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Outline: Curves on Surfaces 1. The Darboux Frame Let S be a regular oriented surface, and let ~x(t) be a curve on S. The Darboux frame for ~x(t) at a point p consists of the following three vectors: ~ to the surface at p. 1. The unit normal vector N 2. The unit tangent vector T~ the curve at p. ~ =N ~ × T~ , which is tangent to the surface and normal to the curve. 3. The third vector U ~,N ~ } are a right-handed frame. When imagining a surface, we usually The three vectors {T~ , U ~ as pointing towards us, in which case U ~ is 90◦ counterclockwise from T~ . think of N 2. Geodesic and Normal Curvature Let S be a regular oriented surface, and let ~x(t) be a curve on S. The geodesic curvature κg (t) and normal curvature κn (t) of ~x(t) are defined by the formula dT~ ~ + κg U ~. = κn N ds That is, ~ ~ · dT κn = N ds Note: Recall that and ~ ~ · dT . κg = U ds dT~ dT~ = κP~ for a space curve. This is often the easiest way to compute . ds ds 3. Normal Curvature and the Second Fundamental Form Normal curvature is related to the second fundamental form. Specifically, if ~x(t) is any curve on a surface S, then II(T~ ) = κn at each point on the curve, where T~ is the unit tangent vector to ~x(t), and κn is the normal curvature. In particular, if ~x(t) is a curve whose unit tangent vector T~ points in one of the principle directions at a point p, then the normal curvature κn at p is equal to one of the principle curvatures of S at p. For example, on a surface of revolution, the principle curvatures are simply the normal curvatures of the usual coordinate lines.