Skip to main content

Plane curves

  • Chapter
  • First Online:
Book cover Geometry Revealed
  • 4088 Accesses

Abstract

Here are three examples of “facts” – but rather more of “results”, of “theorems” – that illustrate the difference between the mathematician and the “person in the street”. They are excellent for explaining the nature of mathematics to a non-professional.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 89.95
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • [B] Berger, M. (1987, 2009).Geometry I, II. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • [BG] Berger, M., & Gostiaux, B. (1987). Differential geometry: Manifolds, curves and surfaces. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • A’Campo, N. (1980). Sur la première partie du seizième problème de Hilbert. In Séminaire Bourbaki 1978–79: Vol. 770. Springer lecture notes in mathematics (pp. 208–227). Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • A’Campo, N. (2000a). Generic immersion of curves, knots, monodromy and gordian number. Publications mathm´ atiques de líInstitut des hautes études scientifiques, 770, 208–227

    MathSciNet  Google Scholar 

  • A’Campo, N. (2000b). Planar trees, slalom curves and hyperbolic knots. Publications mathm´ atiques de líInstitut des hautes études scientifiques

    Google Scholar 

  • Alias, J. (1984). La voie ferrée. Paris: Eyrolles

    Google Scholar 

  • Angenent, S. (1991). On formation of singularities in the curve shortening flow. Journal of Differential Geometry, 33, 601–633

    MATH  MathSciNet  Google Scholar 

  • Angenent, S. (1992). Shrinking doughnuts. In N.G. Lloyd, L.A. Peletier, & J. Serrin (Eds.), Nonlinear diffusion equations and their equilibrium states, 3. Boston: Birkhäuser

    Google Scholar 

  • Arnold, V. (1978). Mathematical methods of classical mechanics. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • Arnold, V. (1994a). Topological invariants of plane curves and caustics. University Lecture Series. Providence, RI: American Mathematical Society

    Google Scholar 

  • Arnold, V. (1994b). Plane curves, their invariants, perestroikas and classifications. Advances in Soviet Mathematics, 21, 33–91

    Google Scholar 

  • Arnold, V. (1995). The geometry of spherical curves and the algebra of quaternions. Russian Mathematical Surveys, 50, 1–68

    Article  MathSciNet  Google Scholar 

  • Arnold, V. (1996). Remarks on the extactic points of plane curves. In The Gelfand mathematical seminars (pp. 11–22). Boston: Birkhäuser

    Google Scholar 

  • Barth, W., & Bauer, T. (1996). Poncelet theorems. Expositiones Mathematicae, 14, 125–144

    MATH  MathSciNet  Google Scholar 

  • Benoist, Y. & Hulin, D. (2004). Itération de pliages de quadrilatères. Inventiones Mathematicae, 157, 147–194

    Article  MATH  MathSciNet  Google Scholar 

  • Bérard, P., Besson, G., & Gallot, S. (1985). Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy. Inventiones Mathematicae, 80, 295–308

    Article  MATH  MathSciNet  Google Scholar 

  • Berger, M. (1972). Enveloppes de droites. Bulletin de l’Association des Professeurs de Mathématiques de l’Enseignement Public, 283, 311–314

    Google Scholar 

  • Berger, M. (1993). Encounter with a geometer: Eugenio Calabi. In P. de Bartolomeis, F. Tricerri, & E. Vesentini (Eds.), Conference in honour of Eugenio Calabi, manifolds and geometry (pp. 20–60). Pisa: Cambridge University Press

    Google Scholar 

  • Berger, M. (1999). Riemannian geometry during the second half of the twentieth century. Providence, RI: American Mathematical Society

    Google Scholar 

  • Berger, M. (2003). A panoramic view of Riemannian geometry. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • Blaschke, W. (1949). Kreis und Kugel. New York: Chelsea

    Google Scholar 

  • Bol, G. (1950). Projektive Differentialgeometrie. Göttingen: Vandenhoeck and Ruprecht

