Page 218 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 18: Theory of Space Curves

From our earlier formulas, we have Notes
x’(0) = s’(0)T(0),

x”(0) = s”(0)T(0) + k(0)s’(0) N(0),
2
x”’(0) = s”’(0)T(0) + s”(0)s’(0)k(0)N(0) + (k(0)s’(0) )’N(0) + k(0)s’(0) (–k(0)s’(0)T(0) + w(0)s’(0)b(0)).
2
2
Substituting these equations in the Taylor series expression, we find:
 t 2 t 3 
3
2
x(t) = x(0) + ts'(0)  s"(0)  ["'(0) k(0) s'(0) ] ... T(0)




 2 6 
t  2 2 t 3 2 



+  k(0)s'(0)  [s"(0)s'(0)k(0) (k(0)s'(0) ] ... N(0)
 2 6 
t  3 3 
+  k(0)w(0)s'(0)  ... b(0).

 6 
If the curve is parametrized by arclength, this representation is much simpler:
 k(0) 2   k(0) k'(0)   k(0)w(0) 
2
3
3
3
x(s) = x(0)   s  s  ... T(0)   s  s  ... N(0)   s  ... b(0).



 6   2 6   6 
Relative to the Frenet frame, the plane with equation x = 0 is the normal plane; the plane with
1
x = 0 is the rectifying plane, and the plane with x = 0 is the osculating plane. These planes are
3
2
orthogonal respectively to the unit tangent vector, the principal normal vector, and the binormal
vector of the curve.
18.7 Plane Curves and a Theorem on Turning Tangents
The general theory of curves developed above applies to plane curves. In the latter case there
are, however, special features which will be important to bring out. We suppose our plane to be
oriented. In the plane, a vector has two components and a frame consists of an origin and an
ordered set of two mutually perpendicular unit vectors forming a right-handed system. To an
oriented curve C defined by x(s) the Frenet frame at s consists of the origin x(s), the unit tangent
vector T(s) and the unit normal vector N(s). Unlike the case of space curves, this Frenet frame is
uniquely determined, under the assumption that both the plane and the curve are oriented.
The Frenet formulas are
x’ = T,

T’ = kN, (1)
N’ = –kT .
The curvature k(s) is defined with sign. It changes its sign when the orientation of the plane or
the curve is reversed.
The Frenet formulas in (1) can be written more explicitly. Let
x(s) = (x (s), x (s)) ...(2)
1
2
Then,
'
'
T(s) = (x (s),x (s)),
2
1

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