(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 363760, 9871]*) (*NotebookOutlinePosition[ 364405, 9893]*) (* CellTagsIndexPosition[ 364361, 9889]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Point inside a polygon", "Title"], Cell[TextData[{ "Solution 1 to 3 are from ", StyleBox["MathGroup", FontSlant->"Italic"], " September 2000\nSolution 4 is from Paul Wellin" }], "Text"], Cell["Please feel free to add further relevant solutions", "Section"], Cell["Solution 1 (fastest and works on non-convex)", "Section"], Cell[CellGroupData[{ Cell["Point location w.r. to a simple polygon", "Section"], Cell[TextData[{ "Adriano Pascoletti\n", StyleBox[ButtonBox["pascolet at dimi.uniud.it", ButtonData:>{ URL[ "mailto:pascolet at dimi.uniud.it"], None}], ButtonBoxOptions -> {DefaultStyle -> "Hyperlink"}] }], "Text"], Cell[TextData[{ "PLSPolygon (Point Location w.r.to a simple polygon) takes a list of 2D \ points (poly), a query point (q) and returns ", StyleBox["inside", FontSlant->"Italic"], " if q is in the interior of poly, ", StyleBox["outside", FontSlant->"Italic"], " if q is on the boundary or in the exterior." }], "Text"], Cell[TextData[{ "The approach follows the ", Cell[BoxData[ \(TraditionalForm\`O(n)\)]], " method suggested constructive proof of Theorem 2.1, Sect. 2.2 of\n\ F.P.Preparata, M.I.Shamos: Computational Geometry, Springer 1985. \n\nIf a \ horizontal line goes from -\[Infinity] to q then count the edges crossing the \ halfline strictly before q: if there is an odd number of intersections then q \ belongs to the interior of poly.\nA symbolic sufficiently small clockwise \ rotation of the halfline around q lets us assume that the halfline doesn't \ pass through any vertex.\nIn view of this it is enough considering \n(1) non \ horizontal edges ", Cell[BoxData[ \(TraditionalForm\`\((a, b)\)\)]], " s.t. \n(2) one vertex has the y-coordinate \[GreaterEqual] ", Cell[BoxData[ \(TraditionalForm\`q\_y\)]], " and the other ", Cell[BoxData[ \(TraditionalForm\`\(\(<\)\(q\_y\)\)\)]], ".\n If ", Cell[BoxData[ \(TraditionalForm\`a\ b\)]], " is such an edge one can assume ", Cell[BoxData[ \(TraditionalForm\`a\_y > b\_y\)]], "(see 3) and ", Cell[BoxData[ \(TraditionalForm\`a\ b\)]], " intersects the halfline through q before q iff the signed area of the \ triangle ", Cell[BoxData[ \(TraditionalForm\`a\ b\ q\)]], " is positive. \n\nThe signed double area is computed by means of ", Cell[BoxData[ FormBox[ RowBox[{"det", "(", GridBox[{ {\(a\_x\), \(a\_y\), "1"}, {\(b\_x\), \(b\_y\), "1"}, {\(q\_x\), \(q\_y\), "1"} }], ")"}], TraditionalForm]]], "\nThe algorithm doesn't require floating point divisions.