    Google Scholar 

  • Boy, W. (1903). Curvatura Integra. Mathematische Annalen, 57, 151–184

    Article  MATH  MathSciNet  Google Scholar 

  • Brieskorn, E., & Knörrer, H. (1986). Plane algebraic curves. Boston: Birkhäuser

    Google Scholar 

  • Buchin, S. (1983). Affine differential geometry. New York: Gordon and Breach

    Google Scholar 

  • Burago, Y., & Zalgaller, V. (1988). Geometric inequalities. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • Cartan, E. (1946–1951). Leçons sur la géométrie des espaces de Riemann (2nd ed.). Paris: Gauthier-Villars

    MATH  Google Scholar 

  • Chenciner, A. (2006). Courbes algébriques. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • Cheeger, J. (1999). Differentiability of Lipschitz functions on a metric measure space. GAFA, Geometric and Functional Analysis, 9, 428–517

    Article  MATH  MathSciNet  Google Scholar 

  • Chmutov, S., & Duzhin, S. (1997). Explicit formulas for Arnold’s generic curve invariants. In The Arnold-Gelfand mathematical seminars (pp. 123–138). Boston: Birkhäuser

    Google Scholar 

  • Coolidge, J. (1959). A treatise on algebraic plane curves. Dover: Oxford University Press

    Google Scholar 

  • Darboux, G. (1879). De l’emploi des fonctions elliptiques dans la théorie du quadrilatère plan. Bulletin des sciences mathématiques, 3, 109–128

    Google Scholar 

  • Darboux, G. (1880). Sur le contact des coniques et des surfaces. Comptes Rendus, AcadÈmie des sciences de Paris, 91, 969–971

    Google Scholar 

  • De Turck, D., Gluck, H., Pomerleano, D., & Shea Vick, R. (2007). The four vertex theorem and its converse. Notices of the American Mathematical Society, 54, 191–207

    Google Scholar 

  • Dieudonné, J. (1960). Foundations of modern analysis. New York: Academic press

    Google Scholar 

  • Dieudonné, J. (1985). History of algebraic geometry. Monterey, CA: Wadsworth

    Google Scholar 

  • Dillen, F.J.E., Verstraelen, L.C.A. (Eds.). (2000). Handbook of differential geometry. Amsterdam: Elsevier

    Google Scholar 

  • Dombrowski, P. (1999). Wege in euklidischen Ebenen. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • Emch, A. (1900). Illustration of the elliptic integral of the first kind by a certain link work. Annals of Mathematics, 1, 81–92

    Article  MathSciNet  Google Scholar 

  • Emch, A. (1901). An application of elliptic functions to Peaucellier link-work (inversor). Annals of Mathematics, 2, 60–63

    Article  MathSciNet  Google Scholar 

  • Fabricius-Bjerre, F. (1962). On the double tangents of plane closed curves. Mathematica Scandinavica, 11, 113–116

    MATH  MathSciNet  Google Scholar 

  • Fischer, G. (1986a). Mathematische modelle. Braunschweig, Germany: Vieweg

    Google Scholar 

  • Fischer, G. (1986b). Mathematical models: Photograph volume and commentary. Braunschweig, Germany: Vieweg

    Google Scholar 

  • Fulton, W. (1969). Algebraic curves. New York: Benjamin

    Google Scholar 

  • Gage, M. (1986b). On an area-preserving evolution equation for plane curves. Contemporary Mathematics, 51, 51–62

    MathSciNet  Google Scholar 

  • Gage, M., & Hamilton, R. (1986a). The heat equation shrinking plane curves. Journal of Differential Geometry, 23, 6996

    MathSciNet  Google Scholar 

  • Giannopoulos, A., & Milman, V. (2001). Euclidean structure in finite dimensional normed spaces. In W.B. Johnson & J. Lindenstrauss (Eds.), Handbook of Geometry of Banach Spaces (Vol. 1). Dordrecht: Kluwer

    Chapter  Google Scholar 

  • Gibbons, G., & Rasheed, D. (1995). Nuclear Physics, B454, 185

    MathSciNet  Google Scholar 

  • Gluck, H. (1971). The converse of the four vertex theorem. Líenseignement mathématique, XVII, 295–309

