Point Pattern Matching Algorithm for Planar Point Sets under Euclidean Transform

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="139014.fig.002a" src="https://static.hindawi.com/articles/jam/volume-2012/139014/figures/139014.fig.002a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b><svg height="11.9625" id="M72" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 14.925 11.9625" width="14.925" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M457 332q0 -81 -41 -161.5t-105.5 -131.5t-129.5 -51q-46 0 -79 30.5t-33 86.5q0 18 7 51q18 95 46 187q6 19 6 33q0 7 -7 7q-24 0 -78 -64l-20 23q32 48 73 77t78 29q33 0 33 -46q0 -38 -10 -70q-28 -92 -39 -152q-7 -38 -7 -57q0 -78 69 -78q68 0 118 75.5t50 194.5
q0 43 -12 57q-11 12 -11 24q0 19 15 35.5t34 16.5q18 0 30.5 -34.5t12.5 -81.5z" id="x1D463"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.225,11.762)"><path d="M483 97q-42 -50 -88.5 -79.5t-68.5 -29.5q-37 0 -17 93l22 102h-2q-54 -79 -144 -149q-59 -46 -100 -46q-24 0 -43 29t-19 86q0 78 34.5 153t94.5 120q41 31 94 51.5t98 20.5q29 0 72 -9q26 -6 39 -6l2 -4q-30 -117 -67 -323q-8 -41 2 -41q16 0 79 58zM374 387
q-32 15 -73 15q-52 0 -83 -23q-48 -36 -78 -108.5t-30 -152.5q0 -33 8.5 -50.5t20.5 -17.5q31 0 107 79t99 132q15 40 29 126z" id="x1D44E"></path></g>
</svg>, <svg height="11.9625" id="M73" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 14.4875 11.9625" width="14.4875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D463"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.225,11.762)"><path d="M452 333q0 -82 -46 -164t-116 -128q-81 -53 -158 -53q-26 0 -51 13t-37 32q-26 41 -6 134l91 431q7 40 2.5 47t-34.5 7h-33l2 25q36 4 72.5 13t58.5 15.5t28 6.5q10 0 4 -29l-91 -391h2q65 77 130 116.5t105 39.5q36 0 56.5 -32t20.5 -83zM365 316q0 76 -35 76
q-31 0 -93.5 -49t-107.5 -110q-12 -37 -17 -70q-9 -65 12 -96.5t59 -31.5q31 0 56 14q53 29 89.5 105t36.5 162z" id="x1D44F"></path></g>
</svg>, and <svg height="11.9625" id="M74" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 12.15 11.9625" width="12.15" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D463"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.225,11.762)"><use xlink:href="#x1D456"></use></g>
</svg> are not collinear.</td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="139014.fig.002b" src="https://static.hindawi.com/articles/jam/volume-2012/139014/figures/139014.fig.002b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b><svg height="11.9625" id="M75" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 14.925 11.9625" width="14.925" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D463"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.225,11.762)"><use xlink:href="#x1D44E"></use></g>
</svg>, <svg height="11.9625" id="M76" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 14.4875 11.9625" width="14.4875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D463"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.225,11.762)"><use xlink:href="#x1D44F"></use></g>
</svg>, and <svg height="11.9625" id="M77" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 12.15 11.9625" width="12.15" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D463"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.225,11.762)"><use xlink:href="#x1D456"></use></g>
</svg> are collinear.</td></tr></table></td></tr></table>

<table>Two pairs of congruent complete graphs.</table>

Journal of Applied Mathematics

fig2

Figure 2

Figure 2: Point Pattern Matching Algorithm for Planar Point Sets under Euclidean Transform 