\n\n" }], "Text"], Cell[BoxData[ \(\(PLSPolygon[poly : {\(({_, _})\) .. }, q : {x_, y_}] /; Length[poly] \[GreaterEqual] 3 := Module[{edges = \((Append[poly, First[poly]] // Partition[#, 2, 1] &)\), temp}, \[IndentingNewLine] (*1 : \ drop\ horizontal\ edges\ *) \[IndentingNewLine]temp = DeleteCases[ edges, {{x1_, yy_}, {x2_, yy_}}]; \[IndentingNewLine] (*2 : \ drop\ edges\ with\ both\ vertex\ ordinates\ \[GreaterEqual] y\ or\ with\ both\ < \ y\ *) \ \[IndentingNewLine]temp = DeleteCases[ temp, {{x1_, y1_}, {x2_, y2_}} /; Min[y1, y2] \[GreaterEqual] y \[Or] Max[y1, y2] < y]; \[IndentingNewLine] (*3 : \ edges\ in\ temp\ intersect\ a\ horizontal\ line\ through\ q; \ direct\ them\ downwards\ *) \[IndentingNewLine]temp = Map[If[#\[LeftDoubleBracket]1, 2\[RightDoubleBracket] > #\[LeftDoubleBracket]2, 2\[RightDoubleBracket], #, Reverse[#]] &, temp]; \[IndentingNewLine] (*4 : \ an\ edge\ \((a, b)\)\ with\ y \((a)\) > y \((b)\)\ intersects\ the\ 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0 1 0 r .48434 .13567 Mdot 1 0 0 r .92097 .34546 Mdot 0 1 0 r .59874 .38737 Mdot 1 0 0 r .87542 .55545 Mdot .04313 .03785 Mdot 0 1 0 r .41496 .39495 Mdot .6127 .33048 Mdot 1 0 0 r .6582 .0595 Mdot .89037 .09071 Mdot 0 1 0 r .56068 .07541 Mdot 1 0 0 r .07252 .54594 Mdot .75519 .12627 Mdot 0 1 0 r .56439 .43935 Mdot 1 0 0 r .81043 .38472 Mdot .94185 .08107 Mdot .91121 .43319 Mdot .92253 .0723 Mdot 0 1 0 r .52006 .06732 Mdot .33364 .34573 Mdot 1 0 0 r .83806 .03691 Mdot 0 1 0 r .41948 .2841 Mdot 1 0 0 r .30119 .5654 Mdot 0 1 0 r .37077 .34208 Mdot .52221 .46822 Mdot 1 0 0 r .78259 .50664 Mdot .68799 .11258 Mdot .81695 .45466 Mdot 0 1 0 r .75298 .2833 Mdot 1 0 0 r .87062 .41145 Mdot 0 1 0 r .25674 .24507 Mdot .56079 .0948 Mdot .39488 .23725 Mdot 1 0 0 r .16512 .41461 Mdot 0 1 0 r .1175 .27576 Mdot 1 0 0 r .77056 .10162 Mdot 0 1 0 r .5715 .41145 Mdot 1 0 0 r .96418 .19889 Mdot .85972 .32796 Mdot .17105 .34814 Mdot .13054 .07374 Mdot .27663 .54061 Mdot .85002 .43259 Mdot .69205 .47489 Mdot 0 1 0 r .47894 .22442 Mdot .55073 .08936 Mdot 1 0 0 r .38525 .55258 Mdot .75637 .59166 Mdot .78996 .17022 Mdot .7684 .42185 Mdot .90644 .44617 Mdot .62116 .1028 Mdot .79971 .40151 Mdot 0 1 0 r .36834 .1661 Mdot 1 0 0 r .9259 .57284 Mdot 0 1 0 r .6525 .29513 Mdot .47077 .3775 Mdot .12558 .23485 Mdot 1 0 0 r .10933 .42883 Mdot 0 1 0 r .34542 .2471 Mdot .29558 .2877 Mdot .55322 .42916 Mdot 1 0 0 r .36535 .44544 Mdot .90827 .35545 Mdot 0 1 0 r .54184 .07302 Mdot .56374 .21844 Mdot .59215 .10409 Mdot 1 0 0 r .88744 .52722 Mdot .14519 .33051 Mdot 0 1 0 r .78567 .32146 Mdot 1 0 0 r .05966 .50559 Mdot .46407 .10344 Mdot 0 1 0 r .74029 .24698 Mdot 1 0 0 r .88705 .27819 Mdot 0 1 0 r .39875 .40545 Mdot 1 0 0 r .95499 .52665 Mdot .83312 .36152 Mdot 0 1 0 r .41506 .3373 Mdot .26478 .28651 Mdot .50382 .41398 Mdot 1 0 0 r .1434 .55991 Mdot .69436 .12161 Mdot .10755 .0834 Mdot 0 1 0 r .2541 .04725 Mdot 1 0 0 r .34347 .44033 Mdot 0 1 0 r .34326 .37297 Mdot 1 0 0 r .92092 .06397 Mdot .36448 .45024 Mdot .11162 .30636 Mdot .92563 .14202 Mdot .82305 .04894 Mdot 0 1 0 r .44562 .33195 Mdot 1 0 0 r .70346 .09294 Mdot 0 1 0 r .72747 .23943 Mdot 1 0 0 r .61972 .03862 Mdot 0 1 0 r .49718 .22126 Mdot .30006 .2022 Mdot 1 0 0 r .17773 .4522 Mdot 0 1 0 r .35535 .16732 Mdot 1 0 0 r .78946 .03105 Mdot .26754 .46487 Mdot .84003 .49294 Mdot 0 1 0 r .4207 .44501 Mdot .41823 .19007 Mdot .69345 .38116 Mdot 1 0 0 r .66697 .55456 Mdot .09754 .37162 Mdot .1936 .36238 Mdot 0 1 0 r .77368 .19851 Mdot 1 0 0 r .03968 .51409 Mdot .44215 .06028 Mdot .22643 .51212 Mdot 0 1 0 r .19842 .19932 Mdot 1 0 0 r .36261 .04826 Mdot 0 1 0 r .75393 .35822 Mdot 1 0 0 r .92059 .4621 Mdot .08429 .58098 Mdot .27743 .51146 Mdot .96296 .23844 Mdot 0 1 0 r .10764 .17816 Mdot .21308 .06902 Mdot 1 0 0 r .09177 .26799 Mdot .74714 .03783 Mdot .84154 .35978 Mdot 0 1 0 r .57253 .44241 Mdot .50274 .3406 Mdot 1 0 0 r .79482 .11327 Mdot 0 1 0 r .55836 .48241 Mdot 1 0 0 r .73434 .13621 Mdot .30474 .57487 Mdot .74759 .50168 Mdot .22091 .42579 Mdot 0 1 0 r .55832 .46175 Mdot .15295 .18689 Mdot 1 0 0 r .78739 .45301 Mdot .28761 .43102 Mdot 0 1 0 r .23866 .03968 Mdot 1 0 0 r .76108 .1195 Mdot .42005 .53032 Mdot 0 1 0 r .22653 .241 Mdot .66192 .4232 Mdot 1 0 0 r .89799 .27005 Mdot .89054 .52543 Mdot .7009 .44817 Mdot .35603 .09276 Mdot 0 1 0 r .57176 .29037 Mdot .54485 .24367 Mdot 1 0 0 r .30796 .46326 Mdot 0 1 0 r .33 .23307 Mdot .52308 .37285 Mdot 1 0 0 r .88616 .30667 Mdot 0 1 0 r .32037 .16093 Mdot 1 0 0 r .24804 .48738 Mdot .39858 .4948 Mdot .33371 .56587 Mdot .67389 .07571 Mdot .95389 .50219 Mdot .12594 .38926 Mdot 0 1 0 r .43284 .28761 Mdot 1 0 0 r .7942 .52991 Mdot .12665 .08362 Mdot 0 1 0 r .29492 .18614 Mdot 1 0 0 r .2167 .38086 Mdot .95076 .0543 Mdot .94487 .49739 Mdot 0 1 0 r .57599 .16341 Mdot 1 0 0 r .63497 .53544 Mdot .8783 .11678 Mdot .65729 .06234 Mdot 0 1 0 r .77617 .33144 Mdot 1 0 0 r .24826 .37864 Mdot .95818 .40544 Mdot 0 1 0 r .14542 .32411 Mdot .68707 .24838 Mdot 1 0 0 r .90492 .54716 Mdot 0 1 0 r .71252 .22317 Mdot 1 0 0 r .93626 .07885 Mdot 0 1 0 r .16034 .08885 Mdot .3251 .14732 Mdot 1 0 0 r .25825 .57598 Mdot .64402 .11407 Mdot 0 1 0 r .45829 .27363 Mdot .41958 .33934 Mdot .47631 .4721 Mdot 1 0 0 r .29797 .04432 Mdot 0 1 0 r .76545 .2528 Mdot 1 0 0 r .36926 .10107 Mdot .07674 .05871 Mdot .4092 .0513 Mdot .8926 .57378 Mdot .10791 .50789 Mdot .65816 .60171 Mdot 0 1 0 r .44009 .42291 Mdot .22369 .35716 Mdot 1 0 0 r .04433 .11265 Mdot .72358 .48897 Mdot .72257 .09742 Mdot .93434 .26526 Mdot .37712 .60026 Mdot .88141 .23563 Mdot .94413 .57805 Mdot .96502 .26577 Mdot .86003 .09924 Mdot 0 1 0 r .33067 .26797 Mdot .44374 .28024 Mdot 1 0 0 r .13079 .51472 Mdot 0 1 0 r .42322 .19668 Mdot 1 0 0 r .38341 .05483 Mdot .67686 .12834 Mdot 0 1 0 r .42528 .39349 Mdot .32355 .13199 Mdot .52007 .18694 Mdot .35563 .15786 Mdot 1 0 0 r .53126 .52509 Mdot .47182 .0877 Mdot 0 1 0 r .22441 .2862 Mdot 1 0 0 r .05188 .41137 Mdot .11743 .37539 Mdot 0 1 0 r .60487 .24378 Mdot .71022 .34964 Mdot 1 0 0 r .90421 .14452 Mdot .30875 .56007 Mdot .60447 .04162 Mdot .76488 .40221 Mdot .27265 .48768 Mdot .25743 .48104 Mdot .77704 .42906 Mdot .05684 .22393 Mdot .74897 .04678 Mdot .91562 .45245 Mdot .16791 .40691 Mdot 0 1 0 r .22921 .13189 Mdot .2399 .29147 Mdot 1 0 0 r .89666 .17573 Mdot 0 1 0 r .61164 .27893 Mdot 1 0 0 r .15559 .37743 Mdot 0 1 0 r .3628 .39517 Mdot 1 0 0 r .87436 .5003 Mdot .56196 .57002 Mdot 0 1 0 r .84134 .30546 Mdot 1 0 0 r .78921 .55233 Mdot .90193 .45693 Mdot 0 1 0 r .64511 .17451 Mdot .69653 .35412 Mdot 1 0 0 r .42902 .48695 Mdot 0 1 0 r .77608 .20748 Mdot .79359 .2371 Mdot .6443 .43396 Mdot .4546 .44585 Mdot 1 0 0 r .74614 .53757 Mdot .86884 .47974 Mdot .88102 .26119 Mdot .10344 .53132 Mdot .9553 .40817 Mdot 0 1 0 r .43455 .38589 Mdot .28257 .08313 Mdot 1 0 0 r .02934 .50285 Mdot 0 1 0 r .4827 .47957 Mdot .21196 .29484 Mdot 1 0 0 r .81461 .07469 Mdot .73357 .4529 Mdot 0 1 0 r .09228 .14104 Mdot 1 0 0 r .84094 .57708 Mdot .18747 .48376 Mdot .7613 .07485 Mdot 0 1 0 r .20838 .10467 Mdot .35057 .29286 Mdot 1 0 0 r .90201 .05062 Mdot 0 1 0 r .34504 .39392 Mdot .44312 .17497 Mdot .15688 .12817 Mdot .60472 .12936 Mdot .39952 .27918 Mdot 1 0 0 r .53625 .59223 Mdot 0 1 0 r .53479 .30601 Mdot .3726 .13756 Mdot .7497 .26025 Mdot .18803 .06197 Mdot 1 0 0 r .42294 .5713 Mdot .26222 .04044 Mdot 0 1 0 r .10171 .20647 Mdot 1 0 0 r .7953 .46938 Mdot .92103 .10738 Mdot .2144 .36911 Mdot 0 1 0 r .54532 .43212 Mdot .65435 .38079 Mdot 1 0 0 r .03434 .15519 Mdot 0 1 0 r .30556 .27232 Mdot 1 0 0 r .26085 .49885 Mdot 0 1 0 r .14135 .23943 Mdot 1 0 0 r .81412 .53147 Mdot 0 1 0 r .85534 .22808 Mdot .73622 .35408 Mdot 1 0 0 r .08384 .3626 Mdot 0 1 0 r .79139 .27579 Mdot 1 0 0 r .84565 .5974 