    Google Scholar 

  • Gluck, H. (1972). The generalized Minkowski problem in differential geometry in the large. Annals of Mathematics, 96, 245–276

    Article  MathSciNet  Google Scholar 

  • Greenberg, M. (1967). Lectures on algebraic topology. New York: Benjamin

    Google Scholar 

  • Griffiths, P., & Harris, J. (1978). Principles of algebraic geometry. New York: John Wiley

    Google Scholar 

  • Gromov, M. (1980). Paul Levy’s isoperimetric inequality. Prepublication M/80/320, Institut des Hautes Études Scientifiques. Appears as Appendix C in Gromov (1999)

    Google Scholar 

  • Gromov, M. (1999). Metric structures for Riemannian and non-Riemannian spaces. Boston: Birkhäuser

    Google Scholar 

  • Guggenheimer, H. (1963). Differential geometry. New York: McGraw Hill

    Google Scholar 

  • Guieu, L., Mourre, E. & Ovsienko, V. (1996). Theorem on six vertices of a plane curve via Sturm theory. In Arnold-Gelfand mathematical seminars (pp. 257–266). Boston: Birkhäuser

    Google Scholar 

  • Harnack, A. (1876). Über Vieltheiligkeit der ebenen algebraischen Curven. Mathematische Annalen, 10, 189–199

    Article  MathSciNet  Google Scholar 

  • Haupt, O., & Künneth, H. (1967). Endliche Ordnung. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • Hellgouarch, Y. (2000). Rectificatif à l’article de H. Darmon. Gazette des mathÈmaticiens (Soc. Math. France), 85, 31–32

    Google Scholar 

  • Hilbert, D., & Cohn-Vossen, S. (1952). Geometry and the imagination. New York: Chelsea

    Google Scholar 

  • Hofer, H., & Zehnder, E. (1994). Symplectic capacities. Boston: Birkhäuser

    Google Scholar 

  • Hopf, H. (1935). Über die Drehung der Tangenten und Sehnen ebener Kurven. Compositio Mathematica, 2, 50–62

    MATH  MathSciNet  Google Scholar 

  • Humbert, G. (1888). Sur les courbes algébriques planes rectifiables. Journal de Mathématiques Pures et Appliquée, IV, 133–151

    Google Scholar 

  • Hurwitz, M. (1902). Sur quelques applications géométriques des séries de Fourier. Annales Scientifiques de l’École Normale Supérieure, 19, 357–408

    MathSciNet  Google Scholar 

  • Itenberg, I., & Viro, O. (1996). Patchworking algebraic curves disproves the Ragsdale conjecture. The Mathematical Intelligencer, 18(4), 19–28

    Article  MATH  MathSciNet  Google Scholar 

  • Kapovich, M., & Millson, J. (1996). The symplectic geometry of polygons in Euclidean space. Journal of Differential Geometry, 44, 479–513

    MATH  MathSciNet  Google Scholar 

  • Kaufman, L. (1994). Knots and physics. Singapore: World Scientific

    Book  Google Scholar 

  • Kneser, A. (1912). Bermerkungen über die Anzahl der Extreme der Krümmung auf geschlossenene Kurven und über verwandte Fragen in einer nicht-euklidischen Geometrie (pp. 170–180). Leipzig-Berlin: H. Weber Festschrift

    Google Scholar 

  • Knöthe, H. (1957). Contributions to the theory of convex bodies. Michigan Mathematical Journal, 4, 39–52

    Article  MathSciNet  Google Scholar 

  • Lafontaine, J. (1996). Introduction aux variétés différentielles. Grenoble: Presses Universitaires de Grenoble

    Google Scholar 

  • Laumon, G. (1976). Degré de la variété duale d’une hypersurface à singularités isolées. Bulletin de la Société MathÈmatique de France, 104, 51–63

