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192:, that is, the point is not changed if all coordinates are multiplied by the same nonzero constant". Most of the other coordinates in that section don't necessarily add up to one; I think they've been chosen to have no denominators or common factors. I changed it back to 1:1:1 to have more consistency in that section. I also added an explicit note that that they're homogeneous so readers can know that they're not necessarily normalized. It seems like the larger section of
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section will be clarified. A section on "triangular BS" deserves to be added to the "Applications" section. However, for mentioning "triangular Bézier surfaces" in the lead, would require to have a wikilink for them, which is not the case. Nevertheless, I agree that my formulation is not correct, and I'll replace "understanding and manipulating Bézier surfaces" by "defining some sorts of Bézier surfaces".
1032:"Trilinear coordinates" redirects to "barycentric coordinates", but they're not the same thing. Essentially, if I put three masses (m,p,q) at the vertices of a given triangle, the point described by those barycentric coordinates is the center of mass of that system. On the other hand, trilinear coordinates describe the relative distances from the three sides of a given triangle. See for example, the
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The (-16 : 33 : 28) point can be verified as circumcenter by any number of methods, so the (17 : 52 : 50) point can’t possibly be the circumcenter. I find the 1 – cos {β, γ, α} cos {γ, α, β} point does not pass any of the tests for circumcenter. It is not on the Euler line, not on any of the triangle
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OK, You are writing for pure mathematicians, I wrote my contribution along Möbius' ideas, which may not fit into Your idea of the article. By the way, the physics needed for understanding of my contributions is highschool level. In order to clear the way for a version You have in mind, I shall remove
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I would like to discuss the appropriateness of this example to illustrate the use of barycentric coordinates relative to a triangle. I have been using barycentric coordinates off-and-on for some time, and there are many geometric uses which are much more accessible. Moreover, the use of baycentric
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the mathematical computations, as an explanation of their meaning. This would allows things to remain understandable for people ignoring physics. For everybody, it is useful to distinguish what is physics, and what is mathematics. In particular, using physics terminology in what is presented as a
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It is a good idea that You moved the general definition on the top. But I regret that You removed my remarks on the geometric meanings. They were essential findings of Möbius and are essential properties for its usage in triangle geometry. I think the new comment on CAD should be replaced/added.
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Like so many math articles on
Knowledge written by putative experts, this one is heavy on notation and extraordinarily light on conceptual explanation. I submit that the author, while handy with the formalisms, possesses no actual UNDERSTANDING of the material such as can be conveyed to another
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AS you can read, my edit summary contains "rm ... details that clearly do not belong to the lead". This does not mean that the removed text does not belong to the article. On the opposite, the removed text about triangle geometry must be added to the section "On a plane", when the mess of this
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In accurate, theoretic math no, it doesn't. In the real world of approximated floating-point math, obviously yes, as FP can only work on a fixed footprint to store values. As the triangle approaches degeneration, that is det(A) appoaches 0 this algo will be less and less tolerant to error.
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2090:- I separated the special points which don't depend on length/angle (vertices and centroid) from those which do (the rest of them). This is to be clear that in a space which has no notion of length/angle, the coordinates of the former points are still valid (mostly, see next note).
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Hi! I just wanted to say that in section 'Barycentric coordinates on triangles' points ABC and P are extensively referenced to explain something, but that these points are not drawn on any schematic. This leads to understanding the text to be a piece of guesswork at best... Thanks!
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Specifically, in the section describing the coordinates for triangles, it's difficult to tell what subtriangles and vectors and such the author is describing, especially considering they're described with elements the author just introduced.
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The
Knowledge articles must be written for the widest possible audience. This includes people who have had US highschool courses, but also people who have forgot these courses and people from other countries that have never learnt the
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I'm interested in the reliability of barycentric coordinates, both the two variable and three variable forms and alternatives when applied to texture mapping for example. Does the accuracy fall for extremely narrow triangles?
