Let u_p be a unit tangent vector of a regular surface M subset R^3. Then the normal curvature of M in the direction u_p is κ(u_p) = S(u_p)·u_p, where S is the shape operator. Let M subset R^3 be a regular surface, p element M, x be an injective regular patch of M with p = x(u_0, v_0), and v_p = a x_u(u_0, v_0) + b x_v(u_0, v_0), where v_p element M_p. Then the normal curvature in the direction v_p is κ(v p) = (e a^2 + 2f a b + g b^2)/(E a^2 + 2F a b + G b^2), where E, F, and G are the coefficients of the first fundamental form and e, f, and g are the coefficients of the second fundamental form.
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