4188:. Let this ant move North. As it moves, it will pass through the other two paraboloids, like a ghost passing through a wall. These other paraboloids only seem like obstacles due to the self-intersecting nature of the immersion. Let the ant ignore all double and triple points and pass right through them. So the ant moves to the North and falls off the edge of the world, so to speak. It now finds itself on the northern lobe, hidden underneath the third paraboloid of Figure 3. The ant is standing upside-down, on the "outside" of the Roman surface.
98:
4225:
three lines of double points intersect at a triple point which lies on the origin. The triple point cuts the lines of double points into a pair of half-lines, and each half-line lies between a pair of lobes. One might expect from the preceding statements that there could be up to eight lobes, one in each octant of space which has been divided by the coordinate planes. But the lobes occupy alternating octants: four octants are empty and four are occupied by lobes.
4075:
4002:
4144:
4170:
4089:
1237:
4113:
36:
4224:
The Roman surface has four "lobes". The boundaries of each lobe are a set of three lines of double points. Between each pair of lobes there is a line of double points. The surface has a total of three lines of double points, which lie (in the parametrization given earlier) on the coordinate axes. The
4211:
axis. As soon as the ant crosses this axis it will find itself "inside" the
Northern lobe, standing right side up. Now let the ant walk towards the North. It will climb up the wall, then along the "roof" of the Northern lobe. The ant is back on the third hyperbolic paraboloid, but this time under it
4153:
Figure 6 shows three lobes seen sideways. Between each pair of lobes there is a locus of double points corresponding to a coordinate axis. The three loci intersect at a triple point at the origin. The fourth lobe is hidden and points in the direction directly opposite from the viewer. The Roman
858:
155:
3728:
Since this is true of all points of S, then it is clear that the Roman surface is a continuous image of a "sphere modulo antipodes". Because some distinct pairs of antipodes are all taken to identical points in the Roman surface, it is not homeomorphic to
3737:. Furthermore, the map T (above) from S to this quotient has the special property that it is locally injective away from six pairs of antipodal points. Or from RP the resulting map making this an immersion of RP — minus six points — into 3-space.
1735:
4228:
If the Roman surface were to be inscribed inside the tetrahedron with least possible volume, one would find that each edge of the tetrahedron is tangent to the Roman surface at a point, and that each of these six points happens to be a
4084:
On the west-southwest and east-northeast directions in Figure 2 there are a pair of openings. These openings are lobes and need to be closed up. When the openings are closed up, the result is the Roman surface shown in Figure 3.
4191:
Let the ant move towards the
Southwest. It will climb a slope (upside-down) until it finds itself "inside" the Western lobe. Now let the ant move in a Southeastern direction along the inside of the Western lobe towards the
3723:
3417:
3790:
These three hyperbolic paraboloids intersect externally along the six edges of a tetrahedron and internally along the three axes. The internal intersections are loci of double points. The three loci of double points:
3329:
3248:
1232:{\displaystyle {\begin{aligned}U^{2}V^{2}+V^{2}W^{2}+W^{2}U^{2}&=z^{2}x^{2}y^{4}+x^{2}y^{2}z^{4}+y^{2}z^{2}x^{4}=(x^{2}+y^{2}+z^{2})(x^{2}y^{2}z^{2})\\&=(1)(x^{2}y^{2}z^{2})=(xy)(yz)(zx)=UVW,\end{aligned}}}
635:-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each coordinate axis which terminate in six pinch points. The entire surface has
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If the three intersecting hyperbolic paraboloids are drawn far enough that they intersect along the edges of a tetrahedron, then the result is as shown in Figure 4.
2029:
1931:
4236:. These singularities, or pinching points, all lie at the edges of the three lines of double points, and they are defined by this property: that there is no plane
1740:(Note that (*) guarantees that either all three of U, V, W are positive, or else exactly two are negative. So these square roots are of positive numbers.)
4409:
3552:
4402:
4326:
3335:
3254:
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If the continuous surface in Figure 4 has its sharp edges rounded out—smoothed out—then the result is the Roman surface in Figure 5.
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79:
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434:
4098:
A pair of lobes can be seen in the West and East directions of Figure 3. Another pair of lobes are hidden underneath the third (
4122:
One of the lobes is seen frontally—head on—in Figure 4. The lobe can be seen to be one of the four corners of the tetrahedron.
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46:
2174:
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4140:
If the surface in Figure 5 is turned around 180 degrees and then turned upside down, the result is as shown in Figure 6.
4127:
57:
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4389:
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4207:
Then let it move
Northwards, over "the hill", then towards the Northwest so that it starts sliding down towards the
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and then smoothing out the edges as necessary so that it will fit a desired shape (e.g. parametrization).
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1730:{\displaystyle x={\sqrt {\frac {WU}{V}}},\ y={\sqrt {\frac {UV}{W}}},\ z={\sqrt {\frac {VW}{U}}}.\,}
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axis. If the paraboloids are extended, they should also be seen to intersect along the lines
3745:
The Roman surface has four bulbous "lobes", each one on a different corner of a tetrahedron.
2311:
4479:
4249:
3458:
and these two points are different: they lie on opposite sides of the center of the sphere.
2004:
1906:
647:
154:
4581:
4143:
4019:
is shown in cyan and purple. In the image the paraboloids are seen to intersect along the
4543:
27:
Self-intersecting, highly symmetrical mapping of the real projective plane into 3D space
642:. It is a particular type (called type 1) of Steiner surface, that is, a 3-dimensional
639:
4683:
4644:
4600:
4586:
4237:
3443:
199:
4351:
4491:
4336:
4308:
The
Algebra and Geometry of Steiner and other Quadratically Parametrizable Surfaces
4213:
4169:
4088:
4390:
Ashay
Dharwadker, Heptahedron and Roman Surface, Electronic Geometry Models, 2004.
4363:
4204:
axis the ant will be on the "outside" of the
Eastern lobe, standing rightside-up.
4166:, i.e. one-sided. This is not quite obvious. To see this, look again at Figure 3.
4154:
surface shown at the top of this article also has three lobes in sideways view.
3718:{\displaystyle T:(-x,-y,-z)\rightarrow ((-y)(-z),(-z)(-x),(-x)(-y))=(yz,zx,xy).}
636:
163:
4112:
17:
4263:
4163:
4134:
3749:
4133:
One of the lobes of the Roman surface is seen frontally in Figure 5, and its
4659:
4548:
4368:
4253:
2865:(each of which is a noncompact portion of a coordinate axis, in two pieces)
413:
1614:
with one exception: In case 3.b. below, we show this cannot be proved.
