4177:. 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.
87:
4214:
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.
4064:
3991:
4133:
4159:
4078:
1226:
4102:
25:
4213:
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
4200:
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
4142:
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
847:
144:
3717:
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
3726:. 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.
1724:
4217:
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
4073:
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.
4180:
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
3712:
3406:
3779:
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:
3318:
3237:
1221:{\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}}}
624:-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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4098:
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.
2018:
1920:
4225:. 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
1729:(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.)
4398:
3541:
4391:
4315:
3324:
35:
3243:
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108:
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4114:
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.
289:
130:
68:
484:
423:
4087:
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 (
4111:
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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2163:
1234:
46:
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4129:
If the surface in Figure 5 is turned around 180 degrees and then turned upside down, the result is as shown in Figure 6.
4116:
4544:
4378:
3460:
101:
95:
4196:
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).
205:
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1757:
1719:{\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
3734:
The Roman surface has four bulbous "lobes", each one on a different corner of a tetrahedron.
2300:
4468:
4238:
3447:
and these two points are different: they lie on opposite sides of the center of the sphere.
1993:
1895:
636:
143:
4570:
4132:
4008:
is shown in cyan and purple. In the image the paraboloids are seen to intersect along the
4532:
16:
Self-intersecting, highly symmetrical mapping of the real projective plane into 3D space
631:. It is a particular type (called type 1) of Steiner surface, that is, a 3-dimensional
628:
4672:
4633:
4589:
4575:
4226:
3432:
188:
4340:
4480:
4325:
4297:
The
Algebra and Geometry of Steiner and other Quadratically Parametrizable Surfaces
4202:
4158:
4077:
4379:
Ashay
Dharwadker, Heptahedron and Roman Surface, Electronic Geometry Models, 2004.
4352:
4193:
axis the ant will be on the "outside" of the
Eastern lobe, standing rightside-up.
4155:, i.e. one-sided. This is not quite obvious. To see this, look again at Figure 3.
4143:
surface shown at the top of this article also has three lobes in sideways view.
3707:{\displaystyle T:(-x,-y,-z)\rightarrow ((-y)(-z),(-z)(-x),(-x)(-y))=(yz,zx,xy).}
625:
152:
4101:
4252:
4152:
4123:
3738:
4122:
One of the lobes of the Roman surface is seen frontally in Figure 5, and its
4648:
4537:
4357:
4242:
2854:(each of which is a noncompact portion of a coordinate axis, in two pieces)
402:
1603:
with one exception: In case 3.b. below, we show this cannot be proved.
4525:
3439:. But the sphere centered at the origin has this property, that if point
410:
183:
of the projective plane; however, the figure resulting from removing six
176:
280:
3401:{\displaystyle z'=xy=r^{2}\,\cos ^{2}\theta \,\cos \phi \,\sin \phi ,}
4433:
4046:
199:
3313:{\displaystyle y'=zx=r^{2}\,\cos \theta \,\sin \theta \,\cos \phi ,}
3232:{\displaystyle x'=yz=r^{2}\,\cos \theta \,\sin \theta \,\sin \phi ,}
1808:
must be 0 also. This shows that is it impossible for exactly one of
4439:
4157:
4131:
4100:
4076:
4062:
3989:
142:
602:{\displaystyle z=r^{2}\cos \theta \sin \theta \cos ^{2}\varphi }
417:), gives parametric equations for the Roman surface as follows:
192:
4387:
3971:
Let us see the pieces being put together. Join the paraboloids
4166:
3737:
A Roman surface can be constructed by splicing together three
391:{\displaystyle x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}-r^{2}xyz=0.\,}
80:
18:
3454:
converts both of these antipodal points into the same point,
534:{\displaystyle y=r^{2}\sin \theta \cos \varphi \sin \varphi }
473:{\displaystyle x=r^{2}\cos \theta \cos \varphi \sin \varphi }
3722:, but is instead a quotient of the real projective plane
2888:
are zero and the third one has absolute value 1, clearly
1449:{\displaystyle U^{2}V^{2}+V^{2}W^{2}+W^{2}U^{2}-UVW=0.\,}
401:
Also, taking a parametrization of the sphere in terms of
4095:) paraboloid and lie in the North and South directions.
3443:
belongs to the sphere, then so does the antipodal point
2224:{\displaystyle y^{2}={\frac {1-{\sqrt {1-4U^{2}}}}{2}},}
1326:{\displaystyle U^{2}V^{2}+V^{2}W^{2}+W^{2}U^{2}-UVW=0\,}
4414:
Compact topological surfaces and their immersions in 3D
2233:
this ensures that (*) holds. It is easy to verify that
2150:{\displaystyle x^{2}={\frac {1+{\sqrt {1-4U^{2}}}}{2}}}
42:
3836:
3544:
3463:
3327:
3246:
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3066:
3016:
2894:
2815:
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2557:
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2239:
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2095:
2035:
1996:
1960:
1930:
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1851:
1760:
1732:
It is easy to use (*) to confirm that (**) holds for
1630:
1544:
1479:
1360:
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850:
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672:
548:
487:
426:
292:
208:
187:
is one. Its name arises because it was discovered by
4281:. Indiana University - Purdue University Fort Wayne.
