Why is this polygon "self intersected" and invalid?

Question

I have a polygon defined as follows:

segment from 43:29:02 N 003:50:39 E to 43:34:12 N 003:43:04 E arc(CW): center (43:34:49 N 003:58:16 E) from 43:34:12 N 003:43:04 E to 43:45:45 N 003:59:56 E segment from 43:45:45 N 003:59:56 E to 43:40:01 N 004:06:40 E arc(CCW): center(43:34:49 N 003:58:16 E) from 43:40:01 N 004:06:40 E to 43:29:02 N 003:50:39 E

The corresponding shapefile can be found at http://bordel.kataplop.net/mont2.zip

EDIT corresponding WKT:

POLYGON ((3.8441666666666667 43.4838888888888917, 3.7177777777777781 43.5700000000000003,3.7177777777777781 43.5700000000000003, 3.7175416922400166 3.5735823218689617, 3.7174122630334763 43.5767891717197671, 3.7173600981542299 3.5799971695780499, 3.7173852258405691 43.5832053383720393, 3.7174876507934393 3.5864127008216897, 3.7176673541599534 43.5896182797364418, 3.7179242935240802 3.5928210983129603, 3.7182584029045262 43.5960201804331646, 3.7186695927598370 3.5992145509620244, 3.7191577500007265 43.6024032360454115, 3.7197227380096543 3.6055852634076615, 3.7203643966676592 43.6087596626488931, 3.7210825423884666 3.6119254655418658, 3.7218769681598638 43.6150817063284819, 3.7227474435923660 3.6182274220155932, 3.7236937149751603 43.6213616526702097, 3.7247155053393355 3.6244834417140126, 3.7258125145283985 43.6275918362168653, 3.7269844192760679 3.6306858871895287, 3.7282308732913374 43.6337646498753315, 3.7295515073508057 3.6368271840406265, 3.7309459293982550 43.6398725542641515, 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3.8840729428592957 3.7529792515656695, 3.8882582034955018 43.7540532727700295, 3.8924689219104849 3.7550742161236030, 3.8967038055389538 43.7560417679131319, 3.9009615541956921 3.7569556308164564, 3.9052408604841538 43.7578155239962783, 3.9095404102075606 3.7586211831888079, 3.9138588827823697 43.7593723607871041, 3.9181949516539527 3.7600688259191912, 3.9225472847143648 43.7607103645209961, 3.9269145447220355 3.7612967794038497, 3.9312953897232594 43.7618278903168019, 3.9356884734753175 3.7623035340034718, 3.9400924458711075 43.7627235642536689, 3.9445059533651090 3.7630878519495354, 3.9489276394005572 43.7633962851063103, 3.9533561448376653 3.7636487689077285, 3.9577901083827460 43.7638452257359774, 3.9622281670180910 3.7639855951961891, 3.9666689564324482 43.7640698341355332, 3.9711111114519491 3.7640979166568798, 3.9755532664713464 43.7640698341269285, 3.9799940558853892 3.7639855951789798, 3.9844321145202120 43.7638452257101846, 3.9888660780645604 3.7636487688733453, 3.9932945835007279 43.7633962850633438, 3.9988888888888892 3.7625000000000028, 3.9988888888888892 43.7625000000000028, 4.1111111111111107 3.6669444444444395, 4.1111111111111107 43.6669444444444395, 4.1081112312987518 3.6694234695150385, 4.1059413722282461 43.6711419317362726, 4.1037302561558127 3.6728326982839690, 4.1014785562310649 43.6744952517733864, 4.0991869581943616 3.6761290834102880, 4.0968561601681621 43.6777336931488875, 4.0944868724443211 3.6793085898469968, 4.0920798172674075 43.6808532914185932, 4.0896357286140903 3.6823673249835380, 4.0871553519686730 43.6838502270144886, 4.0846394440948499 3.6853015434809535, 4.0820887728037327 43.6867208299904419, 4.0795041167182582 3.6881076519267424, 4.0768862650339983 43.6894615845849543, 4.0742360172765091 3.6907822133038266, 4.0715541830552358 43.6920691335946785, 4.0688415818141008 3.6933219512673716, 4.0660990425788137 43.6945402825530920, 4.0633274037010221 3.6957237542238488, 4.0605275125993510 43.6968720037087905, 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3.9020120451845597 3.7038953617291099, 3.8990388221270043 43.7030029014551289, 3.8960876878508262 3.7020729884658010, 3.8931595457504589 43.7011059077573663, 3.8902552920455964 3.7001019557029906, 3.8873758155029599 43.6990614399601753, 3.8845219971605558 3.6979846793748692, 3.8816947100545103 43.6968720038819711, 3.8788948189485768 3.6957237544024579, 3.8761231800663918 43.6945402827370586, 3.8733806408265834 3.6933219514566531, 3.8706680395807975 43.6920691337892180, 3.8679862053547485 3.6907822135035673, 3.8653359575923583 43.6894615847898251, 3.8627181059030749 3.6881076521366793, 3.8601334498124533 43.6867208302053953, 3.8575827785160706 3.6853015437008381, 3.8550668706368625 43.6838502272392475, 3.8525864939859447 3.6823673252131073, 3.8501424053270101 43.6808532916528804, 3.8477353501443670 3.6793085900859452, 3.8453660624146853 43.6777336933924261, 3.8430352643825345 3.6761290836583598, 3.8407436663397725 43.6744952520258849, 3.8384919664088604 3.6728326985408302, 3.8362808503301578 43.6711419319974112, 3.8341109912532807 3.6694234697803836, 3.8319830495325728 43.6676778377045736, 3.8298976725267480 3.6659055698518159, 3.8278554944027792 43.6641072084053476, 3.8258571359440796 3.6622833034818925, 3.8239032043630350 43.6604344129611803, 3.8219942931179487 3.6585611023132358, 3.8201309817344384 43.6566639444233147, 3.8183138356313546 3.6547435194146729, 3.8165434059512564 43.6528004144691408, 3.8148202293954965 3.6508352236455082, 3.8131448280639657 43.6488485476959980, 3.8115177092995394 3.6468409938806090, 3.8099393655372626 43.6448131757796389, 3.8084102741583297 3.6427657131041897, 3.8069308973488800 43.6406992315050104, 3.8055016819636593 3.6386143623794922, 3.8041230593945863 43.6365117426770084, 3.8027954454442381 3.6343920147025415, 3.8015192402043150 43.6322558259189250, 3.8002948279390880 3.6301038287473517, 3.7991225769738843 43.6279366803666306, 3.7980028395886105 3.6257550425109812, 3.7969359519163612 43.6235595812664769, 3.7959222338471279 3.6213509668663235, 3.7949619889366235 43.6191298734849511, 3.7940555043202560 3.6168969790308338, 3.7932030506322558 43.6146529649385144, 3.7924048819299858 3.6123985159593701, 3.7916612356234385 43.6101343199516833, 3.7909723324099347 3.6078610676696670, 3.7903383762140490 43.6055794525518365, 3.7897595541327438 3.6032901705085649, 3.7892360363857511 43.6009939197090617, 3.7887679762711817 3.5986914003676489, 3.7883555101263844 43.5963833145296107, 3.7879987572940457 3.5940703658565170, 3.7876978200935398 43.5917532594112060, 3.7874527837975256 3.5894327014423624, 3.7872637166137779 43.5871093991689307, 3.7871306696722700 3.5847840605642105, 3.7870536770174819 43.5824573941399649, 3.7870327556059387 3.5801301087303372, 3.7870679053089629 43.5778029132758391, 3.7871591089206373 3.5754765166073952, 3.7873063321709624 43.5731516272305015, 3.7875095237441907 3.5708289531096042, 3.7877686153023324 43.5685092014527555, 3.7880835215138080 3.5661930784965250, 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I'm using shapely to handle my data, and it reports that this polygon is self intersecting. I checked with QGis (it is also complaining about this problem), and I don't see why it would be self-intersecting.