4536:
3450:. But the sphere centered at the origin has this property, that if point
421:
194:
of the projective plane; however, the figure resulting from removing six
187:
291:
3412:{\displaystyle z'=xy=r^{2}\,\cos ^{2}\theta \,\cos \phi \,\sin \phi ,}
4444:
4057:
210:
3324:{\displaystyle y'=zx=r^{2}\,\cos \theta \,\sin \theta \,\cos \phi ,}
3243:{\displaystyle x'=yz=r^{2}\,\cos \theta \,\sin \theta \,\sin \phi ,}
1819:
must be 0 also. This shows that is it impossible for exactly one of
4450:
4168:
4142:
4111:
4087:
4073:
4000:
153:
613:{\displaystyle z=r^{2}\cos \theta \sin \theta \cos ^{2}\varphi }
428:), gives parametric equations for the Roman surface as follows:
203:
4398:
3982:
Let us see the pieces being put together. Join the paraboloids
4177:
3748:
A Roman surface can be constructed by splicing together three
402:{\displaystyle x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}-r^{2}xyz=0.\,}
91:
29:
3465:
converts both of these antipodal points into the same point,
545:{\displaystyle y=r^{2}\sin \theta \cos \varphi \sin \varphi }
484:{\displaystyle x=r^{2}\cos \theta \cos \varphi \sin \varphi }
3733:, but is instead a quotient of the real projective plane
2899:
are zero and the third one has absolute value 1, clearly
1460:{\displaystyle U^{2}V^{2}+V^{2}W^{2}+W^{2}U^{2}-UVW=0.\,}
412:
Also, taking a parametrization of the sphere in terms of
4106:) paraboloid and lie in the North and South directions.
3454:
belongs to the sphere, then so does the antipodal point
2235:{\displaystyle y^{2}={\frac {1-{\sqrt {1-4U^{2}}}}{2}},}
1337:{\displaystyle U^{2}V^{2}+V^{2}W^{2}+W^{2}U^{2}-UVW=0\,}
4425:
Compact topological surfaces and their immersions in 3D
2244:
this ensures that (*) holds. It is easy to verify that
2161:{\displaystyle x^{2}={\frac {1+{\sqrt {1-4U^{2}}}}{2}}}
53:
3847:
3555:
3474:
3338:
3257:
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2007:
1971:
1941:
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1743:
It is easy to use (*) to confirm that (**) holds for
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1371:
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is one. Its name arises because it was discovered by
4292:. Indiana University - Purdue University Fort Wayne.
4637:
4609:
4574:
4565:
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4466:
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4430:
2867:
do not correspond to any point on the Roman surface
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612:
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401:
282:
4071:, through them. The result is shown in Figure 2.
4056:The two paraboloids together look like a pair of
3755:Let there be these three hyperbolic paraboloids:
3541:{\displaystyle T:(x,y,z)\rightarrow (yz,zx,xy),}
3921:Likewise, the other external intersections are
3422:which are the points on the Roman surface. Let
209:The simplest construction is as the image of a
3113:{\displaystyle y=r\,\cos \theta \,\sin \phi ,}
3063:{\displaystyle x=r\,\cos \theta \,\cos \phi ,}
842:{\displaystyle T(x,y,z)=(yz,zx,xy)=(U,V,W),\,}
623:The origin is a triple point, and each of the
4410:
3442:The sphere, before being transformed, is not
2887:) is the point (0, 0, 0), then if any two of
2398:{\displaystyle yz=0=V{\text{ and }}zx=0=W,\,}
662:= 1. Given the sphere defined by the points (
8:
4386:(website of the California State University)
4015:is shown in blue and orange. The paraboloid
2988:{\displaystyle (xy,yz,zx)=(0,0,0)=(U,V,W)\,}
743:we apply to these points the transformation
4256:of the projective plane without cross-caps.
3892:= ±1. Their two external intersections are
4571:
4434:
4417:
4403:
4395:
4319:Geometric Modeling and Algebraic Geometry
4306:A. Coffman, A. Schwartz, and C. Stanton:
4063:Now run the third hyperbolic paraboloid,
3857:
3846:
3826:, the second paraboloid is equivalent to
3554:
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3396:
3386:
3374:
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658:For simplicity we consider only the case
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142:Learn how and when to remove this message
80:Learn how and when to remove this message
4212:and standing upside-down. (Compare with
4200:plane. As soon as it passes through the
3803:= 0, intersect at a triple point at the
3167:to all the points on this sphere yields
105:This article includes a list of general
4276:
2079:{\displaystyle |U|\leq {\frac {1}{2}},}
2553:{\displaystyle xy\leq {\frac {1}{2}},}
2460:{\displaystyle |U|>{\frac {1}{2}}.}
2417:In this remaining subcase of the case
1540:{\displaystyle x^{2}+y^{2}+z^{2}=1,\,}
733:{\displaystyle x^{2}+y^{2}+z^{2}=1,\,}
3438:Relation to the real projective plane
2856:{\displaystyle |W|>{\frac {1}{2}}}
2807:{\displaystyle |V|>{\frac {1}{2}}}
2758:{\displaystyle |U|>{\frac {1}{2}}}
213:centered at the origin under the map
7:
4314:(3) 13 (April 1996), p. 257-286
3018:. Then its parametric equations are
283:{\displaystyle f(x,y,z)=(yz,xz,xy).}
47:research paper or scientific journal
4240:to any surface at the singularity.
4220:Double, triple, and pinching points
3998:. The result is shown in Figure 1.
2709:{\displaystyle U=xy,\ V=yz,\ W=zx.}
2623:{\displaystyle |U|>1/2,\ V=W=0,}
2291:{\displaystyle x^{2}y^{2}=U^{2},\,}
186:, with an unusually high degree of
4266:very similar to the Roman surface.
3153:{\displaystyle z=r\,\sin \theta .}
3002:Derivation of parametric equations
2722:, 0, 0) of the equation (*) with
111:it lacks sufficient corresponding
25:
4137:– balloon-like—shape is evident.
1605:{\displaystyle U=xy,V=yz,W=zx,\,}
158:An animation of the Roman surface
4126:
2998:This covers all possible cases.
2512:{\displaystyle x^{2}+y^{2}=1,\,}
2300:and hence choosing the signs of
1894:{\displaystyle U\neq 0,V=W=0.\,}
96:
34:
4312:Computer Aided Geometric Design
2411:leads to the desired converse.
