TOPICS
Search

Principal Curve


A curve alpha on a regular surface M is a principal curve iff the velocity alpha^' always points in a principal direction, i.e.,

 S(alpha^')=kappa_ialpha^',

where S is the shape operator and kappa_i is a principal curvature. If a surface of revolution generated by a plane curve is a regular surface, then the meridians and parallels are principal curves.


Explore with Wolfram|Alpha

References

Gray, A. "Principal Curves" and "The Differential Equation for the Principal Curves of a Surface." §20.1 and 28.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 459-461 and 642-644, 1997.

Referenced on Wolfram|Alpha

Principal Curve

Cite this as:

Weisstein, Eric W. "Principal Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrincipalCurve.html

Subject classifications