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10.2.1 Parametric surfaces

We assume that the given parametric surface = is a regular and non-periodic NURBS surface patch. Wolter [448] shows that on a regular NURBS surface patch there always exists a shortest path joining any two patch points. If the surface patch is defined on a rectangular or even more generally on a locally convex planar domain, then any shortest path in the patch joining any two patch points must have a continuous tangent with the path being arc length parametrized. If the shortest path (without its end points) does not meet the patch boundary then this shortest path is a geodesic in the sense of Definition 10.1.1 where this is proven in [448] under very weak assumptions. We shall henceforth assume throughout this chapter that the shortest path to be computed will not meet the patch boundary except possibly at its end points.

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

 (10.1)

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
 (10.2)

The scalar function
 (10.3)

is called the geodesic curvature of at , or equivalently
 (10.4)

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

 (10.5)

Thus we have
 (10.6)

and hence substituting (10.5) and (10.6) into (10.4) yields
 (10.7)

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]
 (10.8)

geodesic curvature reduces to
 (10.9)

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
 (10.10)

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

 (10.11)

Since , (10.11) can be rewritten as
 (10.12)

By substituting (10.6) into equations we have
 (10.13) (10.14)

By eliminating from (10.13) using (10.14), and eliminating from (10.14) using (10.13) and employing the Christoffel symbols, we obtain [412]
 (10.15) (10.16)

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]
 (10.17) (10.18) (10.19) (10.20)

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

 (10.21)

subject to the conditions
 (10.22)

where
 (10.23)

and and are given constants. It is well known from calculus of variations that the solution of Euler's equation [166]
 (10.24)

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

Next: 10.2.2 Implicit surfaces Up: 10.2 Geodesic equation Previous: 10.2 Geodesic equation   Contents   Index
December 2009