View source for Curvature line

A line on a surface at each point of which the tangent has one of the principal directions. The curvature lines are defined by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027340/c0273401.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027340/c0273402.png" /> are the coefficients of the first fundamental form of the surface, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027340/c0273403.png" /> those of the second fundamental form. The normals to the surface along curvature lines form a developable surface. The curvature lines on a surface of revolution are the meridians and the parallels of latitude. The curvature lines on a developable surface are its generators (which are straight lines) and the lines orthogonal to them.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J. Struik,   "Differential geometry" , Addison-Wesley  (1950)</TD></TR></table>