Voronoi diagrams are used to establish a navigation- aided framework to view 3D maps on mobile devices.
This can be implemented using various programming languages.
It requires building an algorithm and following each step for establishing points and regions within a space.
The implementation can be mostly in two-dimensional space, for which sets of points within a plane are divided into cells. Each cell contains exactly one region, as
shown in Fig. 1.
Fig. 1. Voronoi diagram within a certain space
The figure is generated with each cell containing a boundary which represents points and edges in a space. Points represent the location of objects and the edges are their boundaries. Although the number of nodes or points is arbitrary, it can seen that they fall within an axis of x and y for which they are normalized to zero and one.
There are many spaces between the points, as depicted in the figure, because there are few nodes within each region. The relationship between the points in the space is such that each cell represents a Euclidean distance to the others and is bound to be smaller than the distance to others within the plane. This assertion, when it covers all points within the entire plane, leads to boundaries for cells and results in Voronoi edges.
This will result in the point’s equidistance from the nearest two sites. As a result, the point where multiple boundaries meet is the Voronoi vertex, which is theoretically equidistant from its three nearest sites. This theory is supported by a lemma which states that the
number of Voronoi vertices and edges are:
2 n1 hand 3 n1 h (1) This argument is true when n represents the number of spots within and h is considered as the number of spots on the convex hull of S . The proof of this is guided by
considering the claim for a finite graph which satisfies Euler’s formula. Consider V , EandF to be the vertices,
edges, and faces of the polyhedron (graph), then mapping members’ functions which can be projected on to a plane implies that a set of n points can be transformed into a finite graph; this is explained by presenting 2E3V for each side of the equality using Euler’s formula to give us Eq. (2):
2 2 and 3 2
V F E F (2)
This function that produces V h # Voronoi vertices and Eh # Voronoi edges. Therefore Voronoi vertices can be represented by 2
F2
h whereas its edges will be resolved in 3
F2
h basedAdamu Abubakar, Sadegh Ameri, Suhaimi Ibrahim, Teddy Mantoro
Copyright © 2014 Praise Worthy Prize S.r.l. - All rights reserved International Review on Computers and Software, Vol. 9, N. 9 on the fact that Fn ; such conditions will give rise 1
to Eq. (3), which is the number of Voronoi vertices and edges:
2 n1 h and 3 n1 h (3)
This situation can be conceptualized in a design that will provide for a navigation aid as the platform for establishing the location and the number of interactions between each location in a well-defined pattern. This will lead to establishing that for a given space an application for a navigation aid should consider the points of interest as Voronoi vertices and the links to other points as Voronoi edges. Thus an increase in either an edge or a vertex leads to a corresponding increase in the other.
In order to support this, a second lemma is established This lemma states that the Voronoi diagram V (S) has
O(n) many edges and vertices. The average number of edges in the boundary of a Voronoi region is less than 6.
In the case of a navigation aid, this implies that locations that can be used are such that the average number of edges in the boundary region should be less than 6. In order to prove this, the Euler formula for planar graphs is applied based on the established Voronoi diagram.
Hence, when V , E and F that is vertices, edges,
faces respectively are connected, and putting these components into the Euler equation, we find Eq. (4):
1
V e f c (4)
Thus when this is applied to a finite Voronoi diagram, each vertex will contain at least three incident edges, and adding them together results in Eq. (5):
3 2
v
e (5)
Substituting this inequality with c = 1 and f = n + 1 yields Eq. (6):
2 2 and 3 3
v n e n (6)
Furthermore, adding up the number of edges contained in the boundaries of all n+1 faces results in Eq. (7):
2e6n 6 (7)
This is because each edge is counted twice. Thus, the average number of edges in a region’s boundary is bounded by a value less than six.
This condition supports a framework where a conceptualization can be well supported.
Since it has been proven that points and paths can be established and the number of these within a certain space is well defined, this generates a well established platform for the design of a navigation aid.
For establishing location and the number of interactions between each location a well-defined pattern
is required. This will lead to establishment of a given space in the navigation aid that should consider the points of interest as Voronoi vertices linked to the other points by Voronoi edges. The system starts by considering a given finite space with finite points or locations (see Fig. 2).
Fig. 2. Points within a region of Euclidean plane
Given a finite number (a set of two or more) of distinct points in a Euclidean plane, as shown in Fig. 2, there is an association of all locations in that space with the closest number(s) of the point set with respect to Euclidean distance.
The result is a tessellation of the plane into a set of the regions associated with members of the point set, as shown in Fig. 3. This is the planar ordinary Voronoi diagram generated by the point set, and the regions constituting the Voronoi diagram are ordinary Voronoi polygons. Since the points are the finite number n of points in the Euclidean plane, this will easily relate the pathfinding technique to the Voronoi metric space.
Fig. 3. Voronoi polygon (tessellation of points within a region) Let P
p ,..., p1 n
be the region given by: 22n andxixj for i j, i, jI .n
is:
i
i j n
V p x xx xx for ji,i, jI (8)
is called the planar ordinary Voronoi polygon associated with p or the Voronoi polygon of i p and the set given i
by:
V p ,...,V p1 n
Adamu Abubakar, Sadegh Ameri, Suhaimi Ibrahim, Teddy Mantoro
is called the planar ordinary Voronoi diagram generated by P or the Voronoi diagram of P.
An application with a 3D map for use in mobile devices could implement this technique. In these applications, the approaches that will suit the implementation of this technique are when path and location are important. Obviously, in all navigation aids, path and location are the most important things to consider. The Voronoi diagram being generated is in a two-dimensional plane, which will be underneath a 3D- model file; its role will be to create a sub-division of the entire region of the 3D dataset into appropriate disjointed data points (nodes) to indicate the path differences between each node in the space. This means that the result of combining the two layers will be a separate layer which establishes the known points (nodes) and distance between the nodes (region). As a result these applications present a well-defined space.