4626:
4598:
4563:
4554:
4500:
4455:
4426:
4419:
2856:
do not correspond to any point on the Roman surface
3858:
3706:
3529:
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3231:
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3051:
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2844:
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2500:
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2012:
1982:
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1914:
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1448:
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1220:
830:
721:
601:
533:
472:
390:
271:
4060:, through them. The result is shown in Figure 2.
4045:The two paraboloids together look like a pair of
3744:Let there be these three hyperbolic paraboloids:
3530:{\displaystyle T:(x,y,z)\rightarrow (yz,zx,xy),}
3910:Likewise, the other external intersections are
3411:which are the points on the Roman surface. Let
198:The simplest construction is as the image of a
3102:{\displaystyle y=r\,\cos \theta \,\sin \phi ,}
3052:{\displaystyle x=r\,\cos \theta \,\cos \phi ,}
831:{\displaystyle T(x,y,z)=(yz,zx,xy)=(U,V,W),\,}
612:The origin is a triple point, and each of the
4399:
3431:The sphere, before being transformed, is not
2876:) is the point (0, 0, 0), then if any two of
2387:{\displaystyle yz=0=V{\text{ and }}zx=0=W,\,}
651:= 1. Given the sphere defined by the points (
8:
4375:(website of the California State University)
4004:is shown in blue and orange. The paraboloid
2977:{\displaystyle (xy,yz,zx)=(0,0,0)=(U,V,W)\,}
732:we apply to these points the transformation
4245:of the projective plane without cross-caps.
3881:= ±1. Their two external intersections are
4560:
4423:
4406:
4392:
4384:
4308:Geometric Modeling and Algebraic Geometry
4295:A. Coffman, A. Schwartz, and C. Stanton:
4052:Now run the third hyperbolic paraboloid,
3846:
3835:
3815:, the second paraboloid is equivalent to
3543:
3462:
3385:
3375:
3363:
3358:
3352:
3326:
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3287:
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2577:
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892:
882:
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859:
851:
849:
827:
741:
718:
703:
690:
677:
671:
647:For simplicity we consider only the case
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486:
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425:
387:
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353:
343:
330:
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307:
297:
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131:Learn how and when to remove this message
69:Learn how and when to remove this message
4201:and standing upside-down. (Compare with
4189:plane. As soon as it passes through the
3792:= 0, intersect at a triple point at the
3156:to all the points on this sphere yields
94:This article includes a list of general
4265:
2068:{\displaystyle |U|\leq {\frac {1}{2}},}
2542:{\displaystyle xy\leq {\frac {1}{2}},}
2449:{\displaystyle |U|>{\frac {1}{2}}.}
2406:In this remaining subcase of the case
1529:{\displaystyle x^{2}+y^{2}+z^{2}=1,\,}
722:{\displaystyle x^{2}+y^{2}+z^{2}=1,\,}
3427:Relation to the real projective plane
2845:{\displaystyle |W|>{\frac {1}{2}}}
2796:{\displaystyle |V|>{\frac {1}{2}}}
2747:{\displaystyle |U|>{\frac {1}{2}}}
202:centered at the origin under the map
7:
4303:(3) 13 (April 1996), p. 257-286
3007:. Then its parametric equations are
272:{\displaystyle f(x,y,z)=(yz,xz,xy).}
36:research paper or scientific journal
4229:to any surface at the singularity.
4209:Double, triple, and pinching points
3987:. The result is shown in Figure 1.
2698:{\displaystyle U=xy,\ V=yz,\ W=zx.}
2612:{\displaystyle |U|>1/2,\ V=W=0,}
2280:{\displaystyle x^{2}y^{2}=U^{2},\,}
175:, with an unusually high degree of
4255:very similar to the Roman surface.
3142:{\displaystyle z=r\,\sin \theta .}
2991:Derivation of parametric equations
2711:, 0, 0) of the equation (*) with
100:it lacks sufficient corresponding
14:
4126:– balloon-like—shape is evident.
1594:{\displaystyle U=xy,V=yz,W=zx,\,}
147:An animation of the Roman surface
4115:
2987:This covers all possible cases.
2501:{\displaystyle x^{2}+y^{2}=1,\,}
2289:and hence choosing the signs of
1883:{\displaystyle U\neq 0,V=W=0.\,}
85:
23:
4301:Computer Aided Geometric Design
2400:leads to the desired converse.
