**5. Experimental result**

as shown in **Figure 2(c)**. **Figure 2(d)** shows that each ridge point will connect with the others in the same triangle and three extra path segments between the vertices and neighbour triangles diagonal vertices. Thereafter, the shortest path can be obtained in the graph *G*′

**Figure 2.** Illustration of the Delaunay triangulation algorithm. (a) Initialise (aerial view). (b) Spot source point and destination point (top view). (c) Insert ridge points (top view). (d) Extra connections (aerial view). (e) Obtain the shortest

Ahuja-Dijkstras algorithm presented as a red line shown in **Figure 2(e)**. Finally, **Figure 2(f)**

A near-shortest path algorithm on the Quadratic surface is the fastest in the literature.

**Theorem 1.** The time complexity of the algorithm in the triangle mesh *G* is *O*(*n*log *n*), where

shows the final result after the *PathShortening*.

path (aerial view). (f) PathShortening (top view).

96 Advanced Path Planning for Mobile Entities

**4. Performance analysis**

*n* denotes the number of triangles.

by

The performance of Delaunay triangulation-based path algorithm has been analysed for evaluating the near-shortest path with several real GIS maps in the Matlab Language. The analysis was performed on an Intel Core2 Quad CPU Q9550@2.83 GHz processor with 8 GB memory. **Figure 3** shows one of the experimental results with a GIS map, where the solid line is the near-shortest path and dashed lines are the shortcuts.

Next, we have compared this algorithm to the KS's algorithm with 1, 3, 5, 7, 9, 19 and 29 Steiner points and summarised the comparison results on the average path length and the average runtime in **Table 2**. In KS's algorithm, each edge of the triangle has been divided into multiple segments to generate more connections for path searching. **Figure 4(a)** and **(b)** illustrates the average running time and path length between two algorithms.

When compared to one Steiner point, the average path length difference of the Delaunay triangulation-based algorithm is 6.14% better than the KS's algorithm, and computation time between the Delaunay triangulation-based algorithm and the KS's algorithm is same. When it increased three Steiner points, the length difference is only 0.28%, but the computation time is 31.71 times faster. When 29 Steiner points for the KS's algorithm are applied, the KS's results can be assumed as the shortest path; however, the length difference is 2.81% longer and computation time is 4216 times faster. This proves that the Delaunay triangulation-based algorithm can solve the NP-hard problem and also obtain fast computing features. From the statistical view, **Figure 5** shows the prediction of the average computation time and length difference if the number of KS's Steiner points is infinity.

**SPs Length difference (%) Runtime difference (X) (%)**

Path Planning on Quadric Surfaces and Its Application http://dx.doi.org/10.5772/intechopen.72573 99

**Table 2.** Comparisons between our algorithm and KS's algorithm on average running time and length difference when

**Figure 4.** Comparison between Delaunay triangulation-based algorithm and KS's algorithm on average running time

and path length. (a) Average computation time and (b) average length difference.

 −6.14 0.97 0.28 31.71 1.66 86.40 2.26 162.94 2.68 414.55 2.79 1968.62 2.81 4215.75 999 ≅ ∞ 5.3 3.0E + 10

the Steiner points are 1, 3, 5, 7, 9, 19, 29, …, ∞

**Figure 3.** Near-shortest path searching with a GIS map. (a) Initialise (aerial view) and (b) result (aerial view).


**Table 2.** Comparisons between our algorithm and KS's algorithm on average running time and length difference when the Steiner points are 1, 3, 5, 7, 9, 19, 29, …, ∞

**Figure 4.** Comparison between Delaunay triangulation-based algorithm and KS's algorithm on average running time and path length. (a) Average computation time and (b) average length difference.

**Figure 3.** Near-shortest path searching with a GIS map. (a) Initialise (aerial view) and (b) result (aerial view).

98 Advanced Path Planning for Mobile Entities

**Figure 5.** The prediction of the average computation time and length difference if the number of KS's Steiner points is infinity. (a) Average computation time and (b) average length difference.
