Solutions of Exercises of Introduction to Differential Geometry of Space Curves and Surfaces by Taha Sochi

Author:Taha Sochi [Sochi, Taha]
Language: eng
Format: epub
Publisher: KDP
Published: 2019-02-01T04:00:00+00:00

which is the required result (with minor notational differences).

Show that in any two orthogonal directions at a given point P on a sufficiently smooth surface, the sum of the normal curvatures corresponding to these directions at P is constant.

Answer: This is obviously true at umbilical points. So, we need to show that this is also valid at non-umbilical points.

According to Euler theorem (see Exercise 46 [4↓]) the normal curvature κn at a given point P on a surface of class C2 in a given direction can be expressed as a combination of the principal curvatures, κ1 and κ2, at P as: κn = κ1cos2θ + κ2sin2θ

where θ is the angle between the principal direction of κ1 at P and the given direction. Now, if the angle between the principal direction of κ1 at a given point and the given direction is θ, then the angle between the principal direction of κ1 at that point and the orthogonal direction to the given direction is (π ⁄ 2) − θ. Hence, if we label the normal curvature in the given direction as κn1 and in the orthogonal direction as κn2 then we have: κn1 + κn2 =

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