# Smooth Conjugate Net vs. Curvature-Line Parametrization

so I was wondering what a smooth conjugate net exactly is, intuitively? Also, what exactly is a curvature-line parametrization? What would it mean that a smooth conjugate net is orthogonal? Why is it then a curvature-line parametrization?

Best

Fluffs

This figure is alone quite informative:

Kambe, Tsutomu. Geometrical Theory of Dynamical Systems and Fluid Flows. No. 23. World Scientific, 2004.

A second source, with spectacular images:

Liu, Yang, Weiwei Xu, Jun Wang, Lifeng Zhu, Baining Guo, Falai Chen, and Guoping Wang. "General planar quadrilateral mesh design using conjugate direction field." In ACM Transactions on Graphics (TOG), vol. 30, no. 6, p. 140. ACM, 2011. (PDF download.)

• So a coordinate net is the same as a conjugate net? The curvature-line was defined, s.t. its tangents are in the direction of the principal lines, so are the principal lines tangents to the curvature lines? I´m really confused about this. – Fluffy12 Jul 8 '15 at 13:07
• @Fluffy12: The term "coordinate net" is generic, whereas "conjugate net" is a coordinate net with every point crossed by conjugate directions. – Joseph O'Rourke Jul 8 '15 at 13:13
• @Fluffy12: And "principle directions" (along principle curvatures) are a special case of conjugate directions. – Joseph O'Rourke Jul 8 '15 at 13:22