Curves on Surfaces

Differential Geometry · Axiom Academy

Understanding how curves live on surfaces and how to measure their geometric properties A curve on a surface is obtained by composing the surface parameterization with curve parameters. If X(u,v) parameterizes a surface, then a curve on that surface can be written as: Here, u(t) and v(t) are smooth functions that trace a path in the parameter domain, and their composition with X produces a curve lying entirely on the surface. Two special families of curves on any surface are the coordinate curves , obtained by holding one parameter constant: The u-curves are obtained by fixing v and varying u, while v-curves are obtained by fixing u and varying v. These curves form a coordinate grid on the surface and are fundamental to understanding its local structure. To find the velocity vector of a curve on a surface, we apply the chain rule to the composition. The velocity is expressed in terms of the surface's tangent vectors: Here, X u and X v are the partial derivatives of the surface parameterization (tangent vectors along coordinate curves). The velocity depends on both the rates of change u'(t) and v'(t), and the surface geometry. 4. Arc Length Using the First Fundamental Form The first fundamental form encodes how to measure lengths and angles on a surface. For a curve on a surface, the arc length is computed as:

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