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14.6. Principal Curvatures, Gaussian Cur- Vature, Mean

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14.6. Principal Curvatures, Gaussian Cur- vature, Mean Curvature Principal . We will now study how the normal curvature at a point varies when a unit tangent vector varies. In general, we will see that the normal curvature has a max- Home Page imum value κ1 and a minimum value κ2, and that the corre- Title Page sponding directions are orthogonal. This was shown by Euler in 1760. JJ II J I The quantity K = κ1κ2 called the Gaussian curvature and the Page 712 of 733 quantity H = (κ1 + κ2)/2 called the mean curvature, play a Go Back very important role in the theory of surfaces. Full Screen We will compute H and K in terms of the first and the sec- Close ond fundamental form. We also classify points on a surface Quit according to the value and sign of the Gaussian curvature. Recall that given a surface X and some point p on X, the vectors Xu,Xv form a basis of the tangent space Tp(X). Principal . −→ Given a unit vector t = Xux + Xvy, the normal curvature is given by −→ κ ( t ) = Lx2 + 2Mxy + Ny2, N Home Page since Ex2 + 2F xy + Gy2 = 1. Title Page JJ II Usually, (Xu,Xv) is not an orthonormal frame, and it is useful to replace the frame (Xu,Xv) with an orthonormal frame. J I −→ −→ Page 713 of 733 One verifies easily that the frame ( e1 , e2 ) defined such that Go Back −→ Xu −→ EXv − FXu Full Screen e1 = √ , e2 = . E pE(EG − F 2) Close is indeed an orthonormal frame. Quit With respect to this frame, every unit vector can be written −→ −→ −→ −→ −→ as t = cos θ e1 + sin θ e2 , and expressing ( e1 , e2 ) in terms of Principal . Xu and Xv, we have √ −→ w cos θ − F sin θ E sin θ t = √ Xu + Xv, w E w Home Page √ where w = EG − F 2. Title Page −→ JJ II We can now compute κN ( t ), and we get J I Page 714 of 733 −→ w cos θ − F sin θ2 κN ( t ) = L √ Go Back w E (w cos θ − F sin θ) sin θ E sin2 θ Full Screen + 2M + N . w2 w2 Close Quit We leave as an exercise to show that the above expression can be written as Principal . −→ κN ( t ) = H + A cos 2θ + B sin 2θ, where Home Page GL − 2FM + EN H = , Title Page 2(EG − F 2) L(EG − 2F 2) + 2EF M − E2N JJ II A = 2 , 2E(EG − F ) J I EM − FL B = √ . Page 715 of 733 E EG − F 2 Go Back Full Screen Close Quit √ Letting C = A2 + B2, unless A = B = 0, the function Principal . f(θ) = H + A cos 2θ + B sin 2θ has a maximum κ1 = H + C for the angles θ0 and θ0 + π, and a minimum κ = H − C for the angles θ + π and θ + 3π , 2 0 2 0 2 Home Page A B where cos 2θ0 = C and sin 2θ0 = C . Title Page The curvatures κ1 and κ2 play a major role in surface theory. JJ II J I Page 716 of 733 Go Back Full Screen Close Quit Principal . Definition 14.6.1 Given a surface X, for any√ point p on X, letting A, B, H be defined as above, and C = A2 + B2, un- less A = B = 0, the normal curvature κN at p takes a max- imum value κ1 and and a minimum value κ2 called principal curvatures at p, where κ1 = H +C and κ2 = H −C. The direc- Home Page tions of the corresponding unit vectors are called the principal directions at p. Title Page JJ II The average H = κ1+κ2 of the principal curvatures is called the 2 J I mean curvature, and the product K = κ1κ2 of the principal curvatures is called the total curvature, or Gaussian curvature. Page 717 of 733 Go Back Full Screen Close Quit π Observe that the principal directions θ0 and θ0 + 2 correspond- ing to κ1 and κ2 are orthogonal. Note that Principal . 