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This being said, I have tagged these sections because I think that they deserve to be improved, but I will not edit them soon, because I am working on articles that have a much larger audience. Go ahead yourself !
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Please comment on content, not on contributors. If you see something that you think could be improved, you can either just change it yourself, or suggest the changes here. It might help to review
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I don’t know the original author’s intent here. The errant expression seems to be a typographical misplacement of some sort. It can be simply deleted or replaced by some other helpful expression.
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Good idea. I'm a little hesitant to add a new source to the page before I check out the ones that are already here some more. If it was a typo then it might have been copied from one of them.
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intelligent but as-yet unaware individual. The whole thing just seems so overwhelmingly self-serving, festooned with largely useless (as
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section. I didn't add anything about this since I was mostly trying to simplify the section, but I think it's worth carefully mentioning for readers who care.
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in the section on converting from cartesian coordinates could be expanded to explain how the area interpretation of barycentric coordinates can be recovered.
995:(I cannot verify the reliability of the above equation, but my initial tests have shown no issues (I'll cite a verifiable implementation for test somewhere))
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However, it seems to me that this map is onto the convex hull of the polytope. Even the example given of the quadrilateral fails for the non-convex case.
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coordinates is independent of calculus. I feel the example does not really aid the understanding of barycentric coordinates relative to a triangle.
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918:{\displaystyle \lambda _{2}={\frac {(x_{2}-x_{1})(y-y_{1})-(x-x_{1})(y_{2}-y_{1})}{(x_{2}-x_{1})(y_{3}-y_{1})-(x_{3}-x_{1})(y_{2}-y_{1})}}\,,}
650:{\displaystyle \lambda _{1}={\frac {(x_{1}-x_{3})(y-y_{3})-(x-x_{3})(y_{1}-y_{3})}{(x_{2}-x_{1})(y_{3}-y_{1})-(x_{3}-x_{1})(y_{2}-y_{1})}}\,,}
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Another example: the centroid in barycentric coords is (1,1,1). On the other hand, the incenter in trilinear coordinates is (1,1,1).
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Thanks, I was mistaken the first time. It might help to add that reference there. This page needs more inline citations in general.
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It's not wrong. From the article: "The use of colons in the notation of the tuple means that barycentric coordinates are a sort of
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This is an excellent and helpful article, but there is an error in the
Barycentric Coordinates given for triangle circumcenters:
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The first and third expressions are indisputable, but the middle expression is clearly incorrect. For example, the triangle {
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should appear. I think,from the present lead a non-mathematician gets no idea of the advantages of barycentric coordinates.--
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I am astonished reading Your rating of my efforts improving the article a "mess". Collaboration looks in a different way.--
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Errr, nevermind. I changed the redirect myself. But there's still a stub there if anyone wants to embellish it...
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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points are affinely independent when the dimension of the smallest affine subspace which contains them is
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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of three points in an affine space over a field of characteristic three (e.g. of three points in the
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https://web.archive.org/web/20120419171900/http://faculty.evansville.edu:80/ck6/encyclopedia/ETC.html
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in order to avoid confusion with the points of the triangle. This is in line with the notation on a
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edge perpendicular bisectors, and not a distance of R from any triangle vertex (R = circumradius =
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on
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The equation concluded in this section is wrong and doesn't satisfy the actual requirement of
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It can be found in section 1.8 (Linear
Varieties Generated by Points). It says that a set of
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Seems like someone just made a typo at some point. I tried to fix this but it was undone...
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To bring the false remark in line with this definition, we can just subtract one from the
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Barycentric coordinates are used for the generation of a special kind of Bezier surfaces,
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2536:= 1-cos β cos γ : 1-cos γ cos α : 1-cos α cos β
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If you found an error with any archives or the URLs themselves, you can fix them with
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The first remark on the meaning of affine independence, "this means that there is no
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The area interpretation of the barycentric coordinates can be recovered by applying
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As a cosine equivalent to the sine expression for circumcenter, I would recommend:
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It might be good to change the phrasing to more closely match that in the wiki for
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is wrong, since these coordinates do not sum up to one. It should be 1/3:1/3:1/3.
