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

3.2. Examples for the integration of the natural equation of a plane curve 99
where a and b are nonvanishing constants. A possibility would be to express R in terms
of s and then integrate. However, this would lead to complications, due to the the presence
of the square root and the sign ambiguity. We prefer, instead, to introduce a new
parameter t, through the relations
s = a sin t, R = b cos t.
It is, then, very easy to find the contingency angle in terms of this new parameter. Indeed,
we have
�
α =
�
1
ds =
R
1
at
a cos t dt =
b cos t b .
We can proceed now with the determination of the coordinates x and y, in terms of the
parameter t:
�
x =
�
cos α ds = cos at
�
a b (a − b)t
· a · cos t dt = sin +
b 2 a − b b
b
�
(a + b)t
sin ,
a + b b
�
y =
�
sin α ds = sin at
�
b (a − b)t
· a · cos t dt = −a cos +
b 2 a − b b
b
�
(a + b)t
cos .
a + b b
The clothoid. This is the curve whose natural equation is
R = a2
. (3.2.5)
s
Thus, for the clothoid (also known as the Cornu’s spiral), the radius of curvature is
proportional to the inverse of the arc length. From this point of view, it is, to some
extent, similar to the logarithmic spiral, where the radius of curvature was proportional
to the arc length, rather than to its inverse. We include this curve here to show that even
in the case of a very simple expression of the curvature in terms of the arc length (in this
case the curvature is proportional to the arc length, i.e. it is a linear function), we might
not be able to find a parametric representation in terms of elementary functions.
The contingency angle is easily found:
therefore the coordinates are
�
x =
α = s2
, (3.2.6)
2a
cos s2
�
ds, x =
2a
sin s2
ds. (3.2.7)
2a