    MATH  MathSciNet  Google Scholar 

  • Lebesgue, H. (1950). Leçons sur les constructions géométriques (Reprint Jacques Gabay, 1987). Paris: Gauthier-Villars

    MATH  Google Scholar 

  • Lucas, E. (1960). Récréations mathématiques. Paris: Gauthier-Villars

    Google Scholar 

  • Marchaud, A. (1936). Les surfaces du second ordre en géométrie finie. Journal de Mathématiques Pures et Appliquée, 18, 293–300

    Google Scholar 

  • Marchaud, A. (1965). Sur les droites de la surface du troisième ordre en géométrie finie. Journal de Mathématiques Pures et Appliquée, 44, 49–69

    MATH  MathSciNet  Google Scholar 

  • McDuff, D. (2000). A glimpse into symplectic geometry. In V. Arnold, M. Atiyah, P. Lax, & B. Mazur (Eds.), Mathematics: Frontiers and perspectives. Providence, RI: American Mathematical Society

    Google Scholar 

  • McDuff, D., & Salamon, D. (1998). Introduction to symplectic topology. Oxford, UK: Oxford

    Google Scholar 

  • Mukhopadhyaya, S. (1909). New methods in the geometry of plane arcs. Bulletin of the Calcutta Mathematical Society, 1, 31–37

    Google Scholar 

  • Osserman, R. (1978). The isoperimetric inequality. Bulletin of the American Mathematical Society, 84, 1182–1238

    Article  MATH  MathSciNet  Google Scholar 

  • Osserman, R. (1979). Bonnesen-Fenchel isoperimetric inequalities. The American Mathematical Monthly, 86, 1–29

    Article  MATH  MathSciNet  Google Scholar 

  • Osserman, R. (1985). The four-or-more vertex theorem. The American Mathematical Monthly, 92, 332–337

    Article  MATH  MathSciNet  Google Scholar 

  • Oxtoby, J. (1980). Measure and category. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • Petitot, J., & Tondut, Y. (1998). Vers une neuro-géométrie. Fibrations corticales, structures de contact et contours subjectifs modaux. Mathématiques, Infomatiques et Sciences Humaines, EHESS, 145, 5–101

    Google Scholar 

  • Pohl, W. (1975). A theorem of Géométrie Finie. Journal of Differential Geometry, 10, 435–466

    MATH  MathSciNet  Google Scholar 

  • Porteous, I. (1994). Geometric differentiation (for the intelligence of curves and surfaces). Cambridge, UK: Cambridge University Press

    Google Scholar 

  • Porter, T. (1933). A history of the classical isoperimetric problem. In Contributions to the calculus of variations. Chicago: University of Chicago Press

    Google Scholar 

  • Raphaël, E. (1992). Peut-on faire flotter des troncs d’arbre en apesanteur? Bulletin de la Société Française de Physique, 92, 21

    Google Scholar 

  • Raphaël, E., di Meglio, J.-M., Berger, M., & Calabi, E. (1992). Convex particles at interfaces. Journal de Physique I, 2, 571–579

    Article  Google Scholar 

  • Raphaël, E., & Williams, D. (1993). Three-dimensional convex particles at interfaces. Journal of Colloid and Interface Science, 155, 509–511

    Article  Google Scholar 

  • Rideau, F. (1989). Les systèmes articulés. Pour la Science, 136, 94–101

    Google Scholar 

  • Risler, J.-J. (1993). Construction d’hypersurfaces réelles (d’après Viro), Séminaire Bourbaki. Astérisque, 216, 69–87

    MathSciNet  Google Scholar 

  • Ronga, F. (1998). Klein’s paper on real flexes vindicated. In B. Jakubczyk, W. Pawlucki, & J. Stasica (Eds.), Singularities symposium – Lojasiewicz 70. Warszawa: Banach Center Publications 44

    Google Scholar 

  • Santalo, L. (1976). Integral geometry and geometric probability. Reading, MA: Addison-Wesley

    Google Scholar 

  • Schmidt, E. (1939). Über das isoperimetrische Problem im Raum von n Dimensionen. Mathematische Zeitschrift, 44, 689–788