I've narrowed down the problem to the 2 "single points" of the second arc (beginning and end). It looks like the 2 segments are considered as crossing the second arc... Which is not true. I'm not producing the data but I can ask for fixing it in case it is wrong. But in this case, I don't understand the problem (if any).

Is this an issue of some buggy python library (shared by shapely and qgis) or is it something else ?

Answer 1

If I zoom in here...

I see this:

I think the issue stems from the limitation that a shapefile cannot store arcs. Instead it does a linear approximation. When a polyline intersects (what is intended to be) a circular arc at a tangent the zig zag approximation results in an intersection.

I'd recommend storing the geometry in something that supports circular arcs, like a file geodatabase. There is an open source API available for it. When was Esri File Geodatabase API made available?

Answer 2

There is something strange about the specification of this polygon. The first arc has parameters

center (43:34:49 N 003:58:16 E) from 43:34:12 N 003:43:04 E to 43:45:45 N 003:59:56 E

These have decimal coordinates {3.971111111, 43.58027778}, {3.717777778, 43.57}, and {3.998888889, 43.7625}, respectively. The Haversine formula for spherical distances says the distance from the first point to the center is 0.232751 radians whereas the distance from the second point to the center is 0.14124 radians. Because those distances are different, the specified arc cannot exist. (If such a specification makes sense to some software, it must be applying an idiosyncratic interpretation. I would be interested in knowing what that interpretation is.)

Answer 3

Well it displays in UDiG though whether this indicates validity I'm not sure. For any one wanting to check the WKT is:

MULTIPOLYGON (((3.8441666666666667 43.48388888888889, 3.717777777777778 43.57, 3.717777777777778 43.57, 3.7175416922400166 43.57358232186896, 3.7174122630334763 43.57678917171977, 3.71736009815423 43.57999716957805, 3.717385225840569 43.58320533837204, 3.7174876507934393 43.58641270082169, 3.7176673541599534 43.58961827973644, 3.71792429352408 43.59282109831296, 3.7182584029045262 43.596020180433165, 3.718669592759837 43.599214550962024, 3.7191577500007265 43.60240323604541, 3.7197227380096543 43.60558526340766, 3.720364396667659 43.60875966264889, 3.7210825423884666 43.611925465541866, 3.721876968159864 43.61508170632848, 3.722747443592366 43.61822742201559, 3.7236937149751603 43.62136165267021, 3.7247155053393355 43.62448344171401, 3.7258125145283985 43.627591836216865, 3.726984419276068 43.63068588718953, 3.7282308732913374 43.63376464987533, 3.7295515073508057 43.63682718404063, 3.730945929398255 43.63987255426415, 3.7324137246514693 43.64289983022496, 3.7339544557162703 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