Mémoire sur la représentation des surfaces et les projections des cartes géographiques. Gauthier-Villars, Paris

    Google Scholar 

  • Truesdell C, Toupin R (1960) The classical field theories. In: Handbuch der Physik, vol III/I. Springer, Berlin

    Book  Google Scholar 

  • Uhlig F (1973) Simultaneous block diagonalization of two real symmetric matrices. Linear Algebra Appl 7:281–289

    Article  MathSciNet  MATH  Google Scholar 

  • Uhlig F (1976) A canonical form for a pair of real symmetric matrices that generate a nonsingurlar pencil. Linear Algebra Appl 14:189–210

    Article  MathSciNet  MATH  Google Scholar 

  • Uhlig F (1979) A recurring theorem about pairs of quadratic forms and extensions: a survey. Linear Algebra Appl 25:189–210

    Article  MathSciNet  Google Scholar 

  • Weber H (1867) Über ein Prinzip der Abbildung der Teile einer krummen Oberfläche auf einer Ebene. Journal für die reine und angewandte Mathematik 67:229–247

    Article  MATH  Google Scholar 

  • Weyl H (1918) Reine Infinitesimalgeometrie. Math Z 2:384–411

    Article  MathSciNet  MATH  Google Scholar 

  • Weyl H (1921) Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung, Nachr. Königl. Ges. Wiss. Göttingen, Math.-Phys. Klasse, Göttingen

    Google Scholar 

  • Wray T (1974) The seven aspects of a general map projection. Supplement 2, Canadian cartographer 11, monographe no.11, Cartographica. B.V. Gustell Publ./University of Toronto Press, Toronto

    Google Scholar 

  • Yano K (1970) On Riemannian manifolds admitting an infinitesimal conformal transformation. Math Z 113:205–214

    Article  MathSciNet  MATH  Google Scholar 

  • Yanushaushas AI (1982) Three-dimensional analogues of conformal mappings (in Russian). Iz da te l’stvo Nauka, Novosibirsk

    Google Scholar 

  • Zadro M, Carminelli A (1966) Rapprezentazione conforme del geoide sull ellissoide internazionale. Bollettino di Geodesia e Scienze Affini 25:25–36

    Google Scholar 

  • Zund JD (1987) The tensorial form of the Cauchy-Riemann equations. Tensor New Ser 44: 281–290

    MathSciNet  MATH  Google Scholar 

  • Zund JD, Moore WA (1987) Conformal geometry, Hotine’s conjecture, and differential geodesy, Department of Mathematical Sciences, New Mexico State University, Scientific report no. 1, Las Cruces

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Geodesy and Geoinformatics, Stuttgart University, Stuttgart, Germany

    Erik W. Grafarend

Authors
  1. Erik W. Grafarend
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erik W. Grafarend .

Editor information

Editors and Affiliations

  1. Geomathematische Gruppe, TU Kaiserslautern FB Mathematik, Kaiserslautern, Germany

    Willi Freeden

  2. University of Central Florida Dept. Mathematics, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Inst. Analysis, TU Braunschweig FB 1 Mathematik und Informatik, Braunschweig, Germany

    Thomas Sonar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this entry

Cite this entry

Grafarend, E.W. (2014). Theory of Map Projection: From Riemann Manifolds to Riemann Manifolds. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27793-1_53-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27793-1_53-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Online ISBN: 978-3-642-27793-1

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Chapter history

  1. Latest

    : From Riemann Manifolds to Riemann Manifolds
    Published:
    04 March 2015

    DOI: https://doi.org/10.1007/978-3-642-27793-1_53-2

  2. Original

    Theory of Map Projection: From Riemann Manifolds to Riemann Manifolds
    Published:
    09 October 2014

    DOI: https://doi.org/10.1007/978-3-642-27793-1_53-1

Search

Navigation