That is, principal curvatures have no intrinsic sign. Intersecting $S$ with normal planes at $O$ means looking at the curves Principal curvatures depend on a continuous choice of unit normal field changing the sign of the unit normal field changes the signs of the principal curvatures.
The other one is normal to this.Ĭonsider a surface $S: z=f(x,y)$ touching the $(x,y)$-plane at the origin $O$. It's really clear that the Dupin indicatrix is reflection-symmetric along the plane determined by the longitude and the axis of revolution, thus one direction of principal curvature is determined. According to that book, there's no differential geometric tool (such as the second fundamental form, etc) introduced, so I need an intuitive and elementary explanation for that fact.Īfter googling, I found an explanation which depends on the knowledge of Dupin indicatrix, which is intuitive and computation-free, though not that elementary. It's said that the proposition follows clearly from symmetry.
It's claimed (say, on Stillwell's Geometry of Surfaces) that for a pseudosphere (or generally a surface of revolution) the extremal curvatures are obtained when the normal section coincides or is perpendicular to the plane determined by the longitude and the axis of revolution. 2R, (Solution)This surface admits an obvious parametrization, given by (u v) : (u v u2+ v2) for all (u v)2R2: To compute the Gaussian curvature using (2) we'll need to ndE F G L M andN. The principal curvatures of the surface at a point is defined as the maximal and the minimal curvature among all normal sections.
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