3870:{\displaystyle yz={y \over z}}
3741:Structure of the Roman surface
3709:
3682:
3676:
3673:
3664:
3661:
3652:
3646:
3637:
3634:
3625:
3619:
3610:
3607:
3598:
3595:
3592:
3589:
3562:
3532:
3505:
3502:
3499:
3481:
3446:to the real projective plane,
3163:Then, applying transformation
2981:
2963:
2957:
2939:
2933:
2906:
2836:
2828:
2787:
2779:
2738:
2730:
2578:
2570:
2437:
2429:
2056:
2048:
1802:{\displaystyle U^{2}V^{2}=0\,}
1207:
1198:
1195:
1186:
1183:
1174:
1168:
1135:
1132:
1126:
1113:
1080:
1077:
1038:
832:
814:
808:
781:
775:
757:
654:Derivation of implicit formula
274:
247:
241:
223:
1:
2562:and thus in this case, where
2308:appropriately will guarantee
3426:range from 0 to 2π, and let
1837:Suppose that exactly two of
1765:is 0. From (*) this implies
1469:We prove that there exists (
4317:Bert Jüttler, Ragni Piene:
4706:
1958:{\displaystyle z\neq 0,\,}
1851:Without loss of generality
1811:and hence at least one of
1620:In the case where none of
4162:The Roman surface is non-
3006:Let a sphere have radius
2521:it is easy to check that
190:. This mapping is not an
4286:"Steiner Roman Surfaces"
4148:Figure 6. Roman surface.
4093:Figure 3. Roman surface.
1994:{\displaystyle x=y=0,\,}
1350:, suppose we are given (
62:overly technical phrases
54:help improve the article
4556:Sphere with three holes
4196:axis, always above the
2334:{\displaystyle xy=U.\,}
290:This gives an implicit
184:three-dimensional space
174:is a self-intersecting
126:more precise citations.
4332:restricted online copy
4180:on top of the "third"
4173:
4150:
4119:
4095:
4081:
4008:
3871:
3750:hyperbolic paraboloids
3719:
3542:
3413:
3325:
3244:
3154:
3114:
3064:
2989:
2857:
2808:
2759:
2710:
2624:
2554:
2513:
2461:
2399:
2335:
2292:
2236:
2162:
2080:
2031:contradicting (***).)
2025:
2024:{\displaystyle U=0,\,}
1995:
1959:
1927:
1926:{\displaystyle z=0,\,}
1895:
1803:
1731:
1606:
1541:
1461:
1338:
1233:
843:
734:
614:
546:
485:
403:
284:
159:
4474:Real projective plane
4459:Pretzel (genus 3) ...
4182:hyperbolic paraboloid
4172:
4146:
4115:
4091:
4077:
4060:joined back-to-back.
4004:
3872:
3720:
3543:
3414:
3326:
3245:
3155:
3115:
3065:
2990:
2858:
2809:
2760:
2718:Hence the solutions (
2711:
2625:
2555:
2514:
2462:
2400:
2336:
2293:
2237:
2163:
2081:
2037:In the subcase where
2026:
1996:
1960:
1928:
1896:
1804:
1732:
1607:
1542:
1462:
1339:
1234:
844:
735:
615:
547:
486:
404:
285:
180:real projective plane
157:
4629:Euler characteristic
3845:
3553:
3472:
3336:
3255:
3174:
3125:
3075:
3025:
2903:
2824:
2775:
2726:
2652:
2566:
2525:
2473:
2425:
2347:
2312:
2248:
2175:
2104:
2044:
2005:
1969:
1939:
1907:
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1769:
1639:
1553:
1488:
1369:
1246:
859:
751:
681:
557:
496:
435:
301:
217:
4384:National Curve Bank
4290:National Curve Bank
4260:Tetrahemihexahedron
3810:For example, given
3461:The transformation
56:by rewriting it in
4456:Number 8 (genus 2)
4361:Weisstein, Eric W.
4301:General references
4174:
4151:
4120:
4096:
4082:
4009:
3867:
3715:
3538:
3409:
3321:
3240:
3150:
3110:
3060:
2985:
2853:
2804:
2767:and likewise, (0,
2755:
2706:
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2550:
2509:
2457:
2395:
2331:
2288:
2232:
2158:
2076:
2021:
1991:
1955:
1923:
1891:
1799:
1755:defined this way.
1727:
1602:
1537:
1457:
1334:
1229:
1227:
839:
730:
610:
542:
481:
399:
280:
160:
58:encyclopedic style
45:is written like a
4677:
4676:
4673:
4672:
4507:
4506:
4327:978-3-540-72184-0
4321:. Springer 2008,
3865:
2851:
2802:
2753:
2690:
2672:
2601:
2545:
2452:
2371:
2227:
2221:
2156:
2150:
2071:
1721:
1720:
1699:
1692:
1691:
1670:
1663:
1662:
1632:is 0, we can set
852:But then we have
644:linear projection
152:
151:
144:
90:
89:
82:
16:(Redirected from
4697:
4592:Triangulatedness
4572:
4435:
4431:Without boundary
4419:
4412:
4405:
4396:
4374:
4373:
4352:Steiner Surfaces
4294:
4293:
4281:
4130:
3876:
3874:
3873:
3868:
3866:
3858:
3724:
3722:
3721:
3716:
3547:
3545:
3544:
3539:
3430:range from 0 to
3418:
3416:
3415:
3410:
3379:
3378:
3368:
3367:
3346:
3330:
3328:
3327:
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3184:
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3157:
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3116:
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3069:
3067:
3066:
3061:
2994:
2992:
2991:
2986:
2862:
2860:
2859:
2854:
2852:
2844:
2839:
2831:
2813:
2811:
2810:
2805:
2803:
2795:
2790:
2782:
2764:
2762:
2761:
2756:
2754:
2746:
2741:
2733:
2715:
2713:
2712:
2707:
2688:
2670:
2629:
2627:
2626:
2621:
2599:
2592:
2581:
2573:
2559:
2557:
2556:
2551:
2546:
2538:
2518:
2516:
2515:
2510:
2498:
2497:
2485:
2484:
2466:
2464:
2463:
2458:
2453:
2445:
2440:
2432:
2407:this shows that
2404:
2402:
2401:
2396:
2372:
2369:
2340:
2338:
2337:
2332:
2297:
2295:
2294:
2289:
2283:
2282:
2270:
2269:
2260:
2259:
2241:
2239:
2238:
2233:
2228:
2223:
2222:
2220:
2219:
2201:
2192:
2187:
2186:
2167:
2165:
2164:
2159:
2157:
2152:
2151:
2149:
2148:
2130:
2121:
2116:
2115:
2089:if we determine
2085:
2083:
2082:
2077:
2072:
2064:
2059:
2051:
2030:
2028:
2027:
2022:
2000:
1998:
1997:
1992:
1964:
1962:
1961:
1956:
1932:
1930:
1929:
1924:
1903:It follows that
1900:
1898:
1897:
1892:
1808:
1806:
1805:
1800:
1791:
1790:
1781:
1780:
1736:
1734:
1733:
1728:
1722:
1716:
1708:
1707:
1697:
1693:
1687:
1679:
1678:
1668:
1664:
1658:
1650:
1649:
1611:
1609:
1608:
1603:
1546:
1544:
1543:
1538:
1526:
1525:
1513:
1512:
1500:
1499:
1466:
1464:
1463:
1458:
1437:
1436:
1427:
1426:
1414:
1413:
1404:
1403:
1391:
1390:
1381:
1380:
1343:
1341:
1340:
1335:
1314:
1313:
1304:
1303:
1291:
1290:
1281:
1280:
1268:
1267:
1258:
1257:
1238:
1236:
1235:
1230:
1228:
1167:
1166:
1157:
1156:
1147:
1146:
1119:
1112:
1111:
1102:
1101:
1092:
1091:
1076:
1075:
1063:
1062:
1050:
1049:
1034:
1033:
1024:
1023:
1014:
1013:
1001:
1000:
991:
990:
981:
980:
968:
967:
958:
957:
948:
947:
931:
930:
921:
920:
908:
907:
898:
897:
885:
884:
875:
874:
848:
846:
845:
840:
739:
737:
736:
731:
719:
718:
706:
705:
693:
692:
648:Veronese surface
634:
630:
626:
619:
617:
616:
611:
603:
602:
575:
574:
551:
549:
548:
543:
514:
513:
490:
488:
487:
482:
453:
452:
427:
419:
408:
406:
405:
400:
382:
381:
369:
368:
359:
358:
346:
345:
336:
335:
323:
322:
313:
312:
289:
287:
286:
281:
147:
140:
136:
133:
127:
122:this article by
113:inline citations
100:
99:
92:
85:
78:
74:
71:
65:
38:
37:
30:
21:
4705:
4704:
4700:
4699:
4698:
4696:
4695:
4694:
4680:
4679:
4678:
4669:
4633:
4610:Characteristics
4605:
4567:
4561:
4503:
4462:
4426:
4423:
4364:"Roman Surface"
4359:
4358:
4346:
4303:
4298:
4297:
4284:Coffman, Adam.