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Sin2α : sin2β : sin2γ
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1064:"Affine Coordinates" redirects to this page but do not appear in the article. --
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BS, which are rather different from the "usual" ones (tensor product BS). So:
1369:. No special action is required regarding these talk page notices, other than
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would derisively have termed it) "alphabet soup." And that's a crying shame.
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More abstractly, generalized barycentric coordinates express a polytope with
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First, it's impossible for any set of points to satisfy this condition.
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that contains all the points, or, equivalently that the points define a
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287:{\displaystyle 0<\lambda _{i}<1\;\forall \;i{\text{ in }}1,2,3}
2634:/2sin α, typically). I don’t know what the errant point represents.
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1 – cos {β, γ, α} cos {γ, α, β} = {51/128, 39/32, 75/64} --: -->
1765:-dimensional affine space, has one affine subspace of dimension
350:{\displaystyle 0<\lambda _{i}\;\forall \;i{\text{ in }}1,2,3}
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This is based on “sin 2α = 2sin α cos α” and “sin {α, β, γ} = {
1893:. Also, it's tempting to shorten "affine subspace of dimension
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wiki which gives a correct definition of affine independence:
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When you have finished reviewing my changes, please set the
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2401:. Paris: Hermann, Publishers in arts and science. p. 326.
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for additional information. I made the following changes:
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http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
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sin 2{α, β, γ} = {-√15/8, 33√15/128, 7√15/32} --: -->
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Does the article title really need a disambiguator? —
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2188:{\displaystyle a_{1},\ldots ,a_{m}}
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1109:Generalised Barycentric Coordinates
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109:and see a list of open tasks.
2503:{\displaystyle \lambda _{i}}
1784:{\displaystyle \mathbf {A} }
1754:{\displaystyle \mathbf {A} }
1698:{\displaystyle \mathbf {A} }
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2076:{\displaystyle \gamma }
2036:{\displaystyle \alpha }
1594:to this linear system.
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190:homogeneous coordinates
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1818:
1790:
1788:
1787:
1782:
1780:
1768:
1764:
1760:
1758:
1757:
1752:
1750:
1735:
1720:
1708:
1704:
1702:
1701:
1696:
1694:
1671:
1664:
1662:
1661:
1656:
1654:
1653:
1635:
1634:
1599:ScientistBuilder
1556:law of the lever
1536:
1526:My contributions
1464:
1425:
1416:
1389:
1388:
1367:
1355:
