Chapter 4: Geometry

The angle between these vectors, «, is given by
Ó× « v ½ ¡ v v ¾
v ½ v ¾ ½ ¡ v ¾
Ê ¾¨
Ó×´ ½ µÓ×´ ¾ µ Ó×´ ½ ¾ µ · ×Ò´ ½ µ ×Ò´ ¾ µ
where ×Ò ¾ ½
¾
¾
¾ ØÒ ½ Ö
½
·Ó×´ ½ µÓ×´ ¾ µ ×Ò ¾ ½ ¾
¾
¡
. (The formula using
is more accurate numerically when È ½ and È ¾ are close.) The great circle
distance between È ½ and È ¾ is then ʨ«.
EXAMPLE The angle between È New York with ´ ½ ¼ Æ ½ ¿ Æ µ and È Beijing
with ´ ¾ ¿¿ Æ ¾ ¾¿ Æ µ is « Æ . Using ʨ ¿the great circle
distance between New York and Beijing is about 11,000 km.
4.20 DIFFERENTIAL GEOMETRY
4.20.1 CURVES
4.20.1.1 De nitions
1. A regular parametric representation of class , ½, is a vector valued
function f Á Ê ¿ , where Á Ê is an interval that satisfies (i) f is of class
(i.e., has continuous th order derivatives), and (ii) f ¼´Øµ ¼, for all Ø ¾ Á.
In terms of a standard basis of Ê ¿ , we write x f´Øµ ´ ½´Øµ, ¾´Øµ ¿´Øµµ,
where the real valued functions , ½ ¾ ¿ are the component functions of f.
2. An allowable change of parameter of class is any function  Á,
where  is an interval and ´Âµ Á, that satisfies ¼´ µ ¼, for all ¾ Â.
3. A regular parametric representation f is equivalent to a regular parametric
representation g if and only if an allowable change of parameter exists
so that ´Á µÁ , and g´ µf´´ µµ, for all ¾ Á .
4. A regular curve of class is an equivalence class of regular parametric
representation under the equivalence relation on the set of regular parametric
representations defined above.
5. The arc length of any regular curve defined by the regular parametric representation
f, with Á ℄, is defined by
Ä ¬
¬
f
¼´Ùµ Ù (4.20.1)
An arc length parameter along is defined by
Ø
× «´Øµ ¦ ¬
¬
f
¼´Ùµ Ù (4.20.2)
© 2003 by CRC Press LLC