4283:
4282:
4278:
4273:
4246:
4222:
4160:
4011:The paraboloid
3843:
3842:
3743:
3735:RP = S / (x~-x)
3551:
3550:
3470:
3469:
3440:
3370:
3359:
3339:
3334:
3333:
3278:
3258:
3253:
3252:
3197:
3177:
3172:
3171:
3123:
3122:
3073:
3072:
3023:
3022:
3014:, and latitude
3004:
2901:
2900:
2822:
2821:
2773:
2772:
2724:
2723:
2650:
2649:
2564:
2563:
2523:
2522:
2489:
2476:
2471:
2470:
2423:
2422:
2370: and
2345:
2344:
2310:
2309:
2274:
2261:
2251:
2246:
2245:
2211:
2193:
2178:
2173:
2172:
2140:
2122:
2107:
2102:
2101:
2042:
2041:
2003:
2002:
1967:
1966:
1937:
1936:
1905:
1904:
1858:
1857:
1782:
1772:
1767:
1766:
1709:
1680:
1651:
1637:
1636:
1551:
1550:
1517:
1504:
1491:
1486:
1485:
1428:
1418:
1405:
1395:
1382:
1372:
1367:
1366:
1305:
1295:
1282:
1272:
1259:
1249:
1244:
1243:
1226:
1225:
1158:
1148:
1138:
1117:
1116:
1103:
1093:
1083:
1067:
1054:
1041:
1025:
1015:
1005:
992:
982:
972:
959:
949:
939:
932:
922:
912:
899:
889:
876:
866:
857:
856:
749:
748:
710:
697:
684:
679:
678:
656:
632:
628:
624:
594:
566:
555:
554:
505:
494:
493:
444:
433:
432:
425:
417:
373:
360:
350:
337:
327:
314:
304:
299:
298:
215:
214:
202:when he was in
196:singular points
172:Steiner surface
148:
137:
131:
128:
118:Please help to
117:
101:
97:
86:
75:
69:
66:
51:
39:
35:
28:
23:
22:
18:Steiner surface
15:
12:
11:
5:
4703:
4701:
4693:
4692:
4682:
4681:
4675:
4674:
4671:
4670:
4668:
4667:
4662:
4656:
4650:
4647:
4641:
4639:
4635:
4634:
4632:
4631:
4626:
4621:
4613:
4611:
4607:
4606:
4604:
4603:
4598:
4589:
4584:
4578:
4576:
4569:
4563:
4562:
4560:
4559:
4553:
4552:
4551:
4541:
4540:
4539:
4534:
4526:
4525:
4524:
4515:
4513:
4509:
4508:
4505:
4504:
4502:
4501:
4498:Dyck's surface
4495:
4489:
4488:
4487:
4482:
4470:
4468:
4467:Non-orientable
4464:
4463:
4461:
4460:
4457:
4454:
4448:
4441:
4439:
4432:
4428:
4427:
4424:
4422:
4421:
4414:
4407:
4399:
4393:
4392:
4387:
4379:Roman Surfaces
4375:
4356:
4345:
4344:External links
4342:
4341:
4340:
4329:, p. 30 (
4315:
4302:
4299:
4296:
4295:
4275:
4274:
4272:
4269:
4268:
4267:
4257:
4245:
4242:
4221:
4218:
4159:
4156:
4054:
4053:
4039:
3980:
3979:
3965:
3951:
3937:
3919:
3918:
3904:
3878:
3877:
3864:
3861:
3856:
3853:
3850:
3788:
3787:
3777:
3767:
3742:
3739:
3726:
3725:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3548:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3439:
3436:
3420:
3419:
3408:
3405:
3402:
3399:
3395:
3392:
3389:
3385:
3382:
3377:
3373:
3366:
3362:
3358:
3355:
3352:
3349:
3345:
3342:
3331:
3320:
3317:
3314:
3311:
3307:
3304:
3301:
3297:
3294:
3291:
3285:
3281:
3277:
3274:
3271:
3268:
3264:
3261:
3250:
3239:
3236:
3233:
3230:
3226:
3223:
3220:
3216:
3213:
3210:
3204:
3200:
3196:
3193:
3190:
3187:
3183:
3180:
3161:
3160:
3149:
3146:
3143:
3140:
3136:
3133:
3130:
3120:
3109:
3106:
3103:
3100:
3096:
3093:
3090:
3086:
3083:
3080:
3070:
3059:
3056:
3053:
3050:
3046:
3043:
3040:
3036:
3033:
3030:
3003:
3000:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2850:
2847:
2842:
2838:
2834:
2830:
2801:
2798:
2793:
2789:
2785:
2781:
2752:
2749:
2744:
2740:
2736:
2732:
2705:
2702:
2699:
2696:
2693:
2687:
2684:
2681:
2678:
2675:
2669:
2666:
2663:
2660:
2657:
2619:
2616:
2613:
2610:
2607:
2604:
2598:
2595:
2591:
2587:
2584:
2580:
2576:
2572:
2549:
2544:
2541:
2536:
2533:
2530:
2507:
2504:
2501:
2496:
2492:
2488:
2483:
2479:
2456:
2451:
2448:
2443:
2439:
2435:
2431:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2367:
2364:
2361:
2358:
2355:
2352:
2329:
2326:
2323:
2320:
2317:
2286:
2281:
2277:
2273:
2268:
2264:
2258:
2254:
2231:
2226:
2218:
2214:
2210:
2207:
2204:
2199:
2196:
2190:
2185:
2181:
2169:
2168:
2155:
2147:
2143:
2139:
2136:
2133:
2128:
2125:
2119:
2114:
2110:
2087:
2086:
2075:
2070:
2067:
2062:
2058:
2054:
2050:
2019:
2016:
2013:
2010:
1989:
1986:
1983:
1980:
1977:
1974:
1953:
1950:
1947:
1944:
1921:
1918:
1915:
1912:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1797:
1794:
1789:
1785:
1779:
1775:
1738:
1737:
1725:
1719:
1715:
1712:
1705:
1702:
1696:
1690:
1686:
1683:
1676:
1673:
1667:
1661:
1657:
1654:
1647:
1644:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1535:
1532:
1529:
1524:
1520:
1516:
1511:
1507:
1503:
1498:
1494:
1455:
1452:
1449:
1446:
1443:
1440:
1435:
1431:
