14.4. Normal Curvature and the Second Fun- damental Form

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Now, if C: t ↦→ X(u(t), v(t)) is a curve on a surface X, assuming that C is parameterized by arc length, which implies that (s ′ ) 2 = E(u ′ ) 2 + 2F u ′ v ′ + G(v ′ ) 2 = 1, we have X ′ (s) = X u u ′ + X v v ′ , X ′′ (s) = κ −→ n , and −→ t = X u u ′ + X v v ′ is indeed a unit tangent vector to the curve and to the surface, but −→ n is the principal normal to the curve, and thus it is not necessarily orthogonal to the tangent plane T p (X) at p = X(u(t), v(t)). Thus, if we intend to study how the curvature κ varies as the curve C passing through p changes, the Frenet frame ( −→ t , −→ n , −→ b ) associated with the curve C is not really adequate, since both −→ n and −→ b will vary with C (and −→ n is undefined when κ = 0). Normal Curvature . . . Geodesic Curvature . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 686 of 711 Go Back Full Screen Close Quit
Thus, it is better to pick a frame associated with the normal to the surface at p, and we pick the frame ( −→ t , −→ n g , N) defined as follows.: Definition 14.4.1 Given a surface X, given any curve C: t ↦→ X(u(t), v(t)) on X, for any point p on X, the orthonormal frame ( −→ t , −→ n g , N) is defined such that Normal Curvature . . . Geodesic Curvature . . . Home Page −→ t = Xu u ′ + X v v ′ , N = X u × X v ‖X u × X v ‖ , −→ ng = N × −→ t , where N is the normal vector to the surface X at p. The vector −→ ng is called the geodesic normal vector (for reasons that will become clear later). Title Page ◭◭ ◮◮ ◭ ◮ Page 687 of 711 Go Back Full Screen Close For simplicity of notation, we will often drop arrows above vectors if no confusion may arise. Quit
Page 1 and 2: 14.4. Normal Curvature and the Seco
Page 3: The vector −→ t is the unit tan
Page 7 and 8: Using the abbreviations X uu = ∂2
Page 9 and 10: Thus, it is clear that the normal c
Page 11 and 12: Some authors (including Gauss himse
Page 13 and 14: Properties of the surface expressib
Page 15 and 16: Furthermore, the principal normal o
Page 17 and 18: We found that the tangential part o
Page 19 and 20: Definition 14.4.4 Given a surface X
Page 21 and 22: However, Normal Curvature . . . κ
Page 23 and 24: Doing so, and remembering that κ
Page 25 and 26: Finally, we get and A = u ′′ 1
Page 27 and 28: Looking at the formulae [α β; γ]
Page 29: From all this, we get 2[α β; γ]

14.4. Normal Curvature and the Second Fun- damental Form

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