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Complex Analysis and Differential Geometry




                    Notes          and if k  < 0 and k  < 0, we get the ellipse of equation
                                         1       2
                                                                   x 2    y 2  1.
                                                                    1     1  
                                                                        
                                                                    k 1    k 2
                                   When k  = k , i.e. an umbilical point, the Dupin indicatrix is a circle.
                                         1
                                             2
                                   (2) If LN – M  = 0 and L  + M  + N  > 0, then k  = 0 or k  = 0, but not both.
                                                     2
                                             2
                                                              2
                                                          2
                                                                       1
                                                                              2
                                   This is the case of a parabolic point.
                                   In this case, the Dupin indicatrix degenerates to two parallel lines, since the equation is either
                                                                     k x  = ±1
                                                                        2
                                                                      1
                                   or
                                                                     k y  = ±1.
                                                                       2
                                                                      2
                                   (3) If LN – M  < 0 then k  and k  have different signs. This is the case of a hyperbolic point.
                                             2
                                                            2
                                                      1
                                   In this case, the Dupin indicatrix consists of the two hyperbolae of equations
                                                                   x 2   y 2  1,
                                                                    1    1  
                                                                    k 1  k 2
                                   if k  > 0 and k  < 0, or of equation
                                              2
                                     1
                                                                    x 2   y 2
                                                                             1,
                                                                      1   1
                                                                     k 1   k 2

                                   if k  < 0 and k  > 0.
                                     1
                                              2
                                   These hyperbolae share the same asymptotes, which are the asymptotic directions as defined,
                                   and are given by the equation
                                                                Lx  + 2Mxy + Ny  = 0.
                                                                              2
                                                                  2
                                   Therefore, analyzing the shape of the Dupin indicatrix leads us to rediscover the classification of
                                   points on a surface in terms of the principal curvatures.
                                   It also lends some intuition to the meaning of the words elliptic, hyperbolic, and parabolic (the
                                   last one being a bit misleading).
                                   The analysis of (u, v) also shows that in the elliptic case, in a small neighborhood of X(u, v), all
                                   points of X are on the same side of the tangent plane. This is like being on the top of a round hill.
                                   In the hyperbolic case, in a small neighborhood of X(u, v), there are points of X on both sides of
                                   the tangent plane. This is a saddle point, or a valley (or mountain pass).

                                   30.5 Clairaut’s Theorem

                                   Clairaut’s theorem, published in 1743 by Alexis Claude Clairaut in his Théorie de la figure de la
                                   terre, tirée des principes de l’hydrostatique, synthesized physical and  geodetic evidence that  the
                                   Earth is an oblate rotational ellipsoid. It is a general mathematical law applying to spheroids of
                                   revolution. It was initially used to relate the gravity at any point on the Earth’s surface to the




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