Blaga P. Lectures on the differential geometry of - tiera.ru

3.2. Examples for the integration of the natural equation of a plane curve 97
that can be integrated. We remind that in the case of plane curves the problem is not the
integrability of the natural equation, but, rather, the possibility of expressing the solution
in terms of elementary functions. Usually this is not the case, and this makes even more
interesting the (few) cases in which this possibility does exist. For convenience, we shall
use, for the rest of this paragraph, the curvature radius R = 1/k instead of the curvature.
The logarithmic spiral. In this case the curvature radius is given by
R = a · s, (3.2.1)
where a is a constant. Let α be the contingency angle. Then, as we know, we have
dα
ds
1 1
= =
R(s) as ,
therefore, by integrating and neglecting the integration constant, we get
α = 1
a ln s, whence s = eaα .
To get the coordinates, we have to integrate the system of differential equations
dx
ds
= cos α,
dy
= sin α.
ds
We have (neglecting, again, the integration constants)
�
x =
�
cos αds = a cos αe aα dα = aeaα
a2 (a cos α + sin α)
+ 1
and, analogously,
�
y =
sin α ds = aeaα
a2 (a sin α − cos α).
+ 1
Hence, the parametric equations of the logarithmic spiral are
⎧
⎪⎨
x = aeaα
a2 (a cos α + sin α)
+ 1
⎪⎩ y = aeaα
a2 (a sin α − cos α)
+ 1
. (3.2.2)
The logarithmic spiral (see figure 3.1) is most conveniently described through its polar