Normal plane (geometry)

Summary

Saddle surface with normal planes in directions of principal curvatures.

A normal plane is any plane containing the normal vector of a surface at a particular point.

The normal plane also refers to the plane that is perpendicular to the tangent vector of a space curve; (this plane also contains the normal vector) see Frenet–Serret formulas.

Normal section

The normal section of a surface at a particular point is the curve produced by the intersection of that surface with a normal plane.[1][2][3]

The curvature of the normal section is called the normal curvature.

If the surface is bow or cylinder shaped, the maximum and the minimum of these curvatures are the principal curvatures.

If the surface is saddle shaped the maxima of both sides are the principal curvatures.

The product of the principal curvatures is the Gaussian curvature of the surface (negative for saddle shaped surfaces).

The mean of the principal curvatures is the mean curvature of the surface; if (and only if) the mean curvature is zero, the surface is called a minimal surface.

See also

References

  1. ^ Ruane, Irving Adler, with diagrams by Ruth Adler ; introduction to the Dover edition by Peter (2012). A new look at geometry (Dover ed.). Mineola, N.Y.: Dover Publications. p. 273. ISBN 978-0486498515.
  2. ^ Irving Adler (2013-08-30). A New Look at Geometry. p. 273. ISBN 9780486320496. Retrieved 2016-04-01.
  3. ^ Alfred Gray (1997-12-29). Modern Differential Geometry of Curves and Surfaces with Mathematica, Second ... p. 365. ISBN 9780849371646. Retrieved 2016-04-01.