Page 234 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 19: Serret-Frenet Formulae

19.3 Keywords Notes

Moving frame along f: Let f: ]a, b[  E (or f : [a, b]  E ) be a curve of class C , with p  n. A family
n
p
n
(e (t), . . ., e (t)) of orthonormal frames, where each e : ]a, b[  E is C continuous for i = 1, . . .,
n
n–i
n
1
i
n – 1 and en is C -continuous, is called a moving frame along f.
1
Frenet n-frame or Frenet frame: A moving frame (e (t), . . ., e (t)) along f so that for every k, with
1
n
1  k  n, the kth derivative f (t) of f(t) is a linear combination of (e (t), . . ., e (t)) for every t  ]a,
(k)
1
k
b[, is called a Frenet n-frame or Frenet frame.
Linear isometry: Let f: ]a, b[  E (or f: [a, b]  E ) be curve of class C , with p  n, so that the
n
n
p
derivatives f (t), . . ., f (n–1) (t) of f(t) are linearly independent for all t  ]a, b[ . Let h : E  E be a
(1)
n
n

rigid motion, and assume that the corresponding linear isometry is R. Let f = h o f.
19.4 Self Assessment
1. Let f: ]a, b[  E (or f : [a, b]  E ) be a curve of class C , with p  n. A family (e (t), . . ., e (t))
n
n
p
n
1
of orthonormal frames, where each e : ]a, b[  E is C continuous for i = 1, . . ., n – 1 and
n
n–i
i
en is C -continuous, is called a ....................
1
2. A moving frame (e (t), . . ., e (t)) along f so that for every k, with 1  k  n, the kth derivative
n
1
f (t) of f(t) is a linear combination of (e (t), . . ., e (t)) for every t  ]a, b[, is called a
(k)
1
k
....................
3. Let f: ]a, b[  E (or f: [a, b]  E ) be curve of class C , with p  n, so that the derivatives
n
p
n
f (t), . . ., f (n–1) (t) of f(t) are linearly independent for all t  ]a, b[ . Let h : E  E be a rigid
n
(1)
n

motion, and assume that the corresponding .................... is R. Let f = h o f.
4. Let  , . . .,  be functions defined on some open ]a, b[ containing 0 with  C n–i–1 continuous
1
i
n–1
for i = 1, . . ., n – 1, and with  (t) > 0 for i = 1, . . ., n – 2 and all t  ]a, b[. Then, there is curve
i
f : ]a, b[  E of class C , with p  n, satisfying the ....................
p
n
5. Let X(t) be the matrix whose columns are the vectors e (t), . . ., e (t) of the Frenet frame
1
n
along f. Consider the system of ODE’s, ....................
19.5 Review Questions
1. Show that the helix
8
 :[0,10]  E : s  (2cos( s ),2sin( s ), s )
5 5 5
is a unit speed curve and has constant curvature and torsion.
2. Why do we always have k[]  0?
3. For all s, ’’(s) . (s) = 0; so the acceleration is always perpendicular to the acceleration
along unit-speed curves. What about a’(t) . a’’(t) on arbitrary speed curves?
4. Derive the Frenet-Serret equations for an arbitrary-speed regular curve and show that the
'
following hold for a curve with speed  '. '    v 0 : :


T   '/v, N  B T, B  '   ''

'   ''
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