Diﬀerential geometry 119
y = sin(v),
It is easy to see that the latter equation makes the elevation change from zero
to 1, in accordance with the turning of the helix.
Since the parametric space of the cylinder is simply the rectangle of the
developed surface, it is easy to see some special sub-cases. If the two points
are located at the same rotational position (v = constant), but at diﬀerent
heights, the geodesic curve is a straight line. If the two points are on the same
height (u = constant), but at diﬀerent rotational angles, the geodesic curve is
a circular arc.
The last two sections demonstrated the diﬃculties of ﬁnding the geodesic
curves even on regular surfaces like the sphere or the cylinder. On a general
three-dimensional surface these diﬃculties increase signiﬁcantly and may ren-
der using the diﬀerential equation of the geodesic curve unfeasible.
8.3 Geodesic curvature
Let us consider the parametric curve
(t)=x(t)i + y(t)j + z(t)k
on the surface
S(u, v)=x(u, v)i
+ y(u, v)j + z(u, v)k.
denote the unit normal of the surface. The curvature vector of a three-
dimensional curve is deﬁned as
is the tangent vector computed as
and also assumed to be a unit vector (a unit speed curve) for the simplicity
of the derivation. Then the unit bi-normal vector is
= n × t.