Page 42 - Basic Structured Grid Generation

Classical differential geometry of space-curves 31

2
2 −
from eqn (2.5), which may be integrated to give t = (a +c ) 1 2 s and the intrinsic form

s s cs
x = a cos √ , y = a sin √ , z = √ . (2.6)
2
2
2
a + c 2 a + c 2 a + c 2
√
2
2
Thus the length of this curve in one complete turn about the z-axis is 2π a + c ,
the distance advanced in the direction of the z-axis during this turn being 2πc.
Differentiating eqn (2.1) with respect to t at a point P on the curve gives a vector
˙ r = i˙x + j ˙y + k˙z, (2.7)

2
2
2
s
which is tangential to the curve. The magnitude of this vector is ˙ = ˙ x +˙y +˙z ,
so that a unit tangent vector is
˙ r dr dx dy dz
t = = = i + j + k . (2.8)
˙ s ds ds ds ds
Differentiating the identity
t · t = 1
with respect to s gives
dt
t · = 0,
ds
which shows that the vector dt/ds is perpendicular to t. The direction of this vector at
a point P , given by the unit vector n, is the direction of the principal normal to the
curve at that point, and we write
dt
= κn, (2.9)
ds
where κ is called the curvature at that point. We usually assume that κ is a non-
negative function of s, although there are circumstances where it would be convenient
to let it take negative values in order to allow n to be a continuous vector func-
tion of s.Ifdt/ds = 0, which would be the case everywhere when the curve is a
straight line, then κ = 0, but n is not uniquely defined. The plane containing the
tangent and the principal normal at a point on a curve is called the osculating plane
at that point.

Introducing the shorthand notation d () = () ,wehave
ds
t = r
and

t = r = ix + jy + kz = κn. (2.10)
Hence

2 2 2
κ =|r |= (x ) + (y ) + (z ) . (2.11)
The Chain Rule for differentiation gives

˙ r = r ˙s (2.12)