Page 235 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes '   ''
k 
v 3


'    '' . '''
  2 .
'    ''
5. Viviani's curve is the intersection of the cylinder (x – a) + y = a and the sphere x + y +
2
2
2
2
2
2
z = 4a and has parametric equation:
2
2
t
8

a :[0,4 ]  E : t  a(1 cost,sin t,2sin ).

2
t
13 3cost 6cos 2

Show that it has curvature and torsion given by k(t)  s and (t)  a(13 3cost) .

a(3 cost) 2 

6. Investigate the following curves for n = 0, 1, 2, 3
s 3 s 3 ns
8
 :[0,2 6]  E : s  ( 6 cos( ), sin( ), sin( ))
6 2 6 2 6
 cos sin 0 


7. Show that for all q  [0, 2] the matrix R ( )   sin cos 0  when applied to the

s
 0 0 1 
 
coordinates of a curve in E rotates the curve through angle  in the (x,y)-plane, that is,
8
round the z-axis. Find a matrix R () representing rotation round the y-axis and hence
y
obtain explicitly the result of rotating the curves in the previous question by 60º round the
y-axis.
8. On plane curves,  = 0 everywhere and we sometimes use the signed curvature k defined
2
,
by

2
k [a](t)   ''(t).J '(t) , where J is the linear operator J :  2   2 : (p,g)  ( g,p).

a'(t) 3
We call 1/k [] the radius of curvature. Find the radius of curvature of some plane curves.
2
9. (i) Find two matrices, R and R from SO(3) which represent, respectively, rotation by
y
z
/3 about the y-axis and rotation by /4 about the z-axis; each rotation must be in a
right-hand-screw sense in the positive direction of its axis. Find the product matrix
R R and show that its transpose is its inverse.
y
z
(ii) By considering (R R ) , or otherwise, show that the curve
-1
z
y
8
:[0, )   E : t  ( 1 cos h t/2  t , 2 cosht/2  t , 3 cosh t/2  3t )
2 2 2 2 2 2 2
lies in a plane and find its curvature function and arc length function.

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