Let
be an arc length
parametrized regular curve on this surface which passes through point
as shown in Fig. 3.6 and denoted by

Let be a unit tangent vector of at , be a unit normal vector of at , be a unit surface normal vector of at and be a unit vector perpendicular to in the tangent plane of the surface, defined by . The component of the curvature vector of is the geodesic curvature vector and is given by

The scalar function

(10.3) |

is called the

The unit tangent vector of the curve
can be obtained by
differentiating (10.1) with respect to the arc length
using the chain rule

Thus we have

and hence substituting (10.5) and (10.6) into (10.4) yields

We can easily observe that the coefficients of , , , , are all functions of the coefficients of the first fundamental form , and and their derivatives, , , , , , . It is interesting to note that the normal curvature depends on both the first and second fundamental forms, while the geodesic curvature depends only on the first fundamental form. Using the Christoffel symbols ( ) defined as follows [412]

geodesic curvature reduces to

According to the definition, we can determine the differential equation that any geodesic on a surface must satisfy by simply setting in (10.9) and obtain

Alternatively, we can derive the differential equation for geodesics by
considering that the surface normal
has the direction of a
normal to the geodesic curve

Since , (10.11) can be rewritten as

By substituting (10.6) into equations we have

By eliminating from (10.13) using (10.14), and eliminating from (10.14) using (10.13) and employing the Christoffel symbols, we obtain [412]

Equations (10.15) and (10.16) are related by the first fundamental form and if we eliminate from both equations, the equations reduce to (10.10) with taken as parameter. These two second order differential equations can be rewritten as a system of four first order differential equations [235]

We can also find this result by means of the general rules of the
calculus of variations [166]. We want to minimize

subject to the conditions

(10.22) |

where

and and are given constants. It is well known from calculus of variations that the solution of Euler's equation [166]

gives an extreme value to the integral (10.21). When (10.23) is substituted in Euler's equation (10.24) we can derive the differential equation for geodesics.

*Example 10.2.1.*
Let us obtain the geodesic equations for a parametric bilinear surface
(hyperbolic paraboloid)
(see Fig. 3.4). We have

thus, the Christoffel symbols become

Finally the geodesic equations for the bilinear surface are given by