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Theory of Curves and Surfaces

Yoshihiko Tazawa
Organization: Tokyo Denki University
Department: Natural Sciences
Education level


This course gives an introduction to classical differential geometry of curves and surfaces. Although the basic structure is traditional, the numerical and experimental approach is adopted in various examples, making use of Mathematica. Therefore this is also a Mathematica literacy through basic differential geometry.

Theory of Curves and Surfaces: An Introduction to Classical Differential Geometry by Mathematica (in Japanese) by Tazawa Pearson Education, 309 pages with CD-ROM

The theory of curves and surfaces was established long ago. Yet applying the general theory to individual objects is not easy. For instance, integrating the curvature over a curve or constructing a curve with assigned curvature can be very difficult even in the simplest cases. This is because it is not possible in general to solve differential equations explicitly. Hence, the examples appearing in this field have been confined to a small group of calculable objects. But, numerically, Mathematica does these calculations easily, and this makes it possible for us to deal with a wide range of examples, as the reader shall recognize throughout this book.

The existing best book on differential geometry by Mathematica was written by Alfred Gray. But the approach of the present course is a little different, and I believe it is sufficiently unique, at least at this moment. In short, we use Mathematica not only to calculate geometric quantities or to draw geometric objects, but also to visualize the basic notions of differential geometry and to perform experiments in geometry making use of Mathematica graphics, animations in particular.

  • Vector functions
  • Mathematica Literacy

Plane Curves:
  • Plane Curves
  • Well Known Plane Curves
  • Generic Plane Curves
  • Arclength and the Moving Frame
  • Curvature
  • The Fundamental Thereom of Plane Curves
  • Global Properties of Plane Curves
  • Plane Curves by Mathematica

Space Curves:
  • Space Curves
  • The Moving Frame, Curvature, and Torsion
  • Geometric Meanings of Curvature and Torsion
  • The Fundamental Theorem of Space Curves
  • Global Properties of Space Curves
  • Space Curves by Mathematica

  • Preliminary Observations
  • Surfaces
  • The Tangent Plane and the Unit Normal Vector
  • The First Fundamental Quantities
  • The Second Fundamental Quantities
  • Mean Curvature and Gaussian Curvature
  • The Structure Equations
  • The Gauss Map
  • Minimal Surfaces
  • Geodesics
  • Surfaces by Mathematica

Visualizing a Flat Torus

*Mathematics > Calculus and Analysis > Differential Geometry