Page 44 - Basic Structured Grid Generation

Classical differential geometry of space-curves 33

Normal
plane
b
Rectifying
n plane
t
Osculating P
plane

Fig. 2.1 Osculating, normal, and rectifying planes at a point P on a curve.

with respect to s gives
db dt dn dn dn
= × n + t × = κn × n + t × = t × , (2.20)
ds ds ds ds ds
since n × n = 0. We deduce that db/ds is a vector orthogonal to t. Moreover, since b
is defined as a unit vector, with b · b = 1, it follows that b · db/ds = 0, so that db/ds
is also orthogonal to b.
Hence db/ds can only be parallel to n, and we can write
db
=−τn,
ds
where the scalar τ is called the torsion of the curve at a point. Not all writers follow
the convention of including a negative sign in this equation; the significance of the sign
can be shown by considering the special case of the circular helix given by eqn (2.6).
Here we obtain

1 s s

t = r = √ −ai sin √ + aj cos √ + ck ,
2
2
2
a + c 2 a + c 2 a + c 2
1 s s

t = r = κn = −ai cos √ − aj sin √ ,
2
2
a + c 2 a + c 2 a + c 2
2
so that
a
κ = (2.21)
2
a + c 2
and
s s
n =−i cos √ − j sin √ .
2
2
a + c 2 a + c 2
Hence
1 s s
b = t × n = √ ci sin √ − cj cos √ + ak
2
2
2
a + c 2 a + c 2 a + c 2
and
c s s c

b = i cos √ + j sin √ =− n.
2
2
2
a + c 2 a + c 2 a + c 2 a + c 2
2