Chapter 4: Geometry

The choice of sign is arbitrary and is any number in Á .
6. A natural representation of class of the regular curve defined by the regular
parametric representation f is defined by g´×µ f´« ½´×µµ, for all × ¾ ¼Ä℄.
7. A property of a regular curve is any property of a regular parametric representation
representing which is invariant under any allowable change of
parameter.
8. Let g be a natural representation of a regular curve . The following quantities
may be defined at each point x g´×µ of :
Binormal line y b´×µ ·x
Curvature
´×µ n´×µ ¡ k´×µ
Curvature vector k´×µ t´×µ
Moving trihedron t´×µ n´×µ b´×µ
Normal plane ´y xµ ¡ t´×µ ¼
Osculating plane ´y xµ ¡ b´×µ ¼
Osculating sphere ´y cµ ¡ ´y cµ Ö ¾ where
c x · ´×µn´×µ ´´×µ´ ¾´×µ ´×µµµb´×µ
and Ö ¾ ¾´×µ ·¾´×µ´´×µ ¾´×µµ
Principal normal line y n´×µ ·x
Principal normal unit
vector
n´×µ ¦k´×µk´×µ, for k´×µ ¼defined to be
continuous along
Radius of curvature ´×µ ½ ´×µ, when ´×µ ¼
Rectifying plane
´y xµ ¡ n´×µ ¼
Tangent line y t´×µ ·x
Torsion
Unit binormal vector
Unit tangent vector
´×µ n´×µ ¡ b´×µ
b´×µ t´×µ ¢ n´×µ
t´×µ g´×µ with
g´×µ g
×
4.20.1.2 Results
The arc length Ä and the arc length parameter × of any regular parametric representation
f are invariant under any allowable change of parameter. Thus, Ä is a property
of the regular curve defined by f.
¬
The arc length parameter satisfies × «¼´Øµ ¦ ¬
f
¼´Øµ ¬, which implies that
¬ ¬ Ø
f
¼´×µ ½, if and only if Ø is an arc length parameter. Thus, arc length parameters
are uniquely determined up to the transformation × × ¦× · × ¼ , where × ¼ is any
constant.
The curvature, torsion, tangent line, normal plane, principal normal line, rectifying
plane, binormal line, and osculating plane are properties of the regular curve
defined by any regular parametric representation f.
If x f´Øµ is any regular representation of a regular curve , the following
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