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CURVATURE OF A CURVE ON A SURFACE

Curvature of a curve on a surface. Let S be a simple surface element defined by the one-to-one mapping

            x = x(u, v)

            y = y(u, v)

            z = z(u, v)

of a region R of the uv-plane into xyz-space. Let there be given a curve C: u = u(s), v = v(s) on a surface S. Let

be the position vector to point P on curve C. We wish to find an expression for the curvature k of curve C at point P in terms of the first and second fundamental coefficients E, F, G, L, M, N.

To apply space curve theory we can think of C as merely a curve in space, ignoring the fact that it is lying on a surface. Let be the unit tangent to C at P, be the principal normal of C at P and be the unit surface normal at P. Then we have

Taking the dot product of both sides of 1) with we get

where is the angle between the principal normal vector and the surface normal . Now because , it follows that

Combining 2) and 3) and remembering that we get

or

Thus

Theorem. If two curves passing through a point P on a surface S have the same osculating plane at P, and their common direction at P is not an asymptotic direction, they have the same curvature at P.

This theorem says, in particular, that the curvature at point P of a given curve C on a surface S is equal to the curvature at P of the plane curve in which the osculating plane of C at P cuts S. All curves passing through P with the same osculating plane have the same curvature. Thus we can restrict ourselves to the consideration of curvatures of plane sections of S.

Curvature of a normal section. When = 0 in 5) we have the case of a normal section. Thus it follows that the curvature of a normal section is given by

References.

1. Graustein. Differential Geometry.