Mdot .26988 .44758 Mdot 0 1 0 r .21511 .2457 Mdot .25935 .32147 Mdot 1 0 0 r .88576 .57729 Mdot .9747 .42653 Mdot 0 1 0 r .76822 .36695 Mdot 1 0 0 r .18439 .49897 Mdot .88909 .16796 Mdot 0 1 0 r .42438 .17398 Mdot 1 0 0 r .82906 .40927 Mdot .60919 .5021 Mdot .95961 .41578 Mdot .36312 .08361 Mdot 0 1 0 r .7683 .19916 Mdot 1 0 0 r .12758 .36605 Mdot 0 1 0 r .85875 .22578 Mdot 1 0 0 r .12909 .54343 Mdot .11435 .46274 Mdot .92091 .07354 Mdot 0 1 0 r .20147 .32387 Mdot .52034 .50348 Mdot 1 0 0 r .34862 .51852 Mdot .88735 .03046 Mdot 0 1 0 r .36522 .13183 Mdot 1 0 0 r .54804 .55077 Mdot .57312 .53657 Mdot 0 1 0 r .44427 .2138 Mdot .69057 .33988 Mdot .33899 .27428 Mdot .60004 .48105 Mdot .39429 .22983 Mdot .42238 .18626 Mdot 1 0 0 r .85016 .33026 Mdot 0 1 0 r .09757 .27166 Mdot 1 0 0 r .93902 .32888 Mdot .70856 .16892 Mdot 0 1 0 r .41479 .38203 Mdot .15925 .23625 Mdot 1 0 0 r .94673 .19731 Mdot .44489 .50029 Mdot .63155 .52694 Mdot .82106 .04833 Mdot 0 1 0 r .26106 .3262 Mdot 1 0 0 r .42248 .46598 Mdot .38711 .59985 Mdot 0 1 0 r .34872 .22341 Mdot .4243 .30005 Mdot 1 0 0 r .61636 .08358 Mdot .03332 .52193 Mdot 0 1 0 r .48092 .45124 Mdot 1 0 0 r .06281 .35371 Mdot .05984 .55486 Mdot 0 1 0 r .40747 .43068 Mdot 1 0 0 r .215 .53562 Mdot 0 1 0 r .17021 .13357 Mdot 1 0 0 r .76872 .09872 Mdot .75931 .13763 Mdot .44381 .47922 Mdot .35882 .4415 Mdot .80366 .42473 Mdot .3493 .52347 Mdot .34655 .5774 Mdot 0 1 0 r .31031 .19885 Mdot 1 0 0 r .31051 .05163 Mdot .87905 .37208 Mdot 0 1 0 r .11933 .11993 Mdot .73264 .2676 Mdot 1 0 0 r .32681 .05029 Mdot .94954 .15905 Mdot 0 1 0 r .85921 .17498 Mdot .61453 .32147 Mdot 1 0 0 r .07936 .35417 Mdot .28904 .40191 Mdot 0 1 0 r .70902 .38067 Mdot 1 0 0 r .95494 .23214 Mdot 0 1 0 r .42231 .35813 Mdot 1 0 0 r .09971 .46397 Mdot 0 1 0 r .3268 .26729 Mdot .34327 .22545 Mdot 1 0 0 r .97619 .24608 Mdot .36994 .09548 Mdot 0 1 0 r .14079 .10018 Mdot .73161 .37792 Mdot 1 0 0 r .08524 .34992 Mdot .46638 .57993 Mdot .35243 .57316 Mdot 0 1 0 r .48764 .37688 Mdot 1 0 0 r .90633 .24412 Mdot .41174 .51682 Mdot .60334 .58074 Mdot .09228 .32046 Mdot 0 1 0 r .60336 .36375 Mdot .69855 .25406 Mdot .48638 .29266 Mdot 1 0 0 r .94315 .48005 Mdot .42494 .54665 Mdot 0 1 0 r .50058 .50403 Mdot 1 0 0 r .09632 .5774 Mdot .03674 .15624 Mdot .1662 .36237 Mdot 0 1 0 r .60121 .24333 Mdot .53907 .38554 Mdot 1 0 0 r .53273 .52678 Mdot .91191 .05087 Mdot 0 1 0 r .81039 .3018 Mdot .44935 .36212 Mdot 1 0 0 r .84345 .42567 Mdot .04821 .41938 Mdot .36669 .52555 Mdot .9281 .44589 Mdot 0 1 0 r .35375 .39839 Mdot 1 0 0 r .78571 .11261 Mdot 0 1 0 r .72875 .18415 