1425:
1421:
1417:
1412:
1408:
1402:
1398:
1394:
1389:
1385:
1379:
1375:
1332:
1329:
1326:
1323:
1320:
1317:
1312:
1308:
1302:
1298:
1294:
1289:
1285:
1279:
1275:
1271:
1266:
1262:
1256:
1252:
1240:
1239:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1165:
1161:
1155:
1151:
1145:
1141:
1137:
1134:
1131:
1128:
1125:
1122:
1120:
1118:
1115:
1110:
1106:
1100:
1096:
1090:
1086:
1082:
1079:
1074:
1070:
1066:
1061:
1057:
1053:
1048:
1044:
1040:
1037:
1032:
1028:
1022:
1018:
1012:
1008:
1004:
999:
995:
989:
985:
979:
975:
971:
966:
962:
956:
952:
946:
942:
938:
935:
933:
929:
925:
919:
915:
911:
906:
902:
896:
892:
888:
883:
879:
873:
869:
865:
864:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
756:
741:
740:
728:
725:
722:
717:
713:
709:
704:
700:
696:
691:
687:
655:
652:
621:
620:
609:
606:
601:
597:
593:
590:
587:
584:
581:
578:
573:
569:
565:
562:
552:
541:
538:
535:
532:
529:
526:
523:
520:
517:
512:
508:
504:
501:
491:
480:
477:
474:
471:
468:
465:
462:
459:
456:
451:
447:
443:
440:
410:
409:
397:
394:
391:
388:
385:
380:
376:
372:
367:
363:
357:
353:
349:
344:
340:
334:
330:
326:
321:
317:
311:
307:
279:
276:
273:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
240:
237:
234:
231:
228:
225:
222:
150:
149:
104:
102:
95:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4702:
4691:
4688:
4687:
4685:
4666:
4663:
4661:
4657:
4655:
4651:
4649:Making a hole
4648:
4646:
4645:Connected sum
4643:
4642:
4640:
4636:
4630:
4627:
4625:
4622:
4619:
4615:
4614:
4612:
4608:
4602:
4601:Orientability
4599:
4597:
4593:
4590:
4588:
4585:
4583:
4582:Connectedness
4580:
4579:
4577:
4573:
4570:
4564:
4557:
4554:
4550:
4547:
4546:
4545:
4542:
4538:
4535:
4533:
4530:
4529:
4527:
4522:
4521:
4520:
4517:
4516:
4514:
4512:With boundary
4510:
4500:(genus 3) ...
4499:
4496:
4493:
4490:
4486:
4485:Roman surface
4483:
4481:
4480:Boy's surface
4477:
4476:
4475:
4472:
4471:
4469:
4465:
4458:
4455:
4452:
4449:
4446:
4443:
4442:
4440:
4436:
4433:
4429:
4420:
4415:
4413:
4408:
4406:
4401:
4400:
4397:
4391:
4388:
4385:
4381:
4380:
4376:
4371:
4370:
4365:
4362:
4357:
4355:
4353:
4349:A. Coffman, "
4348:
4347:
4343:
4338:
4334:
4333:
4328:
4324:
4320:
4316:
4313:
4309:
4305:
4304:
4300:
4291:
4287:
4280:
4277:
4270:
4265:
4261:
4258:
4255:
4251:
4250:Boy's surface
4248:
4247:
4243:
4241:
4239:
4235:
4234:
4226:
4219:
4217:
4215:
4210:
4205:
4203:
4199:
4195:
4189:
4187:
4183:
4179:
4171:
4167:
4165:
4158:One-sidedness
4157:
4155:
4149:
4145:
4141:
4138:
4136:
4131:
4129:
4123:
4118:
4114:
4110:
4107:
4105:
4101:
4094:
4090:
4086:
4080:
4076:
4072:
4070:
4066:
4061:
4059:
4051:
4047:
4043:
4040:
4037:
4033:
4029:
4026:
4025:
4024:
4022:
4018:
4014:
4007:
4003:
3999:
3997:
3993:
3989:
3985:
3977:
3973:
3969:
3966:
3963:
3959:
3955:
3952:
3949:
3945:
3941:
3938:
3935:
3931:
3927:
3924:
3923:
3922:
3916:
3912:
3908:
3905:
3902:
3898:
3895:
3894:
3893:
3891:
3887:
3883:
3862:
3859:
3854:
3851:
3848:
3841:
3840:
3839:
3837:
3833:
3829:
3825:
3821:
3817:
3813:
3808:
3806:
3802:
3798:
3794:
3785:
3781:
3778:
3775:
3771:
3768:
3765:
3761:
3758:
3757:
3756:
3753:
3751:
3746:
3740:
3738:
3736:
3732:
3712:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3679:
3670:
3667:
3658:
3655:
3649:
3643:
3640:
3631:
3628:
3622:
3616:
3613:
3604:
3601:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3559:
3556:
3549:
3535:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3496:
3493:
3490:
3487:
3484:
3478:
3475:
3468:
3467:
3466:
3464:
3459:
3457:
3453:
3449:
3445:
3437:
3435:
3433:
3429:
3425:
3406:
3403:
3400:
3397:
3393:
3390:
3387:
3383:
3380:
3375:
3371:
3364:
3360:
3356:
3353:
3350:
3347:
3343:
3340:
3332:
3318:
3315:
3312:
3309:
3305:
3302:
3299:
3295:
3292:
3289:
3283:
3279:
3275:
3272:
3269:
3266:
3262:
3259:
3251:
3237:
3234:
3231:
3228:
3224:
3221:
3218:
3214:
3211:
3208:
3202:
3198:
3194:
3191:
3188:
3185:
3181:
3178:
3170:
3169:
3168:
3166:
3147:
3144:
3141:
3138:
3134:
3131:
3128:
3121:
3107:
3104:
3101:
3098:
3094:
3091:
3088:
3084:
3081:
3078:
3071:
3057:
3054:
3051:
3048:
3044:
3041:
3038:
3034:
3031:
3028:
3021:
3020:
3019:
3017:
3013:
3009:
3001:
2999:
2996:
2978:
2975:
2972:
2969:
2966:
2960:
2954:
2951:
2948:
2945:
2942:
