




Curvature: Motion in several dimension has two aspects: one is its speed
of motion; the other the shape of the curve it follows. The former is
measured by the speed of motion, ds/dt, where s = (rr)^{1/2}.
The latter is measured at each point by the curvature,,
of the curve traced. This is the magnitude of the rate of change of a unit vector
in the direction of motion with distance s along the curve. Its dimension is
that of an inverse length, and its reciprocal 1/,
is called the radius of curvature.
The curvature is proportional to the component of the acceleration normal to
the velocity with a proportionality constant that can only depend on the speed:
The dimension of av
is L^{2}/T^{3} which implies up
to a constant which constant is 1.
The plane of motion is that normal to av.
The torsion of the curve is the magnitude of the rate of change of a
unit vector in the direction of av
with distance along the curve. There is a formula for it that you might remember
exists but should not remember in detail.