*2.1.3 Internal angles of Delaunay triangles*

These are the internal angles of any triangle in the Delaunay triangulation *ωi*; the angles are measured in the positive direction (counterclockwise), **Figure 3**, using the following procedure:


The sequence to measure the angle *ω<sup>i</sup>* is in the following order: if you select first point 1 (vertex 1) then select point 2 (vertex 2), these points will form a starting line where the angle measurement starts, the point 3 (vertex 3) is where the angle measured ends. The selection and sequence of generating points will indicate the final angle obtained.

**Figure 3.**

*Procedure to measure angles between three neighbors that form a Voronoi polygon and choosing one angle.*

*Analysis and Modeling of Polygonality in Retinal Tissue Based on Voronoi Diagram… DOI: http://dx.doi.org/10.5772/intechopen.106178*

## *2.1.4 Mean distances average*

This metric/algorithm determine the mean of average distances from the inner point in every Voronoi cell to its *n* neighbors and it is calculated by

$$\sum\_{i=1}^{n} \frac{d\_i}{n^2} \tag{1}$$

where *d*<sup>i</sup> is the internal distance defined above. It also represents a way to measure the cell size and it could be interpreted as a coefficient of expansion or contraction, **Figure 4**.

## *2.1.5 Polygonality index*

This metric/algorithm generates the measure Ξ,

$$\Xi = \frac{\mathbf{1}}{\sum\_{i=1}^{n} |\chi\_i - \beta| + \mathbf{1}} \tag{2}$$

where *χ<sup>i</sup>* is the formed angle between consecutive neighbors for Delaunay triangle (irregular polygon, dotted arrow), *β* is the angle between consecutive vertex for a regular polygon (solid arrow), *β* ¼ 360 degree*=n* and *n* is the number of neighbors of the Voronoi cell, **Figure 5**. The angle *χ<sup>i</sup>* is invariant under any rotation movement. The measurement is performed counterclockwise.

If the value of Ξ is close to 1, then the Voronoi polygon is close to regularity, angles *χ<sup>i</sup>* and *β* will have similar value. If Ξ is close to 0, then Voronoi polygon is irregular. The units of Ξ is the inverse in degrees.

**Figure 4.** *Modified Ulam tree graph to measure distances in Voronoi cell.*

### **Figure 5.**

*The solid line is a regular polygon, the irregular polygon is represented by the dotted line, and the neighbors are the black circle. The internal black dot is a mathematical node.*

## *2.1.6 Mean-square deviation of angles*

This metrics/algorithms evaluate *ε*,

$$\varepsilon = \sqrt{\sum\_{i=1}^{n} \left(\chi\_i - \beta\right)^2} \tag{3}$$

the root square of mean deviation from the angles *χi*, with respect to angle *β* ¼ 360 degree*=n* where *n* is the number of neighbors of the Voronoi cell for each Voronoi polygon. The magnitudes *χi*, *β*, and *n* are defined above (**Figure 5**). The metric ε is invariant under any rotation movement.

## *2.1.7 Variation index angle of differences*

This metric algorithm gets *δ* ,

$$\delta = \frac{\mathbf{1}}{\sqrt{\sum\_{i=1}^{n} \left(\chi\_i - \beta\right)^2 + \mathbf{1}}} \tag{4}$$

where *χ<sup>i</sup>* and *β* ¼ 360 degree*=n* are defined as above (**Figure 5**). If the value of *δ* is close to 1, then the Voronoi polygon is close to regularity, if *δ* is close to 0, then Voronoi polygon is irregular.

## **2.2 Software description**

In view of the foregoing, we suggest development of an integrated platform based on computational geometry for bioinformatics and computational biology for analyzing spatial cellular organization which in turn use both Voronoi tessellation and Delaunay triangulation, for the purpose to measure the distance, internal angles, radius of circumscribed circle, amid nearby points mean distances average, angular polygonality, polygonality index, mean-square deviation of angles, and variation index angle of differences. The platform holds two options, either being performed by a user or operating with an automatic formulation. This software allows to create Voronoi polygons and Delaunay triangles from a set of *XY* coordinates, or generated by selecting in an imported image. It locates *XY* coordinates, using an auxiliary window S, **Figure 6**.

## *2.2.1 Voronoi frequency*

This function displays graphics of frequency with respect to the number of sides in Voronoi mosaic with a data reading window, wider enough to avoid loss of data of polygons compared to other platforms [17].

## *2.2.2 Circumscribed circle*

These metrics/algorithms are able to find the magnitude of the circumradius, the coordinates of the center of the circumcircle, and the coordinates of the vertices formed in each Delaunay triangle. *R* is the radius of the circle circumscribing a Delaunay triangle, some examples (a) and (b), **Figure 7**.

*Analysis and Modeling of Polygonality in Retinal Tissue Based on Voronoi Diagram… DOI: http://dx.doi.org/10.5772/intechopen.106178*

### **Figure 6.**

*Platform based on computational geometry for Voronoi polygons and Delaunay triangles to biological structures.*

## **Figure 7.** *Circumcircle and circumradius for each Delanuay triangle in a Voronoi polygon.*

## *2.2.3 Distances selected*

These metrics/algorithms are able to get the distance *d*<sup>k</sup> between each pair of points selected by the user. First select the icon distances, located in Diagram Feature screen, after that choose a point of the polygon, by the icon select points, then select again the icon select point for another interest point, finally activate the icon selected file distances from Tools menu to get the data file with its coordinates and the distance that separates them. You might select different pairs of points to find out their distances in a single file, activating the icon selected for several couple points, for example (A), (B), (C), **Figure 8**.

## *2.2.4 Angles selected*

These metrics/algorithms generate a file formed from each pair of points selected by the user. The angle *υ* is relative to the horizontal axis and it works as the rangefinder. First, activate the *angles tree* icon (**Figure 6**), and then you can select different points for the same file using the select point button, to generate the Angles Selected file, **Figure 9**.

**Figure 8.** *Selection of distances to be chosen by the user in a Voronoi polygon.*

**Figure 9.** *Selection of angles between neighbors to be chosen by the user in a Voronoi polygon.*

**Internal angles selected**. This metric algorithm generates a file for the angles *γi*of each Delaunay triangle selected by the user. First, activate radio button internal angles (**Figure 6**), then select points radio button to form the Delaunay triangle, selecting three points. The select order of each point defines the angle to be measured. First, if the black point is selected, then you can choose the white point. These two points form a line from which we start measuring the angle and ends at the line formed between the third point, the striped circle, forming an angle which is measured from for example, tree forms to obtain several internal angles in a Delaunay triangle, *γk*, *γp*, *γi*, **Figure 10**, (A),(B) and (C), respectively.

For example, for four tides polygon, if the option 2 is selected, the angle *γ<sup>i</sup>* is shown in **Figure 11**.

**Figure 10.** *Options to select nearby points to get an angle between them in a Voronoi mosaic.* *Analysis and Modeling of Polygonality in Retinal Tissue Based on Voronoi Diagram… DOI: http://dx.doi.org/10.5772/intechopen.106178*

**Figure 11.** *Delaunay triangle, an angle selected between neighbors with option 2, Figure 10.*