    Article  MathSciNet  Google Scholar 

  • Schwartz, H. (1884). Beweis des Satzes, dass die Kugel kleinere Oberfläche besitzt als jeder andere Körper gleichen Volumens. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, ou OEuvres complètes, Berlin, 1890, 1–13

    Google Scholar 

  • Sedykh, V. (2000). Discrete variants of the four-vertex theorem (Preprint no. 9615). CERE-MADE, Dauphine: Université Paris IX

    Google Scholar 

  • Segre, B. (1957). Some properties of differentiable varieties and transformations. Berlin/ Heidelberg/New York: Springer

    Google Scholar 

  • Segre, B. (1968). Alcune proprieta differenziali in grande delle curve chiuse sghembe. Rendiconti di Matematica, 1, 237–297

    MATH  MathSciNet  Google Scholar 

  • Serre, J.-P. (1970). Cours d’arithmétique. Paris: Presses Universitaires de France

    Google Scholar 

  • Silverman, J. (1986). The arithmetic of elliptic curves. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • Silverman, J., & Tate, J. (1992). Rational points on elliptic curves. Berlin/Heidelberg/New York: Springer

    Google Scholar 

  • Sossinski, A. (1999). Noeuds (Genèse d’une théorie mathématique). Paris: Seuil

    Google Scholar 

  • Tabachnikov, S. (2002). Dual billiards in the hyperbolic plane. Nonlinearity, 15, 1051–1072

    Article  MATH  MathSciNet  Google Scholar 

  • Thom, R. (1962). Sur la théorie des enveloppes. Journal de Mathématiques Pures et Appliquée, 41, 177–192

    MATH  MathSciNet  Google Scholar 

  • Thom, R. (1969). Sur les variétés d’ordre fini. In D. Spencer & S. Iyannaga (Eds.), Global analysis. Papers in Honor of K. Kodaira. Princeton, NJ: Princeton University Press

    Google Scholar 

  • Thorbergsson, G., & Umehara, M. (1999). A unified approach to the four vertex theorems. II. American Mathematical Sociey Translations, 190, 229–252

    MathSciNet  Google Scholar 

  • Thorbergsson, G., & Umehara, M. (2002). Sextactic points on a simple closed curve. Nagoya Mathematical Journal, 167, 55–94

    MATH  MathSciNet  Google Scholar 

  • Thorbergsson, G., & Umehara, M. (2004). A global theory of flexes of periodic functions. Nagoya Mathematical Journal, 173, 85–138

    MATH  MathSciNet  Google Scholar 

  • Turaev, V. (1994). Quantum invariants of knots and 3-manifods. Berlin: de Gruyter

    Google Scholar 

  • Uribe-Vargas, R. (2004a). Four-vertex theorems, Sturm theory and Lagrangian singularities. Mathematical Physics, Analysis and Geometry, 7, 223–237

    Article  MATH  MathSciNet  Google Scholar 

  • Uribe-Vargas, R. (2004b). On singularities, ”perestroikas” and differential geometry of space curves. L’enseignement mathématique, 50, 69–101

    MATH  MathSciNet  Google Scholar 

  • Viro, O. (1990a). Real algebraic plane curves: Constructions with controlled topology. Leningrad Mathematical Journal, 1(5), 1059–1134

    MATH  MathSciNet  Google Scholar 

  • Viro, O. (1990b). Progress in the topology of real algebraic varieties over the last six years. Russian Mathematical Surveys, 41, 55–82

    Article  Google Scholar 

  • Voisin, C. (1996). Symétrie miroir. Paris: Société Mathematique de France

    Google Scholar 

  • Walker, R. (1950). Algebraic curves. Princeton, NJ: Princeton University Press

    Google Scholar 

  • Whitney, H. (1937). On regular closed curves in the plane. Compositio Mathematica, 4, 276–284

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marcel Berger .

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Berger, M. (2010). Plane curves. In: Geometry Revealed. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70997-8_5

Download citation

Publish with us

Policies and ethics