Page 248 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 20: Curves

20.6 Keywords Notes

n
.

Parametrized curve: A parametrized curve is a smooth (C ) function g : I ®  A curve is
regular if ' ¹ 0.
g
Frenet frame equations. The Frenet frame X = (T, N, B) of a curve in  satisfies: (1.2) X’ = X. The
3
Frenet frame equations, form a system of nine linear ordinary differential equations.
Fundamental Theorem: Let  > 0 and  be smooth scalar functions on the interval [0, L]. Then
there is a regular curve  parametrized by arclength, unique up to a rigid motion of  , whose
3
curvature is  and torsion is .
20.7 Self Assessment

1. A ................ is a smooth (C ) function : Ig ®  n . A curve is regular if 'g ¹ 0.

3
'
2. Let  : I   be a curve in  . The unit vector T   is called the ................ of . The
3
curvature  is the scalar    " .
3. The Frenet frame X = (T, N, B) of a curve in  satisfies: (1.2) X’ = X. The ................,
3
Equation (1.2), form a system of nine linear ordinary differential equations.
4. A rigid ................ is a function of the form R(x) = x + Qx where Q is orthonormal with det
0
Q = 1.

20.8 Review Questions

1. A regular space curve :[a,b]   is a helix if there is a fixed unit vector u  such that
3
3
e u is constant. Let k and  be the curvature and torsion of a regular space curve , and

1
suppose that k  0. Prove that  is a helix if and only if  = ck for some constant c.
2. Let  : I   be a smooth curve parameterized by arclength such that   , are
4
, 
linearly independent. Prove the existence of a Frenet frame, i.e., a positively oriented
orthonormal frame X = (e , e , e , e ) satisfying e = , and X = Xw, where w is anti-
3
4
1
1
2
symmetric, tri-diagonal, and w i,i+1 > 0 for i  n – 2. The curvatures of  are the three
functions k = w i,i+1 , i = 1, 2, 3. Note that k , k > 0, but k has a sign.
2
i
3
1
3. Prove the Fundamental Theorem for curves in  : Given functions k , k , k on I with k , k 2
4
1
1
2
3
> 0, there is a smooth curve  parameterized by arclength on I such that k , k , k are the
1
2
3
curvatures of . Furthermore,  is unique up to a rigid motion of  .
4
4. Let  :[a,b]  be a regular plane curve with non-zero curvature k  0, and let
2
 + k N be the locus of the centers of curvature of .
-1
(a) Prove that  is regular provided that k = 0.
(b) Prove that each tangent  of  intersects  at a right angle.
(c) Prove that each regular plane curve :[a,b]   has at most one evolute.
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