10.4.2 Circular arc approximation

The problem of the straight line approximation is that when there are more than one path, it cannot capture the other paths. To make the method more reliable, the following algorithm has been developed [247]. First we pick two points and in the parameter domain, which are on the bisector of the two end points and , such that or as illustrated in Fig. 10.2 (b). Then we determine two circular arcs which pass through the three points and . If and are taken at a large enough distance from , all the geodesic paths in the parameter domain between points and are likely to lie within or close to the region surrounded by the two circular arcs. Notice that the algorithm fails once the circular arcs go outside the domain so these arcs are chosen such that they are entirely within the domain. The coordinates in the parameter domain i.e. , can be obtained by equally distributing the points along the circular arc in the parameter domain. Once we have a set of points in the parameter domain, we can easily evaluate by using the central difference formula for a non-uniform mesh points [117], for

the forward difference formula for

and the backward difference formula for

where is replaced by or , and the step length is evaluated by computing the chord length between the successive points on the surface. Even if the mesh points are equally distributed along the circular arc in the parameter domain as shown in Fig. 10.2 (b), is not in general constant.

We give the flow chart of the algorithm based on circular arc approximation for computing the geodesic path between two given points in Fig. 10.3.

Figure 10.3: Flow chart of the algorithm based on circular arc approximation for computing the geodesic path between two given points (adapted from [247])