How to compare two polygons?

Question

I have these polygon points which are like these:

a = {Polygon[{{1504.1, 456.266}, {1539.8, 402.781}, {1588.17, 
    378.487}, {1528.17, 456.266}}], 

b=  Polygon[{{273.801, 456.266}, {273.801, 306.852}, {545.027, 
    306.852}, {587.642, 456.266}}], 

c=  Polygon[{{1495.53, 456.266}, {1500.18, 386.794}, {1539.8, 
    402.781}, {1504.1, 456.266}}], 

d= Polygon[{{1528.17, 456.266}, {1588.17, 378.487}, {1636.98, 
    345.708}, {1648.84, 456.266}}], 

e= Polygon[{{1569.07, 306.852}, {1494.36, 368.012}, {1031.56, 
    341.018}, {1028.45, 306.852}}], 

f= Polygon[{{1032.95, 456.266}, {1031.56, 341.018}, {1494.36, 
    368.012}, {1500.18, 386.794}, {1495.53, 456.266}}], 

g= Polygon[{{1648.84, 456.266}, {1636.98, 345.708}, {1641.75, 
    306.852}, {2025.09, 306.852}, {2025.09, 456.266}}], 

h= Polygon[{{587.642, 456.266}, {545.027, 306.852}, {1028.45, 
    306.852}, {1031.56, 341.018}, {1032.95, 456.266}}], 

i= Polygon[{{1641.75, 306.852}, {1636.98, 345.708}, {1588.17, 
    378.487}, {1539.8, 402.781}, {1500.18, 386.794}, {1494.36, 
    368.012}, {1569.07, 306.852}}]}

Now, from here, I want to extract the common points of i from a to h. So, suppose, the first value of i {1641.75, 306.852} is common in g. So, I want to extract g and save it. Then the second value {1636.98, 345.708} is common in d. So, then I want to extract d and save it. In this way, I want to find all the polygons that have one point which is common with i.

Please let me know how to do it.

Answer 1

Let's reformulate data a bit, you do not need all those variables and Polygons:

plgs = {

   {{1504.1, 456.266}, {1539.8, 402.781}, {1588.17, 378.487}, {1528.17, 456.266}},

   {{273.801, 456.266}, {273.801, 306.852}, {545.027, 306.852}, {587.642, 456.266}},

   {{1495.53, 456.266}, {1500.18, 386.794}, {1539.8, 402.781}, {1504.1, 456.266}},

   {{1528.17, 456.266}, {1588.17, 378.487}, {1636.98, 345.708}, {1648.84, 456.266}},

   {{1569.07, 306.852}, {1494.36, 368.012}, {1031.56, 341.018}, {1028.45, 306.852}},

   {{1032.95, 456.266}, {1031.56, 341.018}, {1494.36, 368.012}, {1500.18, 386.794}, {1495.53, 
     456.266}},

   {{1648.84, 456.266}, {1636.98, 345.708}, {1641.75, 306.852}, {2025.09, 306.852}, {2025.09, 
     456.266}},

   {{587.642, 456.266}, {545.027, 306.852}, {1028.45, 306.852}, {1031.56, 341.018}, {1032.95, 
     456.266}},

   {{1641.75, 306.852}, {1636.98, 345.708}, {1588.17, 378.487}, {1539.8, 402.781}, {1500.18, 
     386.794}, {1494.36, 368.012}, {1569.07, 306.852}}};

Let's get the picture, where orange is i-th polygon:

Graphics[{{FaceForm[], EdgeForm[Black], Polygon@plgs[[;; -2]]}, 
{FaceForm[Orange], EdgeForm[Black], Polygon@plgs[[-1]]}}]

Length of i-th:

iLength = Length[plgs[[-1]]]

7

Now compare the lengths:

neigh = Select[plgs[[;; -2]], 
Length[Union[Join[#, plgs[[-1]]]]] =!= Length[#] + iLength &];

and you got the neighbores:

Graphics[{{FaceForm[], EdgeForm[Black], Polygon@neigh}, 
{FaceForm[Orange], EdgeForm[Black], Polygon@plgs[[-1]]}}]

This works only if polygons have exactly the same points. If they do not, you can Round your data so points do become equivalent. Or use Other function in Select based, for example, on some EuclideanDistance less than a threshold criteria.