# Problem defining a polygon

I have two polygons that are very similar as:

poly1 =
Polygon[{{1, 19/1650}, {2, 19/1650}, {3, 19/1650}, {4, 19/1650}, {5,   19/1650}, {6, 19/1650}, {7, 19/1650}, {8, 19/1650}, {9, 19/ 1650}, {10, 19/1650}, {11, 19/1650}, {12, 17/1650}, {13, 17/   1650}, {14, 17/1650}, {15, 17/1650}, {16, 17/1650}, {17, 17/   1650}, {18, 17/1650}, {19, 17/1650}, {20, 17/1650}, {21, 17/   1650}, {22, 17/1650}, {23, 17/1650}, {24, 17/1650}, {25, 17/   1650}, {26, 17/1650}, {27, 1/110}, {28, 1/110}, {29, 1/110}, {30,    1/110}, {31, 1/110}, {32, 1/110}, {33, 1/110}, {34, 13/1650}, {35,    13/1650}, {36, 13/1650}, {37, 13/1650}, {38, 13/1650}, {39, 1/   150}, {40, 1/150}, {41, 1/150}, {42, 1/150}, {43, 3/550}, {44, 3/   550}, {45, 3/550}, {46, 7/1650}, {47, 7/1650}, {48, 1/330}, {49, 1/   330}, {50, 1/   550}, {50, -(1/1650)}, {49, -(1/550)}, {48, -(1/330)}, {47, -(1/    330)}, {46, -(1/330)}, {45, -(7/1650)}, {44, -(7/1650)}, {43, -(7/    1650)}, {42, -(3/550)}, {41, -(3/550)}, {40, -(3/550)}, {39, -(1/    150)}, {38, -(1/150)}, {37, -(1/150)}, {36, -(1/150)}, {35, -(1/    150)}, {34, -(1/150)}, {33, -(13/1650)}, {32, -(13/1650)}, {31, -(    13/1650)}, {30, -(13/1650)}, {29, -(13/1650)}, {28, -(13/    1650)}, {27, -(13/1650)}, {26, -(13/1650)}, {25, -(1/    110)}, {24, -(1/110)}, {23, -(1/110)}, {22, -(1/110)}, {21, -(1/    110)}, {20, -(1/110)}, {19, -(1/110)}, {18, -(1/110)}, {17, -(1/    110)}, {16, -(1/110)}, {15, -(1/110)}, {14, -(1/110)}, {13, -(1/    110)}, {12, -(1/110)}, {11, -(1/110)}, {10, -(1/110)}, {9, -(1/    110)}, {8, -(1/110)}, {7, -(1/110)}, {6, -(1/110)}, {5, -(1/    110)}, {4, -(1/110)}, {3, -(1/110)}, {2, -(1/110)}, {1, -(1/    110)}}];


and

poly2 =
Polygon[{{1, 19/1650}, {2, 19/1650}, {3, 19/1650}, {4, 19/1650}, {5,    19/1650}, {6, 19/1650}, {7, 19/1650}, {8, 19/1650}, {9, 19/   1650}, {10, 19/1650}, {11, 19/1650}, {12, 19/1650}, {13, 19/   1650}, {14, 19/1650}, {15, 19/1650}, {16, 17/1650}, {17, 17/   1650}, {18, 17/1650}, {19, 17/1650}, {20, 17/1650}, {21, 17/   1650}, {22, 17/1650}, {23, 17/1650}, {24, 17/1650}, {25, 17/   1650}, {26, 17/1650}, {27, 17/1650}, {28, 1/110}, {29, 1/110}, {30,    1/110}, {31, 1/110}, {32, 1/110}, {33, 1/110}, {34, 1/110}, {35,    13/1650}, {36, 13/1650}, {37, 13/1650}, {38, 13/1650}, {39, 13/   1650}, {40, 1/150}, {41, 1/150}, {42, 1/150}, {43, 1/150}, {44, 3/   550}, {45, 3/550}, {46, 3/550}, {47, 7/1650}, {48, 7/1650}, {49, 1/   330}, {50, 1/330}, {51, 1/   550}, {51, -(1/1650)}, {50, -(1/550)}, {49, -(1/330)}, {48, -(1/    330)}, {47, -(1/330)}, {46, -(7/1650)}, {45, -(7/1650)}, {44, -(3/    550)}, {43, -(3/550)}, {42, -(3/550)}, {41, -(3/550)}, {40, -(1/    150)}, {39, -(1/150)}, {38, -(1/150)}, {37, -(1/150)}, {36, -(1/    150)}, {35, -(13/1650)}, {34, -(13/1650)}, {33, -(13/    1650)}, {32, -(13/1650)}, {31, -(13/1650)}, {30, -(13/    1650)}, {29, -(13/1650)}, {28, -(13/1650)}, {27, -(1/    110)}, {26, -(1/110)}, {25, -(1/110)}, {24, -(1/110)}, {23, -(1/    110)}, {22, -(1/110)}, {21, -(1/110)}, {20, -(1/110)}, {19, -(1/    110)}, {18, -(1/110)}, {17, -(1/110)}, {16, -(1/110)}, {15, -(1/    110)}, {14, -(1/110)}, {13, -(1/110)}, {12, -(1/110)}, {11, -(1/    110)}, {10, -(1/110)}, {9, -(1/110)}, {8, -(1/110)}, {7, -(1/    110)}, {6, -(1/110)}, {5, -(1/110)}, {4, -(1/110)}, {3, -(1/    110)}, {2, -(1/110)}, {1, -(1/110)}}]


Visually they look very similar, only that poly2 is a bit larger. But I can use poly2 to do all sort of functions. For example:

Area[poly2] == 457/550


In contrast, I cannot evaluate poly1 in any expression. Everything I evaluate returns unevaluated.

