14.4. Normal Curvature and the Second Fun- damental Form

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Therefore, we showed that κ −→ n = κ N N + κ g −→ ng , Normal Curvature . . . Geodesic Curvature . . . where κ N = L(u ′ ) 2 + 2Mu ′ v ′ + N(v ′ ) 2 and Home Page κ g −→ ng = (N × (X uu (u ′ ) 2 + 2X uv u ′ v ′ + X vv (v ′ ) 2 )) × N +X u u ′′ + X v v ′′ . Title Page ◭◭ ◮◮ ◭ ◮ The term κ g −→ ng is worth an official definition. Page 700 of 711 Go Back Full Screen Close Quit
Definition 14.4.4 Given a surface X, given any curve C: t ↦→ X(u(t), v(t)) on X, for any point p on X, the quantity κ g appearing in the expression Normal Curvature . . . Geodesic Curvature . . . κ −→ n = κ N N + κ g −→ ng giving the acceleration vector of X at p is called the geodesic curvature of C at p. In the next section, we give an expression for κ g −→ ng in terms of the basis (X u , X v ). Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 701 of 711 Go Back Full Screen Close Quit
Page 1 and 2: 14.4. Normal Curvature and the Seco
Page 3 and 4: The vector −→ t is the unit tan
Page 5 and 6: Thus, it is better to pick a frame
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: We found that the tangential part o
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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