Page 41 - Basic Structured Grid Generation

2

Classical differential geometry

of space-curves

2.1 Vector approach

3
We consider smooth curves in E specified in terms of rectangular cartesian co-
ordinates x, y, z (or y 1 ,y 2 ,y 3 ). Such curves are generated by three smooth functions
of a single real parameter, say t:
x = x(t), y = y(t), z = z(t),

so that the position vector r of points on the curve relative to some origin O is given by
r = r(t) = x(t)i + y(t)j + z(t)k. (2.1)

A frequently-quoted example is the circular helix, which could be specified by
x = a cos t, y = a sin t, z = ct, (2.2)

where a and c are constants. This curve lies on the surface of the circular cylinder
2
2
2
x + y = a , and makes complete turns about the z-axis as t increases by 2π.
Distance along a curve is measured by the arc-length parameter s, where differentials
of arc-length ds satisfy the equation
2
2
2
2
ds = dx + dy + dz = dr · dr, (2.3)
or, in terms of derivatives with respect to the parameter t,
2 2 2 2
˙ s =˙x +˙y +˙z = ˙ r · ˙ r, (2.4)
where the dot denotes d ,or
dt

2
2
2
ds = ˙ x +˙y +˙z dt. (2.5)
An intrinsic definition of a space-curve is given when s is used as a parameter in
eqn (2.1). For example, the circular helix (2.2) gives

2 2 2 2 2
ds = (−a sin t) + (a cos t) + c dt = a + c dt