Mdot .27046 .33099 Mdot .21983 .26128 Mdot .33475 .30921 Mdot 1 0 0 r .38565 .56339 Mdot .86161 .551 Mdot 0 1 0 r .5184 .16681 Mdot 1 0 0 r .83721 .1607 Mdot 0 1 0 r .17553 .24517 Mdot 1 0 0 r .88531 .31872 Mdot .79798 .45069 Mdot 0 1 0 r .12341 .23519 Mdot 1 0 0 r .09304 .29562 Mdot .82916 .50812 Mdot .84942 .06343 Mdot 0 1 0 r .51822 .228 Mdot 1 0 0 r .48758 .10395 Mdot 0 1 0 r .63282 .28091 Mdot 1 0 0 r .94539 .54106 Mdot .77182 .14929 Mdot 0 1 0 r .79367 .32497 Mdot 1 0 0 r .86271 .43449 Mdot .9719 .47819 Mdot 0 1 0 r .76311 .22838 Mdot 1 0 0 r .90267 .21165 Mdot .91015 .32418 Mdot 0 1 0 r .07706 .17731 Mdot 1 0 0 r .41574 .12527 Mdot 0 1 0 r .56569 .10244 Mdot 1 0 0 r .75913 .44827 Mdot 0 1 0 r .59651 .1653 Mdot 1 0 0 r .96352 .32806 Mdot .77905 .44424 Mdot .12461 .49749 Mdot .78336 .56996 Mdot 0 1 0 r .33771 .29819 Mdot 1 0 0 r .8569 .3874 Mdot .40377 .57793 Mdot 0 1 0 r .80365 .23917 Mdot 1 0 0 r .96424 .48175 Mdot 0 1 0 r .26177 .16581 Mdot .22892 .06256 Mdot 1 0 0 r .64147 .0296 Mdot 0 1 0 r .24161 .33841 Mdot .83862 .18926 Mdot 1 0 0 r .14081 .44484 Mdot 0 1 0 r .07907 .22321 Mdot .7793 .17573 Mdot 1 0 0 r .19838 .43973 Mdot 0 1 0 r .39934 .20172 Mdot .37093 .22965 Mdot .41131 .32388 Mdot 1 0 0 r .13297 .09292 Mdot 0 1 0 r .2062 .29041 Mdot 1 0 0 r .46771 .09241 Mdot .94079 .55591 Mdot .6053 .50706 Mdot .8238 .14016 Mdot 0 1 0 r .55004 .31293 Mdot 1 0 0 r .06831 .56833 Mdot .37547 .47711 Mdot 0 1 0 r .64518 .3957 Mdot 1 0 0 r .02835 .27655 Mdot 0 1 0 r .25768 .1009 Mdot 1 0 0 r .87158 .21272 Mdot .07529 .41441 Mdot .42768 .14939 Mdot .1107 .46241 Mdot 0 1 0 r .79859 .24625 Mdot .26311 .35134 Mdot 1 0 0 r .27236 .53723 Mdot .21861 .38692 Mdot .8731 .0892 Mdot .54964 .59513 Mdot .86856 .41656 Mdot .31578 .52331 Mdot .97319 .23293 Mdot 0 1 0 r .2643 .13799 Mdot .56932 .11263 Mdot .17741 .27949 Mdot 1 0 0 r .74693 .47029 Mdot .8905 .53207 Mdot .49838 .53697 Mdot 0 1 0 r .6957 .17424 Mdot .60148 .47686 Mdot .16986 .18302 Mdot 1 0 0 r .70913 .08938 Mdot 0 1 0 r .8303 .26362 Mdot 1 0 0 r .71215 .46035 Mdot 0 1 0 r .58981 .15472 Mdot 1 0 0 r .16664 .37681 Mdot .43621 .47913 Mdot .39591 .51043 Mdot .52191 .55098 Mdot .87374 .57737 Mdot .80242 .40583 Mdot 0 1 0 r .29607 .1296 Mdot .65637 .2519 Mdot .56315 .06931 Mdot 1 0 0 r .80228 .59219 Mdot 0 1 0 r .82721 .21287 Mdot 1 0 0 r .23628 .46656 Mdot 0 1 0 r .68438 .43997 Mdot 1 0 0 r .77628 .59133 Mdot .31227 .53345 Mdot 0 1 0 r .27818 .06944 Mdot 1 0 0 r .41474 .55999 Mdot 0 1 0 r .45196 .26752 Mdot 1 0 0 r .14248 .45947 Mdot .7718 .04471 Mdot 0 1 0 r .55554 .41924 Mdot 1 0 0 r .94573 .05644 