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Figure 1. MIC algorithm developed in the study. For the extraction of the points that are candidates for being the center of MIC, Voronoi diagrams are created for the set of points located along PF subject to the calculation. For this purpose, in the algorithm (Figure 1) firstly, vertices of PF are extracted and stored in an array list (as) by creating mid-points and, if necessary, extra points between the couple of points forming the edges of PF.

Figure 1 MIC algorithm developed in the study. For the extraction of the points that are candidates for being the center of MIC, Voronoi diagrams are created for the set of points located along PF subject to the calculation. For this purpose, in the algorithm (Figure 1) firstly, vertices of PF are extracted and stored in an array list (as) by creating mid-points and, if necessary, extra points between the couple of points forming the edges of PF.

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Figure 2. Sample polygon and first candidate for MIC (n,). Parallel to this, the index information (,) for the edge closest to the point represented by index is also calculated. A this stage, the point represented by index is actually a good candidate for the center of MIC.
Figure 2. Sample polygon and first candidate for MIC (n,). Parallel to this, the index information (,) for the edge closest to the point represented by index is also calculated. A this stage, the point represented by index is actually a good candidate for the center of MIC.
Figure 3. The first approximation for the center of MIC based on the points located along the medial axis. UNCOVERING THE POTENTIAL CORES THAT MAY COVER THE CENTER OF MIC
Figure 3. The first approximation for the center of MIC based on the points located along the medial axis. UNCOVERING THE POTENTIAL CORES THAT MAY COVER THE CENTER OF MIC
Figure 4. An illustrative example for multiple core problem (a polygon feature having three remarkable cores). Depending on the shape of PF there may be more than one core that should be used in the approximation of the center of MIC (Figure 4). Thus, in the algorithm all cores lying along the medial axis are considered in the calculation of MIC. As it can be followed from the algorithm (Figure 1), for this purpose, firstly the closest point (m2) in points] (yellow points in Figure 4 along medial axis) to n; is found. The objective function for the respective point can be given as below:
Figure 4. An illustrative example for multiple core problem (a polygon feature having three remarkable cores). Depending on the shape of PF there may be more than one core that should be used in the approximation of the center of MIC (Figure 4). Thus, in the algorithm all cores lying along the medial axis are considered in the calculation of MIC. As it can be followed from the algorithm (Figure 1), for this purpose, firstly the closest point (m2) in points] (yellow points in Figure 4 along medial axis) to n; is found. The objective function for the respective point can be given as below:
Assume that in Figure 3, n; is the point identified by index and a is the closest edge of the polygon feature to n;. If nz is the closest point to n; compared with the other points located along the medial axis and c is the closest edge of the polygon feature to n2, a better candidate for the center of MIC should lie between n, and np, and the circle drawn around it (with a radius d;) should be tangent to both a and c (Figure 5 and Figure 6). The condition for being tangent to both a and c reveals that the respective point (x;, y1) lies at the intersection of b and the angle bisector of the angle formed by the extensions of lines a and c.
Assume that in Figure 3, n; is the point identified by index and a is the closest edge of the polygon feature to n;. If nz is the closest point to n; compared with the other points located along the medial axis and c is the closest edge of the polygon feature to n2, a better candidate for the center of MIC should lie between n, and np, and the circle drawn around it (with a radius d;) should be tangent to both a and c (Figure 5 and Figure 6). The condition for being tangent to both a and c reveals that the respective point (x;, y1) lies at the intersection of b and the angle bisector of the angle formed by the extensions of lines a and c.
Figure 6. The second approximation for the center of MIC based on the intersection of b and the angle bisector of the angle formed by the extensions of lines a and c.
Figure 6. The second approximation for the center of MIC based on the intersection of b and the angle bisector of the angle formed by the extensions of lines a and c.
If the circle drawn around (x;,, y;) is tangent to only two of the edges of the polygon feature, it may not be tangent to a third edge of PF and it may actually overflow the boundaries of the PFs. In other words, it means that it may not be the maximum inscribed circle (Figure 5 and Figure 6). Figure 7. Sample polygon and third candidate for MIC (blue circle drawn around the small blue point).
If the circle drawn around (x;,, y;) is tangent to only two of the edges of the polygon feature, it may not be tangent to a third edge of PF and it may actually overflow the boundaries of the PFs. In other words, it means that it may not be the maximum inscribed circle (Figure 5 and Figure 6). Figure 7. Sample polygon and third candidate for MIC (blue circle drawn around the small blue point).
Figure 8. The third approximation for the center of MIC based on the intersection of the angle bisector of the angle formed by the extensions of lines a and c with the angle bisector of the angle formed by the extensions of lines d and ¢. Thus, further considerations are necessary to find the real MIC (Figure 7). Assume that in Figure 8, a, c and d is the first three closest edges of the polygon feature to the point (x;, y;). A better candidate (x, yz) for the center of MIC should lie at the intersection of the angle bisector of the angle formed by the extensions of lines a and c, and the angle bisector of the angle formed by the extensions of lines d and c (Figure 8).
