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

Sec. 19] The Natural Trihedron of a Space Curve 239
may be computed from the formulas
t= dr
dF ;
ds
v =
dt
ds
If X f K, Z are the current coordinates of the point of the tangent, then
the equations of the tangent have the form
where T x -- ;
Rectifying
plane
T =~ , v T
X-x = Y
'
3C
2 = -~ ;
y==
Z z
*
if '2
Fig. 84
Normal
plane
Osculating
plane
(I)
from the condition of perpendicularity
of the line and the plane we get an equation of the normal plane:
Z 2) = 0. (2)
If in equations (1) and (2), we replace T x , T
y , 7\ by B x , B
v , B
z and A^.
Wy, ^V^, we get the equations of the binomial and the principal normal and,
respectively, the osculating plane and the rectifying plane.
Example t. Find the basic unit vectors T, v and p of the curve
at the point i = 1.
Write the equations of the tangent, the principal normal and the binormal
at this point.
Solution. We have
and