Mdot 0 1 0 r .70454 .23546 Mdot .73326 .19379 Mdot 1 0 0 r .04397 .3994 Mdot .93318 .20637 Mdot .70791 .46987 Mdot .67882 .16602 Mdot .31698 .51379 Mdot .25066 .50241 Mdot 0 1 0 r .19831 .08341 Mdot 1 0 0 r .45506 .48678 Mdot 0 1 0 r .61898 .26808 Mdot .48554 .45943 Mdot 1 0 0 r .89065 .06171 Mdot 0 1 0 r .72849 .29472 Mdot 1 0 0 r .87049 .26622 Mdot .77151 .11744 Mdot .18639 .40027 Mdot .11651 .55534 Mdot .84562 .49038 Mdot .84205 .08202 Mdot 0 1 0 r .67112 .43606 Mdot .4108 .19915 Mdot .07595 .19706 Mdot 1 0 0 r .90147 .34363 Mdot 0 1 0 r .16151 .16444 Mdot .19679 .07799 Mdot 1 0 0 r .26723 .50213 Mdot .40149 .56446 Mdot 0 1 0 r .10465 .13095 Mdot 1 0 0 r .30879 .03821 Mdot 0 1 0 r .23524 .24447 Mdot 1 0 0 r .44294 .56011 Mdot 0 1 0 r .54033 .41233 Mdot 1 0 0 r .05595 .39005 Mdot 0 1 0 r .48819 .24435 Mdot 1 0 0 r .13069 .0755 Mdot 0 1 0 r .35049 .109 Mdot 1 0 0 r .91011 .60143 Mdot 0 1 0 r .10707 .21078 Mdot .53243 .06605 Mdot 1 0 0 r .02623 .10892 Mdot 0 1 0 r .24745 .05693 Mdot 1 0 0 r .7672 .46836 Mdot .78072 .10073 Mdot 0 1 0 r .25068 .08511 Mdot .74857 .3146 Mdot 1 0 0 r .7387 .44468 Mdot 0 1 0 r .64169 .26818 Mdot .41202 .36476 Mdot .70779 .27066 Mdot .32875 .18307 Mdot .19917 .2337 Mdot .32633 .10324 Mdot 1 0 0 r .92793 .20585 Mdot 0 1 0 r .53534 .23879 Mdot .17102 .1342 Mdot .30847 .18277 Mdot .39865 .42352 Mdot .54598 .342 Mdot .73317 .18442 Mdot 1 0 0 r .15777 .58115 Mdot .04919 .51767 Mdot .80522 .42716 Mdot 0 1 0 r .82622 .31306 Mdot .5027 .35301 Mdot 1 0 0 r .8745 .13629 Mdot .94357 .1433 Mdot .7273 .03117 Mdot .65891 .56445 Mdot 0 1 0 r .35245 .21157 Mdot .13675 .25153 Mdot 1 0 0 r .59549 .05623 Mdot .95519 .27429 Mdot 0 1 0 r .57011 .14247 Mdot 1 0 0 r .17377 .45104 Mdot .7201 .43332 Mdot .64728 .12711 Mdot 0 1 0 r .8218 .32612 Mdot 1 0 0 r .67992 .58772 Mdot .11832 .32403 Mdot .04481 .05236 Mdot .74207 .14155 Mdot .88427 .40474 Mdot 0 1 0 r .17039 .1144 Mdot 1 0 0 r .90529 .15953 Mdot .57648 .57584 Mdot 0 1 0 r .75533 .3124 Mdot 1 0 0 r .83259 .1716 Mdot 0 1 0 r .13186 .21438 Mdot 1 0 0 r .0346 .4494 Mdot 0 1 0 r .42815 .23057 Mdot 1 0 0 r .89249 .15445 Mdot 0 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It works for right or left ordered polygons (it changes the \ sign of I) Denoting by P the point and V_i the set of N vertex of the \ polygon, we can calculate the following I = Sum( angle(V_(i+1)-P, V_i-P), i=1..N-1) + angle(V_N-P,V_1-P) if I != 0 \ then P is inside the polygon. If P == 0 the P is outside. with angle \ returning a value in ]-Pi,Pi] or ]-180,180]. 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