2 2 2 2 2 K = κ1κ2 = (H − C)(H + C) = H − C = H − (A + B ). After some laborious calculations, we get the following (fa- Home Page mous) formulae for the mean curvature and the Gaussian cur- vature: Title Page JJ II GL − 2FM + EN H = , J I 2(EG − F 2) Page 718 of 733 LN − M 2 K = . EG − F 2 Go Back Full Screen Close Quit We showed that the normal curvature κN can be expressed as Principal . κN (θ) = H + A cos 2θ + B sin 2θ −→ −→ over the orthonormal frame ( e1 , e2 ). We also showed that the angle θ such that cos 2θ = A and 0 0 C Home Page B sin 2θ0 = C , plays a special role. Title Page Indeed, it determines one of the principal directions. JJ II −→ −→ −→ −→ J I If we rotate the basis ( e1 , e2 ) and pick a frame (f1 , f2 ) cor- Page 719 of 733 responding to the principal directions, we obtain a particu- Go Back larly nice formula for κN . Indeed, since A = C cos 2θ0 and B = C sin 2θ0, letting ϕ = θ − θ0, we get Full Screen 2 2 Close κN (θ) = κ1 cos ϕ + κ2 sin ϕ. Quit −→ Thus, for any unit vector t expressed as −→ −→ −→ Principal . t = cos ϕf1 + sin ϕf2 with respect to an orthonormal frame corresponding to the principal directions, the normal curvature κN (ϕ) is given by Euler’s formula (1760): Home Page Title Page 2 2 κN (ϕ) = κ1 cos ϕ + κ2 sin ϕ. JJ II J I Recalling that EG − F 2 is always strictly positive, we can Page 720 of 733 classify the points on the surface depending on the value of the Gaussian curvature K, and on the values of the principal Go Back curvatures κ1 and κ2 (or H). Full Screen Close Quit Definition 14.6.2 Given a surface X, a point p on X belongs Principal . to one of the following categories: (1) Elliptic if LN − M 2 > 0, or equivalently K > 0. 2 (2) Hyperbolic if LN − M < 0, or equivalently K < 0. Home Page 2 2 2 2 (3) Parabolic if LN − M = 0 and L + M + N > 0, or Title Page equivalently K = κ1κ2 = 0 but either κ1 6= 0 or κ2 6= 0. JJ II (4) Planar if L = M = N = 0, or equivalently κ1 = κ2 = 0. J I Page 721 of 733 Furthermore, a point p is an umbilical point (or umbilic) if Go Back K > 0 and κ1 = κ2. Full Screen Close Quit Note that some authors allow a planar point to be an umbilical point, but we don’t. Principal . At an elliptic point, both principal curvatures are nonnull and have the same sign. For example, most points on an ellipsoid are elliptic. Home Page At a hyperbolic point, the principal curvatures have opposite Title Page signs. For example, all points on the catenoid are hyperbolic. JJ II At a parabolic point, one of the two principal curvatures is J I zero, but not both. This is equivalent to K = 0 and H 6= 0. Page 722 of 733 Points on a cylinder are parabolic. Go Back Full Screen At a planar point, κ1 = κ2 = 0. This is equivalent to K = H = 0. Points on a plane are all planar points! On a monkey Close saddle, there is a planar point. The principal directions at Quit that point are undefined. -11 x y -0.5 0.5 0 0.5 0 1 -0.5 Principal . -1 2 1 Home Page z 0 Title Page JJ II -1 J I -2 Page 723 of 733 Go Back Full Screen Figure 14.3: A monkey saddle Close Quit For an umbilical point, we have κ1 = κ2 6= 0. This can only happen when√H − C = H + C, which implies 2 2 that C = 0, and since C = A + B , we have A = B = 0. Principal . Thus, for an umbilical point, K = H2. In this case, the function κN is constant, and the principal directions are undefined. All points on a sphere are umbilics. Home Page A general ellipsoid (a, b, c pairwise distinct) has four umbilics. Title Page It can be shown that a connected surface consisting only of JJ II umbilical points is contained in a sphere. J I Page 724 of 733 It can also be shown that a connected surface consisting only Go Back of planar points is contained in a plane. Full Screen Close Quit A surface can contain at the same time elliptic points, parabolic points, and hyperbolic points. This is the case of a torus. Principal . The parabolic points are on two circles also contained in two tangent planes to the torus (the two horizontal planes touching the top and the bottom of the torus on the following picture). Home Page The elliptic points are on the outside part of the torus (with Title Page normal facing outward), delimited by the two parabolic circles. JJ II The hyperbolic points are on the inside part of the torus (with J I normal facing inward). Page 725 of 733 Go Back Full Screen Close Quit Principal . Home Page 2 Title Page 1 JJ II 0 0.75z y 0.5 J I 0.25 0 0 Page 726 of 733 1 -2 Go Back x 2 3 Full Screen Close Figure 14.4: Portion of torus Quit The normal curvature 2 2 Principal . κN (Xux + Xvy) = Lx + 2Mxy + Ny will vanish for some tangent vector (x, y) 6= (0, 0) iff M 2 − LN ≥ 0.
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