1263:The Cooper Union
1202:
1200:
1199:
1194:
1183:
1182:
1156:
1154:
1153:
1148:
1125:of the standard
1012:ScientistBuilder
989:
987:
986:
981:
974:
973:
961:
960:
942:
941:
924:
922:
921:
916:
909:
907:
903:
902:
890:
889:
874:
873:
861:
860:
842:
841:
829:
828:
813:
812:
800:
799:
786:
782:
781:
769:
768:
753:
752:
728:
727:
706:
705:
693:
692:
679:
674:
673:
656:
654:
653:
648:
641:
639:
635:
634:
622:
621:
606:
605:
593:
592:
574:
573:
561:
560:
545:
544:
532:
531:
518:
514:
513:
501:
500:
485:
484:
460:
459:
438:
437:
425:
424:
411:
406:
405:
383:
381:
380:
375:
356:
354:
353:
348:
331:
328:
316:
315:
293:
291:
290:
285:
268:
265:
247:
246:
174:ScientistBuilder
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
2706:
2705:
2701:
2700:
2699:
2697:
2696:
2695:
2676:
2675:
2626:
2606:
2602:
2601:(17 : 52 : 50)
2598:
2528:
2490:
2485:
2484:
2463:
2458:
2457:
2438:
2437:
2418:
2417:
2408:
2391:
2349:
2348:
2328:
2323:
2322:
2299:
2257:
2256:
2237:
2236:
2217:
2216:
2215:, whose number
2197:
2196:
2175:
2156:
2151:
2150:
2131:
2130:
2065:
2064:
2045:
2044:
2025:
2024:
1999:
1998:
1979:
1978:
1959:
1958:
1939:
1938:
1924:
1894:
1845:
1842:affine subspace
1837:
1830:
1823:
1816:
1802:
1771:
1770:
1766:
1762:
1741:
1740:
1733:
1730:affine subspace
1726:
1718:
1715:affine subspace
1706:
1685:
1684:
1674:Euclidean space
1666:
1645:
1626:
1621:
1620:
1614:
1585:
1530:
1528:
1458:
1456:
1434:
1419:
1414:
1382:
1375:have permission
1365:
1349:
1319:this simple FaQ
1304:
1258:
1223:
1168:
1163:
1162:
1127:
1126:
1111:
1076:
1062:
1030:
965:
952:
933:
928:
927:
894:
881:
865:
852:
833:
820:
804:
791:
787:
773:
760:
744:
719:
697:
684:
680:
665:
660:
659:
626:
613:
597:
584:
565:
552:
536:
523:
519:
505:
492:
476:
451:
429:
416:
412:
397:
392:
391:
366:
365:
307:
296:
295:
238:
227:
226:
223:
214:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
2704:
2702:
2694:
2693:
2688:
2678:
2677:
2628:
2607:
2603:
2599:
2596:
2527:
2524:
2523:
2522:
2512:FionaLovesCats
2497:
2493:
2470:
2466:
2445:
2425:
2414:
2413:
2412:
2406:
2389:
2388:
2387:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2335:
2331:
2306:
2302:
2296:
2293:
2286:
2281:
2278:
2275:
2271:
2267:
2264:
2244:
2224:
2204:
2182:
2178:
2174:
2171:
2168:
2163:
2159:
2138:
2111:FionaLovesCats
2072:
2052:
2032:
2012:
2009:
2006:
1986:
1966:
1946:
1923:
1920:
1919:
1918:
1917:
1916:
1906:FionaLovesCats
1887:
1859:FionaLovesCats
1820:
1819:
1814:
1779:
1749:
1693:
1652:
1648:
1644:
1641:
1638:
1633:
1629:
1618:
1613:
1610:
1584:
1581:
1580:
1579:
1564:
1527:
1524:
1523:
1522:
1521:
1520:
1503:
1502:
1455:
1452:
1433:
1430:
1409:
1408:
1401:
1334:
1333:
1325:Added archive
1303:
1300:
1299:
1298:
1257:
1254:
1243:145.53.101.116
1239:
1222:
1221:suggested move
1219:
1204:
1203:
1192:
1189:
1186:
1181:
1178:
1175:
1171:
1146:
1143:
1140:
1137:
1134:
1110:
1107:
1075:
1072:
1061:
1058:
1057:
1056:
1029:
1024:
1023:
1022:
993:
991:
990:
979:
972:
968:
964:
959:
955:
951:
948:
945:
940:
936:
925:
914:
906:
901:
897:
893:
888:
884:
880:
877:
872:
868:
864:
859:
855:
851:
848:
845:
840:
836:
832:
827:
823:
819:
816:
811:
807:
803:
798:
794:
790:
785:
780:
776:
772:
767:
763:
759:
756:
751:
747:
743:
740:
737:
734:
731:
726:
722:
718:
715:
712:
709:
704:
700:
696:
691:
687:
683:
677:
672:
668:
657:
646:
638:
633:
629:
625:
620:
616:
612:
609:
604:
600:
596:
591:
587:
583:
580:
577:
572:
568:
564:
559:
555:
551:
548:
543:
539:
535:
530:
526:
522:
517:
512:
508:
504:
499:
495:
491:
488:
483:
479:
475:
472:
469:
466:
463:
458:
454:
450:
447:
444:
441:
436:
432:
428:
423:
419:
415:
409:
404:
400:
373:
346:
343:
340:
337:
334:
326:
321:
314:
310:
306:
303:
283:
280:
277:
274:
271:
263:
258:
253:
250:
245:
241:
237:
234:
222:
219:
213:
210:
209:
208:
198:FionaLovesCats
185:
184:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2703:
2692:
2689:
2687:
2684:
2683:
2681:
2674:
2673:
2669:
2665:
2660:
2658:
2653:
2651:
2647:
2643:
2638:
2635:
2633:
2623:
2619:
2615:
2611:
2595:
2593:
2588:
2586:
2582:
2578:
2574:
2570:
2566:
2562:
2558:
2554:
2550:
2546:
2542:
2537:
2534:
2531:
2525:
2521:
2517:
2513:
2495:
2491:
2468:
2464:
2443:
2423:
2415:
2409:
2404:
2400:
2399:
2394:
2390:
2385:
2369:
2366:
2363:
2360:
2357:
2333:
2329:
2304:
2300:
2294:
2291:
2284:
2279:
2276:
2273:
2269:
2265:
2262:
2242:
2222:
2202:
2180:
2176:
2172:
2169:
2166:
2161:
2157:
2136:
2128:
2127:
2126:
2125:
2123:
2122:
2121:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2091:
2088:
2086:
2070:
2050:
2030:
2010:
2007:
2004:
1984:
1964:
1944:
1935:
1933:
1929:
1921:
1915:
1911:
1907:
1903:
1897:
1892:
1891:affine spaces
1888:
1885:
1884:
1883:
1879:
1875:
1871:
1870:
1869:
1868:
1864:
1860:
1855:
1852:
1848:
1844:of dimension
1843:
1834:
1826:
1817:
1812:
1808:
1807:
1801:
1800:
1799:
1797:
1792:
1737:
1732:of dimension
1731:
1724:
1717:of dimension
1716:
1712:
1705:of dimension
1683:
1679:
1675:
1669:
1650:
1646:
1642:
1639:
1636:
1631:
1627:
1617:
1611:
1609:
1608:
1604:
1600:
1595:
1593:
1592:Cramer's rule
1588:
1582:
1578:
1574:
1570:
1565:
1561:
1557:
1552:
1551:
1550:
1549:
1545:
1541:
1534:
1525:
1519:
1515:
1511:
1507:
1506:
1505:
1504:
1501:
1497:
1493:
1488:
1487:
1486:
1485:
1481:
1477:
1473:
1469:
1462:
1453:
1451:
1450:
1446:
1442:
1438:
1431:
1429:
1428:
1423:
1418:
1417:
1406:
1402:
1399:
1395:
1394:
1393:
1386:
1380:
1376:
1372:
1368:
1362:
1357:
1353:
1347:
1343:
1339:
1332:
1328:
1324:
1323:
1322:
1320:
1316:
1312:
1307:
1301:
1297:
1293:
1289:
1285:
1281:
1280:
1279:
1278:
1274:
1270:
1266:
1264:
1255:
1253:
1252:
1248:
1244:
1237:
1236:
1232:
1228:
1220:
1218:
1217:
1213:
1209:
1190:
1187:
1179:
1176:
1173:
1160:
1141:
1138:
1135:
1124:
1120:
1116:
1115:
1114:
1108:
1106:
1105:
1101:
1097:
1091:
1090:
1086:
1082:
1081:68.144.80.168
1073:
1071:
1070:
1067:
1059:
1055:
1052:
1048:
1047:
1046:
1045:
1042:
1037:
1035:
1028:
1025:
1021:
1017:
1013:
1009:
1008:
1007:
1005:
1001:
1000:AvinashANBhat
997:
996:
977:
970:
966:
962:
957:
953:
949:
946:
943:
938:
934:
926:
912:
899:
895:
891:
886:
882:
870:
866:
862:
857:
853:
846:
838:
834:
830:
825:
821:
809:
805:
801:
796:
792:
778:
774:
770:
765:
761:
749:
745:
741:
738:
732:
724:
720:
716:
713:
702:
698:
694:
689:
685:
675:
670:
666:
658:
644:
631:
627:
623:
618:
614:
602:
598:
594:
589:
585:
578:
570:
566:
562:
557:
553:
541:
537:
533:
528:
524:
510:
506:
502:
497:
493:
481:
477:
473:
470:
464:
456:
452:
448:
445:
434:
430:
426:
421:
417:
407:
402:
398:
390:
389:
388:
385:
371:
363:
358:
344:
341:
338:
335:
332:
324:
312:
308:
304:
301:
281:
278:
275:
272:
269:
261:
251:
248:
243:
239:
235:
232:
220:
218:
211:
207:
203:
199:
195:
191:
187:
186:
183:
179:
175:
172:I fixed this.