Does anybody have any ideas about why this may be happening?

## 2 Answers

Works for me in version 12.0:

poly1 = Polygon[{{1, 19/1650}, {2, 19/1650}, {3, 19/1650}, {4,
19/1650}, {5, 19/1650}, {6, 19/1650}, {7, 19/1650}, {8,
19/1650}, {9, 19/1650}, {10, 19/1650}, {11, 19/1650}, {12,
17/1650}, {13, 17/1650}, {14, 17/1650}, {15, 17/1650}, {16,
17/1650}, {17, 17/1650}, {18, 17/1650}, {19, 17/1650}, {20,
17/1650}, {21, 17/1650}, {22, 17/1650}, {23, 17/1650}, {24,
17/1650}, {25, 17/1650}, {26, 17/1650}, {27, 1/110}, {28,
1/110}, {29, 1/110}, {30, 1/110}, {31, 1/110}, {32, 1/110}, {33,
1/110}, {34, 13/1650}, {35, 13/1650}, {36, 13/1650}, {37,
13/1650}, {38, 13/1650}, {39, 1/150}, {40, 1/150}, {41,
1/150}, {42, 1/150}, {43, 3/550}, {44, 3/550}, {45, 3/550}, {46,
7/1650}, {47, 7/1650}, {48, 1/330}, {49, 1/330}, {50,
1/550}, {50, -(1/1650)}, {49, -(1/550)}, {48, -(1/
330)}, {47, -(1/330)}, {46, -(1/330)}, {45, -(7/
1650)}, {44, -(7/1650)}, {43, -(7/1650)}, {42, -(3/
550)}, {41, -(3/550)}, {40, -(3/550)}, {39, -(1/
150)}, {38, -(1/150)}, {37, -(1/150)}, {36, -(1/
150)}, {35, -(1/150)}, {34, -(1/150)}, {33, -(13/
1650)}, {32, -(13/1650)}, {31, -(13/1650)}, {30, -(13/
1650)}, {29, -(13/1650)}, {28, -(13/1650)}, {27, -(13/
1650)}, {26, -(13/1650)}, {25, -(1/110)}, {24, -(1/
110)}, {23, -(1/110)}, {22, -(1/110)}, {21, -(1/
110)}, {20, -(1/110)}, {19, -(1/110)}, {18, -(1/
110)}, {17, -(1/110)}, {16, -(1/110)}, {15, -(1/
110)}, {14, -(1/110)}, {13, -(1/110)}, {12, -(1/
110)}, {11, -(1/110)}, {10, -(1/110)}, {9, -(1/110)}, {8, -(1/
110)}, {7, -(1/110)}, {6, -(1/110)}, {5, -(1/110)}, {4, -(1/
110)}, {3, -(1/110)}, {2, -(1/110)}, {1, -(1/110)}}];Area[poly1] Perimeter[poly1]


53/66, $$\frac{\sqrt{680626}}{55}+\frac{68494}{825}$$

• It seems there was a problem before the update. Thank you! May 23, 2019 at 11:55

When I remove just one specific point, {12, 17/1650}. from poly1 to create a new polygon everything works as expected.

newPoly = DeleteCases[poly1, {12, 17/1650}, {2}];
Graphics[
{{FaceForm[None], EdgeForm[Black], poly1}, {FaceForm[None], EdgeForm[Red], newPoly}},
AspectRatio -> Full]


Area[newPoly]


221/275

Perimeter[poly3]


1/825 (67669 + 14 Sqrt[680626] + Sqrt[2722501])

RegionCentroid[newPoly]


{14515/663, 1273/1640925}

I think you have created a polygon that breaks some to region mensuration functions. I also note that RegionMeasure[poly1] reports the error

RegionMeasure::nmet: Unable to compute the measure of region Polygon[{{1,19/1650},{2,19/1650},{3,19/1650},{4,19/1650},{5,19/1650},{6,19/1650},{7,19/1650},{8,19/1650},{9,19/1650},{10,19/1650},<<32>>,{43,3/550},{44,3/550},{45,3/550},{46,7/1650},{47,7/1650},{48,1/330},{49,1/330},{50,1/550},<<50>>}].

I recommend you report this problem to Wolfram Research tech support.

• But how would you know which point to remove so that it works? May 23, 2019 at 11:42
• @NinjaCowAndForks. I coded a binary search. May 23, 2019 at 12:56