Chapter 4: Geometry

3. Curvature: k ´ ¾ · ¾ µ ½
4. Principal normal unit vector: n k k ´ Ó× Ø ×Ò Ø ¼µ
5. Unit tangent vector: t x¼
x ¼ ´¾ · ¾ µ ½ ¾ ´ ×Ò Ø Ó× Ø µ
6. Unit binormal vector:
b t ¢ n ´ ¾ · ¾ µ ½ ¾ ´ ×Ò Ø Ó× Ø µ
b Ø b
× Ø ´¾ · ¾ µ ½´Ó× Ø ×Ò Ø ¼µ
7. Torsion: n ¡ b ´ ¾ · ¾ µ ½
The values of and can be verified using the formulae in (4.20.3). The sign of
(the invariant) determines whether the helix is right handed, ¼, or left handed,
¼.
4.20.2 SURFACES
4.20.2.1 Definitions
1. A coordinate patch of class , ½ on a surface Ë Ê ¿ is a vector valued
function Í Ë, where Í Ê ¾ is an open set, that satisfies (i) f is
class on Í, (ii) Ù´Ù f
f
Úµ ¢ Ú´Ù Úµ ¼, for all ´Ù Úµ ¾ Í, and (iii) f is
one-to-one and bi-continuous on Í.
2. In terms of a standard basis of Ê ¿ we write x f´Ù Úµ ´ ½´Ù Úµ, ¾´Ù Úµ,
¿´Ù Úµµ, where the real valued functions ½ ¾ ¿ are the component functions
of f. The notation x ½ x Ù f
Ù , x ¾ x Ú f
Ú , Ù½ Ù Ù ¾ Ú, is
frequently used.
3. A Monge patch is a coordinate patch where f has the form f´Ù Úµ
´Ù Ú ´Ù Úµµ, where is a real valued function of class .
4. The Ù-parameter curves Ú Ú ¼ on Ë are the images of the lines Ú Ú ¼ in Í.
They are parametrically represented by x f´Ù Ú ¼ µ. The Ú-parameter curves
Ù Ù ¼ are defined similarly.
5. An allowable parameter transformation of class is a one-to-one function
Í Î , where Í Î Ê ¾ are open, that satisfies
½
Ø Ù ´Ù Úµ
½
Ú ´Ù Úµ
¾
Ù ´Ù Úµ
¾
Ú ´Ù Úµ ¼ (4.20.8)
for all ´Ù Úµ ¾ Í, where the real valued functions, ½ and ¾ , defined by
´Ù Úµ ´ ½´Ù Úµ, ¾´Ù Úµµ are the component functions of . One may
also write the parameter transformation as Ù ½ ½´Ù½ Ù ¾ µ, Ù ¾ ¾´Ù½ Ù ¾ µ.
6. A local property of surface Ë is any property of a coordinate patch that is
invariant under any allowable parameter transformation.
7. Let f define a coordinate patch on a surface Ë. The following quantities may
be defined at each point x f´Ù Úµ on the patch:
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