2936:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2898:
2894:
2890:
2886:
2882:
2878:
2874:
2870:
2868:
2863:
2848:
2845:
2840:
2832:
2819:
2814:
2799:
2796:
2791:
2783:
2770:
2765:
2750:
2747:
2742:
2734:
2721:
2716:
2703:
2700:
2697:
2694:
2691:
2685:
2682:
2679:
2676:
2673:
2667:
2664:
2661:
2658:
2655:
2648:) satisfying
2647:
2643:
2639:
2635:
2630:
2617:
2614:
2611:
2608:
2605:
2602:
2596:
2593:
2589:
2585:
2582:
2574:
2560:
2547:
2542:
2539:
2534:
2531:
2528:
2519:
2505:
2502:
2499:
2494:
2490:
2486:
2481:
2477:
2467:
2454:
2449:
2446:
2441:
2433:
2420:
2416:
2412:
2410:
2405:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2365:
2362:
2359:
2356:
2353:
2350:
2341:
2327:
2324:
2321:
2318:
2315:
2307:
2303:
2298:
2284:
2279:
2275:
2271:
2266:
2262:
2256:
2252:
2242:
2229:
2224:
2216:
2212:
2208:
2205:
2202:
2197:
2194:
2188:
2183:
2179:
2153:
2145:
2141:
2137:
2134:
2131:
2126:
2123:
2117:
2112:
2108:
2100:
2099:
2098:
2096:
2092:
2073:
2068:
2065:
2060:
2052:
2040:
2039:
2038:
2036:
2032:
2017:
2014:
2011:
2008:
1987:
1984:
1981:
1978:
1975:
1972:
1965:implies that
1951:
1948:
1945:
1942:
1933:
1919:
1916:
1913:
1910:
1901:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1854:
1852:
1848:
1844:
1840:
1836:
1832:
1830:
1826:
1822:
1818:
1814:
1809:
1795:
1792:
1787:
1783:
1777:
1773:
1764:
1761:Suppose that
1760:
1756:
1754:
1750:
1746:
1741:
1723:
1717:
1713:
1710:
1703:
1700:
1694:
1688:
1684:
1681:
1674:
1671:
1665:
1659:
1655:
1652:
1645:
1642:
1635:
1634:
1633:
1631:
1627:
1623:
1619:
1615:
1612:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1547:
1533:
1530:
1527:
1522:
1518:
1514:
1509:
1505:
1501:
1496:
1492:
1482:
1480:
1476:
1472:
1467:
1453:
1450:
1447:
1444:
1441:
1438:
1433:
1429:
1423:
1419:
1415:
1410:
1406:
1400:
1396:
1392:
1387:
1383:
1377:
1373:
1363:
1362:) satisfying
1361:
1357:
1353:
1349:
1345:
1330:
1327:
1324:
1321:
1318:
1315:
1310:
1306:
1300:
1296:
1292:
1287:
1283:
1277:
1273:
1269:
1264:
1260:
1254:
1250:
1222:
1219:
1216:
1213:
1210:
1204:
1201:
1192:
1189:
1180:
1177:
1171:
1163:
1159:
1153:
1149:
1143:
1139:
1129:
1123:
1121:
1108:
1104:
1098:
1094:
1088:
1084:
1072:
1068:
1064:
1059:
1055:
1051:
1046:
1042:
1035:
1030:
1026:
1020:
1016:
1010:
1006:
1002:
997:
993:
987:
983:
977:
973:
969:
964:
960:
954:
950:
944:
940:
936:
934:
927:
923:
917:
913:
909:
904:
900:
894:
890:
886:
881:
877:
871:
867:
855:
854:
853:
850:
835:
829:
826:
823:
820:
817:
811:
805:
802:
799:
796:
793:
790:
787:
784:
778:
772:
769:
766:
763:
760:
754:
746:
726:
723:
720:
715:
711:
707:
702:
698:
694:
689:
685:
677:
676:
675:
673:
669:
665:
661:
653:
651:
649:
645:
641:
638:
607:
604:
599:
595:
591:
588:
585:
582:
579:
576:
571:
567:
563:
560:
553:
539:
536:
533:
530:
527:
524:
521:
518:
515:
510:
506:
502:
499:
492:
478:
475:
472:
469:
466:
463:
460:
457:
454:
449:
445:
441:
438:
431:
430:
429:
423:
415:
395:
392:
389:
386:
383:
378:
374:
370:
365:
361:
355:
351:
347:
342:
338:
332:
328:
324:
319:
315:
309:
305:
297:
296:
295:
293:
277:
271:
268:
265:
262:
259:
256:
253:
250:
244:
238:
235:
232:
229:
226:
220:
212:
207:
205:
201:
200:Jakob Steiner
197:
193:
189:
185:
181:
177:
173:
169:
168:Roman surface
165:
156:
146:
143:
135:
125:
121:
115:
114:
108:
103:
94:
93:
84:
81:
73:
63:
60:and simplify
59:
55:
49:
48:
43:This article
41:
32:
31:
19:
4544:Möbius strip
4492:Klein bottle
4484:
4383:
4378:
4367:
4350:
4337:Google Books
4335:, p. 30, at
4330:
4318:
4311:
4307:
4289:
4279:
4230:
4227:
4223:
4214:Klein bottle
4208:
4206:
4201:
4197:
4193:
4190:
4185:
4175:
4161:
4152:
4147:
4139:
4132:
4124:
4121:
4116:
4108:
4103:
4099:
4097:
4092:
4083:
4078:
4068:
4064:
4062:
4055:
4049:
4045:
4041:
4035:
4031:
4027:
4020:
4016:
4012:
4010:
4005:
3995:
3991:
3987:
3983:
3981:
3975:
3971:
3967:
3961:
3957:
3953:
3947:
3943:
3939:
3933:
3929:
3925:
3920:
3914:
3910:
3906:
3900:
3896:
3889:
3888:= 1 so that
3885:
3881:
3879:
3835:
3831:
3827:
3823:
3819:
3815:
3811:
3809:
3800:
3796:
3792:
3789:
3783:
3779:
3773:
3769:
3763:
3759:
3754:
3747:
3744:
3734:
3730:
3727:
3462:
3460:
3455:
3451:
3447:
3444:homeomorphic
3441:
3431:
3427:
3423:
3421:
3164:
3162:
3015:
3011:
3010:, longitude
3007:
3005:
2997:
2995:as desired.
