Courses offered in English
Curves and Surfaces


Chapter 1: Curves

  • Parametrizations. Parametrization of conics. Implicit equations of a curve. Planar curves: implicit, explicit equations and equations in polar coordinates.
  • Length of a regular curve. Line integral of a scalar-valued function. Arc length parameter (or natural parameter).
  • Frenet frame. Planes and lines associated with Frenet frame. Projections of a curve on planes.
  • Curvature and torsion. Radius of curvature. Osculating circle. Frenet Serret formulas.
  • Significative curves: helices.

Chapter 2: Surfaces

  • Parametrizations. Parametrizations of quadric surfaces. Implicit and explicit equations. Parametrizations by rotation, translation, etc. Surfaces of revolution.
  • Tangent plane and Normal line at a regular point. Parametric curves.
  • First fundamental form. Measure. Surface integral of scalar function.
  • Second fundamental form. Curvatures of the curves of a surface. Normal curvature.
  • Asymptotic directions. Asymptotic curves.
  • Classification of regular points of a surface: elliptic, hyperbolic and parabolic points. Umbilic points and planar points.
  • Principal curvatures. Gaussian curvature. Mean curvature. Principal directions. Lines of Curvature. Geodesics. Euler formula.
  • Ruled Surfaces. Directrix and rulings. Parametrization of a ruled surface.
  • Measure and shape of a ruled surface: first and second fundamental forms. distribution parameter. Properties of the ruled surfaces.
  • Classification of ruled surfaces: developable surfaces and non-developable surfaces. Properties. Edge of regression. Striction curve.


  • M.P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, NJ, 1976.
  • J. Burgos. Curvas y Superficies. Ed. Garcia-Maroto. 2008.
  • Larson, R., Hostetler, R.P., Edwards, B.H., Calculus II. Houghton Miffin Co., 2005.
  • Lipschutz, M. M. Schaum's outline of theory and problems of differential geometry, 1969.
  • J. E. Marsden, A. J. Tromba. Vector Calculus (4th ed.)Freeman, New York 1996.
  • A. López de la Rica, A. Villa. Geometría diferencial. Clagsa, 1997.
  • S.L. Salas, E. Hille. Calculus. Ed. Reverté.
  • R.E. Larson, R.P. Hostetler, BH Edwards. Calculus with Analytic Geometry (6th ed.). DC Heath and Company, New York, NY, 1998.

Additional material