**Path Planning on Quadric Surfaces and Its Application**

**Path Planning on Quadric Surfaces and Its Application**

DOI: 10.5772/intechopen.72573

Chi-Chia Sun, Gene Eu Jan, Chaomin Lu and Kai-Chieh Yang Kai-Chieh Yang Additional information is available at the end of the chapter

Chi-Chia Sun, Gene Eu Jan, Chaomin Lu and

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72573

#### **Abstract**

In this chapter, recent near-shortest path-planning algorithms with *O*(*n*log *n*) in the quadric plane based on the Delaunay triangulation, Ahuja-Dijkstra algorithm, and ridge points are reviewed. The shortest path planning in the general three-dimensional situation is an NP-hard problem. The optimal solution can be approached under the assumption that the number of Steiner points is infinite. The state-the-art method has at most 2.81% difference on the shortest path length, but the computation time is 4216 times faster. Compared to the other *O*(*n*log *n*) time near-shortest path approach (Kanai and Suzuki, KS's algorithm), the path length of the Delaunay triangulation method is 0.28% longer than the KS's algorithm with three Steiner points, but the computation is about 31.71 times faster. This, however, has only a few path length differences, which promises a good result, but the best computing time. Notably, these methods based on Delaunay triangulation concept are ideal for being extended to solve the path-planning problem on the Quadric surface or even the cruise missile mission planning and Mars rover.

**Keywords:** Delaunay triangulation, Dijkstra algorithm, ridge point, near-shortest path, mission planning, NP-hard

### **1. Introduction**

In the Euclidean plane with obstacles, the shortest path problem is to find an optimal path between source and destination. Shortest path algorithms have already been applied to motion planning of robots and path planning of navigation. Furthermore, it can be applied to electronic design automation (EDA), biological cell transportation and operation research (OR) [1–3].

In [4–6], Jan et al. proposed two *O*(*n*log *n*) time path-planning algorithms to obtain the near-shortest path in the Euclidian and quadric planes, respectively. Compared to the other

approaches of reduced visibility graph, this fast method outperforms the rest of *O*(*n*log *n*) algorithms in the general two-dimensional situation, except the path length compared to the shortest *O*(*n*<sup>2</sup> ) time shortest algorithm of visibility graph.

In the quadratic plane, a survey of the shortest path problem concerning a two or higher dimensional geometric object (e.g. a surface, a polyhedron, space, network) can be found in [7]. The shortest path problem in the general three-dimensional situation is non-deterministic polynomial-time hard (NP-hard) problem [8], and only exponential time algorithms are known. In [9], the shortest path on a polyhedron is its local, which has an important property called unfolding, where the path must enter and leave at the same angle to the intersecting edge.

It follows that any locally optimal shortest path joining two consecutive obstacle vertices can be unfolded at each edge along its edge sequence, thus obtaining a straight segment. Sharir and Schorr [10] proposed an *O*(*n*<sup>3</sup> log *n*) algorithm, which first applied this property to find the exact shortest path on a convex surface, where *n* is the number of edges. Later, Mitchell et al. [11] proposed an *O*(*n*<sup>2</sup> log *n*) algorithm for propagating the shortest path map over a surface by a continuous Dijkstra method for general polyhedron. Chen and Han [12] improved it to an *O*(*n*<sup>2</sup> ) algorithm. Faster algorithms than these cannot be found by far.

4216 times faster. Therefore, it can not only obtain a good near-shortest path length on the quadratic surface, but also improve the computation time. Furthermore, it is worth noting that Delaunay triangulation-based fast algorithms are ideal for being extended to solve the path planning in the polyhedron plane or be applied to cruise missile mission planning in the

**Table 1.** Comparison of different shortest path algorithms in the three-dimensional space, *n* denotes the number of

log *n*) *O*(*n*<sup>2</sup>

Is the path shortest? N/A N/A N/A Near-shortest Near-shortest

**Polyhedron Polyhedron or quadric**

**Sharir [10] Mitchell [11] Chen [12] KS's [9] Delaunay method**

) *O*(*kn*

*n* 9*n*

Path Planning on Quadric Surfaces and Its Application http://dx.doi.org/10.5772/intechopen.72573 93

*n*) *O*(*n*log *n*)

log *k*<sup>2</sup>

This chapter is organised as follows. Section II briefly introduces the concept of shortest path algorithms. In Section III, we will describe the idea of the triangulation-based near-shortest path algorithm, the performance of which is analysed in Section IV. The experimental results are shown in Section V. Section VI explains a possible application for cruise missile mission