2896:
2892:
2888:
2884:
2880:
2876:
2872:
2871:
2866:
2864:
2817:
2815:
2768:
2766:
2719:
2717:
2645:
2641:
2637:
2633:
2631:
2561:
2520:
2468:
2418:
2414:
2413:
2409:this subcase
2408:
2406:
2342:
2305:
2301:
2299:
2243:
2170:
2094:
2090:
2088:
2034:
2033:
1934:
1902:
1855:
1846:
1842:
1838:
1834:
1833:
1828:
1824:
1820:
1816:
1812:
1810:
1762:
1758:
1757:
1752:
1748:
1744:
1742:
1739:
1629:
1625:
1621:
1617:
1616:
1613:
1548:
1483:
1481:) such that
1478:
1474:
1470:
1468:
1364:
1359:
1355:
1351:
1347:
1346:
1344:as desired.
1241:
851:
744:
742:
674:) such that
671:
667:
663:
659:
657:
622:
411:
208:
171:
167:
161:
138:
129:
110:
76:
67:
44:
4587:Compactness
4233:singularity
4176:Imagine an
3880:and either
2816:and (0, 0,
2771:, 0) with
2343:Since also
747:defined by
637:tetrahedral
164:mathematics
124:introducing
4638:Operations
4620:components
4616:Number of
4596:smoothness
4575:Properties
4523:Semisphere
4438:Orientable
4271:References
4264:polyhedron
4164:orientable
3456:(-x,-y,-z)
2421:, we have
2001:and hence
1853:we assume
1549:for which
1348:Conversely
132:March 2018
107:references
70:March 2018
4665:Immersion
4660:cross-cap
4658:Gluing a
4652:Gluing a
4549:Cross-cap
4494:(genus 2)
4478:genus 1;
4453:(genus 1)
4447:(genus 0)
4369:MathWorld
4254:immersion
4117:Figure 4.
4079:Figure 2.
4006:Figure 1.
3799:= 0, and
3668:−
3656:−
3641:−
3629:−
3614:−
3602:−
3593:→
3584:−
3575:−
3566:−
3503:→
3404:ϕ
3401:
3394:ϕ
3391:
3384:θ
3381:
3316:ϕ
3313:
3306:θ
3303:
3296:θ
3293:
3235:ϕ
3232:
3225:θ
3222:
3215:θ
3212:
3145:θ
3142:
3105:ϕ
3102:
3095:θ
3092:
3055:ϕ
3052:
3045:θ
3042:
2632:there is
2535:≤
2206:−
2198:−
2135:−
2061:≤
1946:≠
1867:≠
1831:to be 0.
1439:−
1316:−
608:φ
605:
592:θ
589:
583:θ
580:
540:φ
537:
531:φ
528:
522:θ
519:
479:φ
476:
470:φ
467:
461:θ
458:
414:longitude
371:−
206:in 1844.
192:immersion
4690:Surfaces
4684:Category
4618:boundary
4537:Cylinder
4244:See also
4231:Whitney
3432:π/2
3344:′
3263:′
3182:′
640:symmetry
422:latitude
188:symmetry
4568:notions
4566:Related
4532:Annulus
4528:Ribbon
4382:at the
4238:tangent
4186:z = x y
4135:bulbous
4058:orchids
4017:x = y z
4013:y = x z
3884:= 0 or
3838:. Then
3452:(x,y,z)
2820:) with
1935:(since
1849:are 0.
1242:and so
646:of the
631:-, and
292:formula
178:of the
176:mapping
120:improve
52:Please
4654:handle
4445:Sphere
4325:
4044:= −1,
3805:origin
3428:θ
3424:φ
3016:θ
3012:φ
2689:
2671:
2600:
2469:Since
1698:
1669:
420:) and
211:sphere
166:, the
109:, but
4624:Genus
4451:Torus
4310:. In
4252:– an
4209:x = 0
4202:z = 0
4194:z = 0
4030:= 1,
4021:z = 0
3978:= −1.
3950:= −1;
3917:= −1.
3897:x = y
3795:= 0,
1856:(***)
1484:(**)
849:say.
182:into
4519:Disk
4323:ISBN
4262:– a
3990:and
3964:= 1;
3936:= 1;
3903:= 1;
3818:and
2875:If (
2841:>
2792:>
2743:>
2583:>
2442:>
2304:and
2171:and
2093:and
1365:(*)
204:Rome
4594:or
4558:...
4216:.)
4198:x-y
4178:ant
4048:= −
3970:= −
3942:= −
3909:= −
3398:sin
3388:cos
3372:cos
3310:cos
3300:sin
3290:cos
3229:sin
3219:sin
3209:cos
3139:sin
3099:sin
3089:cos
3049:cos
3039:cos
2097:by
627:-,
596:cos
586:sin
577:cos
534:sin
525:cos
516:sin
473:sin
464:cos
455:cos
294:of
170:or
162:In
4686::
4366:.
4288:.
4184:,
4104:xy
4102:=
4069:xy
4067:=
4034:=
3996:yz
3994:=
3988:xz
3986:=
3974:,
3960:,
3956:=
3946:,
3932:,
3928:=
3913:,
3899:,
3830:=
3824:zx
3822:=
3816:yz
3814:=
3807:.
3784:xy
3782:=
3774:zx
3772:=
3764:yz
3762:=
3731:RP
3448:RP
3434:.
2895:,
2891:,
2883:,
2879:,
2873:4.
2869:.
2644:,
2640:,
2634:no
2419:3.
2415:b.
2035:a.
1888:0.
1845:,
1841:,
1835:3.
1827:,
1823:,
1815:,
1759:2.
1751:,
1747:,
1628:,
1624:,
1618:1.
1454:0.
1358:,
1354:,
670:,
666:,
650:.
633:xz
629:yz
625:xy
396:0.
4418:e
4411:t
4404:v
4372:.
4354:"
4339:)
4100:z
4065:z
4052:.
4050:x
4046:y
4042:z
4038:;
4036:x
4032:y
4028:z
3992:x
3984:y
3976:x
3972:z
3968:y
3962:x
3958:z
3954:y
3948:y
3944:z
3940:x
3934:y
3930:z
3926:x
3915:z
3911:y
3907:x
3901:z
3890:z
3886:z
3882:y
3863:z
3860:y
3855:=
3852:z
3849:y
3836:z
3834:/
3832:y
3828:x
3820:y
3812:x
3801:z
3797:y
3793:x
3786:.
3780:z
3776:,
3770:y
3766:,
3760:x
3713:.
