Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 20] Curvature and Torsion of a Space Curve 243
where (p is the angle of turn of the tangent line (angle of continence) on a
segment of the curve MN, As is the arc length of this segment of the curve.
R is called the radius of curvature. If a curve is defined by the equation
r=r(s), where s is the arc length, then
For the case of a general parametric representation
of the curve we have
2. Torsion. By torsion (second curvature) of a curve at a point M we
mean the number
r-l-ihn -1
Q As-*o As
where is the angle of turn of the binormal (angle of contingence of the
second kind) on the segment of the curve MN. The quantity Q is called the
radius of torsion or the radius of second curvature. If r=r(s), then
where the minus sign
dp_
ds
drd 2 rd s r
ds ds 2 ds 3
is taken when the vectors and v have the same
direction, and the plus
If /=/(/), where t
sign, when not
is an arbitrary
the same.
parameter, then
dr d 2 1
Q
r dV
d/d?" d7 = -
dt ~d,
Whence
Example
Solution. We have
1. Find the curvature and the torsion of the screw-line
r= i a cos t -\-j a sin / + k bt (a > 0).
~= _/ a sin t +ja cos t + kb t
d 2 _ r
== / a cos / / a sin /,
a sin t
a cos /
= _/ a sin/ Ja cos/.
J k
a cost b
-a sin /
= i ab sin / jab cos t + a*k
(1)
(2)