In this section, basic concepts of the Delaunay triangulation algorithm on the quadratic surface will be introduced, such as Euclidean plane, Delaunay triangulation, ridge points, and Dijkstra's single-source shortest path algorithm. Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry [15], as well as the generalisations of these notions to higher dimensions. It can be used to distinguish these spaces from the curved

For the quadratic surface, there is essentially only one Euclidean space with three real number

A Quadric surface is the locus of the points (*x, y, z*), which satisfy a second-degree equation

The classification of the different types of Quadric surfaces is made, first, on the basis of the

*A D E D B F*

types of Quadric surfaces, a total of 15, are obtained by varying the coefficients of it [17].

+ 2*Dxy* + 2*Exy* + 2*Fyz* + *Gx* + *Hy* + *Iz* + *J* = 0, and the different

*<sup>E</sup> <sup>F</sup> <sup>C</sup>*) (1)

spaces of non-Euclidean geometry and Einstein's general theory of relativity [16].

quadratic plane.

triangle mesh.

planning, and Section VII concludes the chapter.

**2. Algorithm backgrounds**

**Euclidean Algorithms**

Time complexity *O*(*n*<sup>3</sup>

Connection No No No 6*k*<sup>2</sup>

log *n*) *O*(*n*<sup>2</sup>

coordinates from the modern viewpoint.

+ *By*<sup>2</sup>

*AQ* <sup>=</sup> (

+ *Cz*<sup>2</sup>

matrix of the quadratic from determining by the symmetric matrix:

in three variables, *Ax*<sup>2</sup>

Kimmel and Sethian [13] presented a fast searching method for solving the Eikonal equation on a rectangular orthogonal mesh in *O*(*M*log *M*) steps, where *M* is the total number of grid points. They extended the fast marching method to triangulated domains with the same computational complexity. As an application, they provide an optimal time algorithm for computing the geodesic distances and thereby extracting shortest paths on triangulated manifolds.

Helgason et al. [14] presented a heuristic algorithm based on geometric concepts for the problem of finding a path composed of line segments from a given destination in the presence of polygonal obstacles. The basic idea involves constructing circumscribing triangles around the obstacles to be avoided. Their heuristic algorithm considers paths composed primarily of line segments corresponding to partial edges of these circumscribing triangles and uses a simple branch-and-bound procedure to find a relatively short path of this type.

Kanai and Suzuki proposed a near-shortest path approach (Kanai and Suzuki, KS's algorithm [9]) based on the Delaunay triangulation, the Dijkstra algorithm, and Steiner points, with computational complexity of *O*(*k*<sup>2</sup> *n*log *k*<sup>2</sup> *n*), where *k* is the number of Steiner points and *n* is the number of the triangles. Although KS's algorithm is an approximation, it has the significant advantages of easy implementation, high approximation accuracy, and numerical robustness. However, to obtain a shorter path, the computation time required by the path planning will increase rapidly when the Steiner points increases. A detailed comparison can be found in **Table 1**.

In this chapter, an *O*(*n* log *n*) time near-shortest path planning is introduced. It combined with the Delaunay triangulation, Ahuja-Dijkstra algorithm, and ridge points for path planning on a quadratic surface. Experimental results show that the average path length of the Delaunay triangulation-based algorithm is 0.28% longer than the KS's algorithm; however, the speed is 31.71 times faster. Furthermore, when performing KS's algorithm with 29 Steiner points, the NP-hard shortest path will be found (extremely close approximation of the shortest path planning). Although the length is 2.81% longer than the shortest, the computation time is


**Table 1.** Comparison of different shortest path algorithms in the three-dimensional space, *n* denotes the number of triangle mesh.

4216 times faster. Therefore, it can not only obtain a good near-shortest path length on the quadratic surface, but also improve the computation time. Furthermore, it is worth noting that Delaunay triangulation-based fast algorithms are ideal for being extended to solve the path planning in the polyhedron plane or be applied to cruise missile mission planning in the quadratic plane.

This chapter is organised as follows. Section II briefly introduces the concept of shortest path algorithms. In Section III, we will describe the idea of the triangulation-based near-shortest path algorithm, the performance of which is analysed in Section IV. The experimental results are shown in Section V. Section VI explains a possible application for cruise missile mission planning, and Section VII concludes the chapter.