3710:)
3707:y
3704:x
3701:,
3698:x
3695:z
3692:,
3689:z
3686:y
3683:(
3680:=
3677:)
3674:)
3671:y
3665:(
3662:)
3659:x
3653:(
3650:,
3647:)
3644:x
3638:(
3635:)
3632:z
3626:(
3623:,
3620:)
3617:z
3611:(
3608:)
3605:y
3599:(
3596:(
3590:)
3587:z
3581:,
3578:y
3572:,
3569:x
3563:(
3560::
3557:T
3536:,
3533:)
3530:y
3527:x
3524:,
3521:x
3518:z
3515:,
3512:z
3509:y
3506:(
3500:)
3497:z
3494:,
3491:y
3488:,
3485:x
3482:(
3479::
3476:T
3463:T
3407:,
3376:2
3365:2
3361:r
3357:=
3354:y
3351:x
3348:=
3341:z
3319:,
3284:2
3280:r
3276:=
3273:x
3270:z
3267:=
3260:y
3238:,
3203:2
3199:r
3195:=
3192:z
3189:y
3186:=
3179:x
3165:T
3148:.
3135:r
3132:=
3129:z
3108:,
3085:r
3082:=
3079:y
3058:,
3035:r
3032:=
3029:x
3008:r
2982:)
2979:W
2976:,
2973:V
2970:,
2967:U
2964:(
2961:=
2958:)
2955:0
2952:,
2949:0
2946:,
2943:0
2940:(
2937:=
2934:)
2931:x
2928:z
2925:,
2922:z
2919:y
2916:,
2913:y
2910:x
2907:(
2897:z
2893:y
2889:x
2885:W
2881:V
2877:U
2849:2
2846:1
2837:|
2833:W
2829:|
2818:W
2800:2
2797:1
2788:|
2784:V
2780:|
2769:V
2751:2
2748:1
2739:|
2735:U
2731:|
2720:U
2704:.
2701:x
2698:z
2695:=
2692:W
2686:,
2683:z
2680:y
2677:=
2674:V
2668:,
2665:y
2662:x
2659:=
2656:U
2646:z
2642:y
2638:x
2636:(
2618:,
2615:0
2612:=
2609:W
2606:=
2603:V
2597:,
2594:2
2590:/
2586:1
2579:|
2575:U
2571:|
2548:,
2543:2
2540:1
2532:y
2529:x
2506:,
2503:1
2500:=
2495:2
2491:y
2487:+
2482:2
2478:x
2455:.
2450:2
2447:1
2438:|
2434:U
2430:|
2392:,
2389:W
2386:=
2383:0
2380:=
2377:x
2374:z
2366:V
2363:=
2360:0
2357:=
2354:z
2351:y
2328:.
2325:U
2322:=
2319:y
2316:x
2306:y
2302:x
2285:,
2280:2
2276:U
2272:=
2267:2
2263:y
2257:2
2253:x
2230:,
2225:2
2217:2
2213:U
2209:4
2203:1
2195:1
2189:=
2184:2
2180:y
2154:2
2146:2
2142:U
2138:4
2132:1
2127:+
2124:1
2118:=
2113:2
2109:x
2095:y
2091:x
2074:,
2069:2
2066:1
2057:|
2053:U
2049:|
2018:,
2015:0
2012:=
2009:U
1988:,
1985:0
1982:=
1979:y
1976:=
1973:x
1952:,
1949:0
1943:z
1920:,
1917:0
1914:=
1911:z
1885:=
1882:W
1879:=
1876:V
1873:,
1870:0
1864:U
1847:W
1843:V
1839:U
1829:W
1825:V
1821:U
1817:V
1813:U
1796:0
1793:=
1788:2
1784:V
1778:2
1774:U
1763:W
1753:z
1749:y
1745:x
1724:.
1718:U
1714:W
1711:V
1704:=
1701:z
1695:,
1689:W
1685:V
1682:U
1675:=
1672:y
1666:,
1660:V
1656:U
1653:W
1646:=
1643:x
1630:W
1626:V
1622:U
1599:,
1596:x
1593:z
1590:=
1587:W
1584:,
1581:z
1578:y
1575:=
1572:V
1569:,
1566:y
1563:x
1560:=
1557:U
1534:,
1531:1
1528:=
1523:2
1519:z
1515:+
1510:2
1506:y
1502:+
1497:2
1493:x
1479:z
1477:,
1475:y
1473:,
1471:x
1451:=
1448:W
1445:V
1442:U
1434:2
1430:U
1424:2
1420:W
1416:+
1411:2
1407:W
1401:2
1397:V
1393:+
1388:2
1384:V
1378:2
1374:U
1360:W
1356:V
1352:U
1331:0
1328:=
1325:W
1322:V
1319:U
1311:2
1307:U
1301:2
1297:W
1293:+
1288:2
1284:W
1278:2
1274:V
1270:+
1265:2
1261:V
1255:2
1251:U
1223:,
1220:W
1217:V
1214:U
1211:=
1208:)
1205:x
1202:z
1199:(
1196:)
1193:z
1190:y
1187:(
1184:)
1181:y
1178:x
1175:(
1172:=
1169:)
1164:2
1160:z
1154:2
1150:y
1144:2
1140:x
1136:(
1133:)
1130:1
1127:(
1124:=
1114:)
1109:2
1105:z
1099:2
1095:y
1089:2
1085:x
1081:(
1078:)
1073:2
1069:z
1065:+
1060:2
1056:y
1052:+
1047:2
1043:x
1039:(
1036:=
1031:4
1027:x
1021:2
1017:z
1011:2
1007:y
1003:+
998:4
994:z
988:2
984:y
978:2
974:x
970:+
965:4
961:y
955:2
951:x
945:2
941:z
937:=
928:2
924:U
918:2
914:W
910:+
905:2
901:W
895:2
891:V
887:+
882:2
878:V
872:2
868:U
836:,
833:)
830:W
827:,
824:V
821:,
818:U
815:(
812:=
809:)
806:y
803:x
800:,
797:x
794:z
791:,
788:z
785:y
782:(
779:=
776:)
773:z
770:,
767:y
764:,
761:x
758:(
755:T
745:T
727:,
724:1
721:=
716:2
712:z
708:+
703:2
699:y
695:+
690:2
686:x
672:z
668:y
664:x
660:r
600:2
572:2
568:r
564:=
561:z
511:2
507:r
503:=
500:y
450:2
446:r
442:=
439:x
426:φ
424:(
418:θ
416:(
393:=
390:z
387:y
384:x
379:2
375:r
366:2
362:x
356:2
352:z
348:+
343:2
339:z
333:2
329:y
325:+
320:2
316:y
310:2
306:x
278:.
275:)
272:y
269:x
266:,
263:z
260:x
257:,
254:z
251:y
248:(
245:=
242:)
239:z
236:,
233:y
230:,
227:x
224:(
221:f
145:)
139:(
134:)
130:(
116:.
83:)
77:(
72:)
68:(
64:.
50:.
20:)
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