171:
170:
169:
166:
163:
157:
142:
138:
137:High-priority
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
62:High‑priority
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2661:
2656:
2654:
2649:
2645:
2641:
2639:
2636:
2631:
2627:
2621:
2617:
2613:
2609:
2591:
2589:
2584:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2538:
2535:
2532:
2529:
2397:
2320:
2106:
2092:
2089:
1936:
1931:
1927:
1925:
1895:
1856:
1853:
1846:
1835:
1824:
1821:
1809:, Springer,
1805:
1796:affine space
1793:
1738:
1727:
1682:affine space
1672:points in a
1667:
1615:
1596:
1589:
1586:
1583:Cramers Rule
1559:
1529:
1471:
1467:
1457:
1441:104.7.12.222
1439:
1435:
1413:
1410:
1385:source check
1364:
1358:
1345:
1341:
1337:
1335:
1308:
1305:
1267:
1259:
1238:
1224:
1205:
1158:
1122:
1118:
1112:
1096:82.88.245.35
1092:
1077:
1066:91.23.217.47
1063:
1038:
1031:
998:
994:
992:
386:
359:
224:
215:
193:
167:
164:
161:
136:
96:
40:WikiProjects
1352:Sourcecheck
1288:Deltahedron
1269:73.49.1.133
1074:reliability
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2680:Categories
2407:2705656758
2107:Definition
1902:hyperplane
1472:triangular
1468:triangular
1422:Report bug
1709:that are
1405:this tool
1398:this tool
1256:Criticism
1034:Mathworld
2664:JFSather
2652:cos γ).
2648:cos β :
2644:cos α :
2395:(1974).
2095:centroid
1874:Apocheir
1569:D.Lazard
1533:D.Lazard
1492:D.Lazard
1461:D.Lazard
1411:Cheers.—
2659:}/2R”.
2657:a, b, c
2592:a, b, c
2398:Algebra
2149:points
2129:"Given
1723:simplex
1540:Ag2gaeh
1510:Ag2gaeh
1476:Ag2gaeh
1338:checked
1315:my edit
1227:Tamfang
139:on the
2539:=
2093:- The
1680:or an
1346:failed
36:scale.
2416:Here
1900:" to
1761:, an
1560:after
1208:Zteve
1123:image
1051:Lunch
1041:Lunch
2668:talk
2516:talk
2403:ISBN
2386:..."
2115:talk
1910:talk
1878:talk
1863:talk
1811:ISBN
1678:flat
1676:, a
1619:Let
1603:talk
1573:talk
1544:talk
1514:talk
1496:talk
1480:talk
1454:Lead
1445:talk
1342:true
1292:talk
1273:talk
1247:talk
1231:talk
1212:talk
1100:talk
1085:talk
1016:talk
1004:talk
362:here
305:<
249:<
236:<
202:talk
178:talk
131:High
2510:".
2195:of
2023:to
1898:- 1
1849:- 1
1827:+ 1
1670:+ 1
1665:be
1379:RfC
1356:).
