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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 f 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
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 f = 0 in 5) we have the case of a normal section. Thus it follows that the curvature of a normal section is given by
1. Graustein. Differential Geometry.
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