Figure 8. The third approximation for the center of MIC based on the intersection of the angle bisector of the angle formed by the extensions of lines a and c with the angle bisector of the angle formed by the extensions of lines d and ¢. Thus, further considerations are necessary to find the real MIC (Figure 7). Assume that in Figure 8, a, c and d is the first three closest edges of the polygon feature to the point (x;, y;). A better candidate (x, yz) for the center of MIC should lie at the intersection of the angle bisector of the angle formed by the extensions of lines a and c, and the angle bisector of the angle formed by the extensions of lines d and c (Figure 8).
Statistics are based on the whole set of irregular & complex PFs. MIC function is called with various names in the respective software programs. In LayoutEditor (Thies, 2018) originally created in order to edit designs for MEMS (Micro-Electro- Mechanical Systems) and IC (Integrated Circuit) fabrication, it is called as “Biggest Inner Circle” (BIC) that is available unde1 “Shape Utilities” menu item. In Flowmap (2013) produced for the analysis of spatial relationships, it is called as “Larges Inscribed Circle” (LIC) that is available under “Point in Polygon Analysis” menu item. In SAGA (2018), it can be calculated via “Largest Circles in Polygons” (LCP) tool available under “Tools Chains” menu item. Since the result of the respective too! involves more than one circle for a PF, the largest circle can be selected by comparing the all the circles calculated by the software with each other. In GeoWizard (2018) extension for ArcGIS Desktop, MIC is available as a tool unde: “Miscellaneous” menu item. The hardware configuration and operating system of the computer used for performance test is a: following; a quad-core processor with 2.66 Ghz frequency and 16 GB of RAM on 64 bit Windows 10 operating system. Figure 9. Various runs of the script for a set of regular (A) and irregular and complex (B) polygon features.
Statistics are based on the whole set of irregular & complex PFs. MIC function is called with various names in the respective software programs. In LayoutEditor (Thies, 2018) originally created in order to edit designs for MEMS (Micro-Electro- Mechanical Systems) and IC (Integrated Circuit) fabrication, it is called as “Biggest Inner Circle” (BIC) that is available unde1 “Shape Utilities” menu item. In Flowmap (2013) produced for the analysis of spatial relationships, it is called as “Larges Inscribed Circle” (LIC) that is available under “Point in Polygon Analysis” menu item. In SAGA (2018), it can be calculated via “Largest Circles in Polygons” (LCP) tool available under “Tools Chains” menu item. Since the result of the respective too! involves more than one circle for a PF, the largest circle can be selected by comparing the all the circles calculated by the software with each other. In GeoWizard (2018) extension for ArcGIS Desktop, MIC is available as a tool unde: “Miscellaneous” menu item. The hardware configuration and operating system of the computer used for performance test is a: following; a quad-core processor with 2.66 Ghz frequency and 16 GB of RAM on 64 bit Windows 10 operating system. Figure 9. Various runs of the script for a set of regular (A) and irregular and complex (B) polygon features.
Table 1. Comparison of MIC script produced for OpenJUMP with some other software programs and scripts For the comparison of performances of the MIC algorithm developed in this study and MIC tool in ET GeoWizarc another table is prepared. With this intention, for each PF, the radius of the MIC computed by the script created for thi study (fmicoj) is compared with the one calculated by ET GeoWizard (7yice:) in order to obtain a comparison paramete (comp). Subsequently, if rico; iS greater than pice, Value of comp iS set tO (Mnicoj - Tmicet)* 100/Mnicet Hf ico 18 smaller tha: Tmice Value of comp is set tO (Mnicer - Nnicoj)*100/Fmico;. Lastly, the number of PFs is counted according to the interval defined for the value of comp and shown in Table 2.
Table 1. Comparison of MIC script produced for OpenJUMP with some other software programs and scripts For the comparison of performances of the MIC algorithm developed in this study and MIC tool in ET GeoWizarc another table is prepared. With this intention, for each PF, the radius of the MIC computed by the script created for thi study (fmicoj) is compared with the one calculated by ET GeoWizard (7yice:) in order to obtain a comparison paramete (comp). Subsequently, if rico; iS greater than pice, Value of comp iS set tO (Mnicoj - Tmicet)* 100/Mnicet Hf ico 18 smaller tha: Tmice Value of comp is set tO (Mnicer - Nnicoj)*100/Fmico;. Lastly, the number of PFs is counted according to the interval defined for the value of comp and shown in Table 2.
Table 2. Comparison of MIC script run in OpenJUMP with ET GeoWizards’ MIC tool. What is evident from Table 2 is that, for all of the convex regular PFs and 96.33% of the irregular & complex PFs, the MIC calculated by the script created for the algorithm developed in this study is larger than the one calculated by MIC tool in ET GeoWizard, which unveils that the MIC algorithm developed in this study is not only faster but also more successful in finding MIC compared with the alternative software.
Table 2. Comparison of MIC script run in OpenJUMP with ET GeoWizards’ MIC tool. What is evident from Table 2 is that, for all of the convex regular PFs and 96.33% of the irregular & complex PFs, the MIC calculated by the script created for the algorithm developed in this study is larger than the one calculated by MIC tool in ET GeoWizard, which unveils that the MIC algorithm developed in this study is not only faster but also more successful in finding MIC compared with the alternative software.
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