1344:or
1329:to
329:in
266:in
2682::
2670:)
2620:+
2616:+
2612:(-
2587:)
2583:-
2579:+
2567:+
2563:-
2551:+
2547:+
2543:(-
2518:)
2492:λ
2367:≤
2361:≤
2270:∑
2170:…
2117:)
2071:γ
2063:,
2051:β
2043:,
2031:α
1977:,
1957:,
1912:)
1904:.
1880:)
1865:)
1857:-
1833:.
1791:.
1725:.
1640:…
1605:)
1575:)
1546:)
1516:)
1498:)
1482:)
1447:)
1392:.
1387:}}
1383:{{
1354:}}
1350:{{
1294:)
1275:)
1249:)
1233:)
1214:)
1185:↠
1177:−
1170:Δ
1139:−
1102:)
1087:)
1018:)
1006:)
967:λ
963:−
954:λ
950:−
935:λ
892:−
863:−
847:−
831:−
802:−
771:−
742:−
733:−
717:−
695:−
667:λ
624:−
595:−
579:−
563:−
534:−
503:−
474:−
465:−
449:−
427:−
399:λ
372:γ
357:.
320:∀
309:λ
257:∀
240:λ
204:)
180:)
2666:(
2650:c
2646:b
2642:a
2640:(
2632:a
2622:c
2618:b
2614:a
2610:a
2608:{
2585:c
2581:b
2577:a
2575:(
2573:c
2569:c
2565:b
2561:a
2559:(
2557:b
2553:c
2549:b
2545:a
2541:a
2514:(
2496:i
2469:i
2465:x
2444:K
2424:E
2410:.
2373:)
2370:m
2364:i
2358:1
2355:(
2334:i
2330:a
2305:i
2301:a
2295:m
2292:1
2285:m
2280:1
2277:=
2274:i
2266:=
2263:g
2243:K
2223:m
2203:E
2181:m
2177:a
2173:,
2167:,
2162:1
2158:a
2137:m
2113:(
2011:C
2008:B
2005:A
1985:C
1965:B
1945:A
1908:(
1896:n
1876:(
1861:(
1847:n
1838:n
1831:n
1825:n
1778:A
1767:n
1763:n
1748:A
1734:n
1719:n
1707:n
1692:A
1668:n
1651:n
1647:A
1643:,
1637:,
1632:0
1628:A
1601:(
1571:(
1542:(
1535::
1531:@
1512:(
1494:(
1478:(
1463::
1459:@
1443:(
1424:)
1420:(
1407:.
1400:.
1290:(
1271:(
1245:(
1229:(
1210:(
1191:.
1188:P
1180:1
1174:n
1159:n
1145:)
1142:1
1136:n
1133:(
1119:n
1098:(
1083:(
1014:(
1002:(
978:.
971:2
958:1
947:1
944:=
939:3
913:,
905:)
900:1
896:y
887:2
883:y
879:(
876:)
871:1
867:x
858:3
854:x
850:(
844:)
839:1
835:y
826:3
822:y
818:(
815:)
810:1
806:x
797:2
793:x
789:(
784:)
779:1
775:y
766:2
762:y
758:(
755:)
750:1
746:x
739:x
736:(
730:)
725:1
721:y
714:y
711:(
708:)
703:1
699:x
690:2
686:x
682:(
676:=
671:2
645:,
637:)
632:1
628:y
619:2
615:y
611:(
608:)
603:1
599:x
590:3
586:x
582:(
576:)
571:1
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558:3
554:y
550:(
547:)
542:1
538:x
529:2
525:x
521:(
516:)
511:3
507:y
498:1
494:y
490:(
487:)
482:3
478:x
471:x
468:(
462:)
457:3
453:y
446:y
443:(
440:)
435:3
431:x
422:1
418:x
414:(
408:=
403:1
345:3
342:,
339:2
336:,
333:1
325:i
313:i
302:0
282:3
279:,
276:2
273:,
270:1
262:i
252:1
244:i
233:0
200:(
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143:.
42::
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