3859:{\displaystyle yz={y \over z}}
3730:Structure of the Roman surface
3698:
3671:
3665:
3662:
3653:
3650:
3641:
3635:
3626:
3623:
3614:
3608:
3599:
3596:
3587:
3584:
3581:
3578:
3551:
3521:
3494:
3491:
3488:
3470:
3435:to the real projective plane,
3152:Then, applying transformation
2970:
2952:
2946:
2928:
2922:
2895:
2825:
2817:
2776:
2768:
2727:
2719:
2567:
2559:
2426:
2418:
2045:
2037:
1791:{\displaystyle U^{2}V^{2}=0\,}
1196:
1187:
1184:
1175:
1172:
1163:
1157:
1124:
1121:
1115:
1102:
1069:
1066:
1027:
821:
803:
797:
770:
764:
746:
643:Derivation of implicit formula
263:
236:
230:
212:
1:
2551:and thus in this case, where
2297:appropriately will guarantee
3415:range from 0 to 2π, and let
1826:Suppose that exactly two of
1754:is 0. From (*) this implies
1458:We prove that there exists (
4306:Bert Jüttler, Ragni Piene:
4695:
1947:{\displaystyle z\neq 0,\,}
1840:Without loss of generality
1800:and hence at least one of
1609:In the case where none of
4151:The Roman surface is non-
2995:Let a sphere have radius
2510:it is easy to check that
179:. This mapping is not an
4275:"Steiner Roman Surfaces"
4137:Figure 6. Roman surface.
4082:Figure 3. Roman surface.
1983:{\displaystyle x=y=0,\,}
1339:, suppose we are given (
51:overly technical phrases
43:help improve the article
4545:Sphere with three holes
4185:axis, always above the
2323:{\displaystyle xy=U.\,}
279:This gives an implicit
173:three-dimensional space
163:is a self-intersecting
115:more precise citations.
4321:restricted online copy
4169:on top of the "third"
4162:
4139:
4108:
4084:
4070:
3997:
3860:
3739:hyperbolic paraboloids
3708:
3531:
3402:
3314:
3233:
3143:
3103:
3053:
2978:
2846:
2797:
2748:
2699:
2613:
2543:
2502:
2450:
2388:
2324:
2281:
2225:
2151:
2069:
2020:contradicting (***).)
2014:
2013:{\displaystyle U=0,\,}
1984:
1948:
1916:
1915:{\displaystyle z=0,\,}
1884:
1792:
1720:
1595:
1530:
1450:
1327:
1222:
832:
723:
603:
535:
474:
392:
273:
148:
4463:Real projective plane
4448:Pretzel (genus 3) ...
4171:hyperbolic paraboloid
4161:
4135:
4104:
4080:
4066:
4049:joined back-to-back.
3993:
3861:
3709:
3532:
3403:
3315:
3234:
3144:
3104:
3054:
2979:
2847:
2798:
2749:
2707:Hence the solutions (
2700:
2614:
2544:
2503:
2451:
2389:
2325:
2282:
2226:
2152:
2070:
2026:In the subcase where
2015:
1985:
1949:
1917:
1885:
1793:
1721:
1596:
1531:
1451:
1328:
1223:
833:
724:
604:
536:
475:
393:
274:
169:real projective plane
146:
4618:Euler characteristic
3834:
3542:
3461:
3325:
3244:
3163:
3114:
3064:
3014:
2892:
2813:
2764:
2715:
2641:
2555:
2514:
2462:
2414:
2336:
2301:
2237:
2164:
2093:
2033:
1994:
1958:
1928:
1896:
1849:
1758:
1628:
1542:
1477:
1358:
1235:
848:
740:
670:
546:
485:
424:
290:
206:
4373:National Curve Bank
4279:National Curve Bank
4249:Tetrahemihexahedron
3799:For example, given
3450:The transformation
45:by rewriting it in
4445:Number 8 (genus 2)
4350:Weisstein, Eric W.
4290:General references
4163:
4140:
4109:
4085:
4071:
3998:
3856:
3704:
3527:
3398:
3310:
3229:
3139:
3099:
3049:
2974:
2842:
2793:
2756:and likewise, (0,
2744:
2695:
2609:
2539:
2498:
2446:
2384:
2320:
2277:
2221:
2147:
2065:
2010:
1980:
1944:
1912:
1880:
1788:
1744:defined this way.
1716:
1591:
1526:
1446:
1323:
1218:
1216:
828:
719:
599:
531:
470:
388:
269:
149:
47:encyclopedic style
34:is written like a
4666:
4665:
4662:
4661:
4496:
4495:
4316:978-3-540-72184-0
4310:. Springer 2008,
3854:
2840:
2791:
2742:
2679:
2661:
2590:
2534:
2441:
2360:
2216:
2210:
2145:
2139:
2060:
1710:
1709:
1688:
1681:
1680:
1659:
1652:
1651:
1621:is 0, we can set
841:But then we have
633:linear projection
141:
140:
133:
79:
78:
71:
4686:
4581:Triangulatedness
4561:
4424:
4420:Without boundary
4408:
4401:
4394:
4385:
4363:
4362:
4341:Steiner Surfaces
4283:
4282:
4270:
4119:
3865:
3863:
3862:
3857:
3855:
3847:
3713:
3711:
3710:
3705:
3536:
3534:
3533:
3528:
3419:range from 0 to
3407:
3405:
3404:
3399:
3368:
3367:
3357:
3356:
3335:
3319:
3317:
3316:
3311:
3276:
3275:
3254:
3238:
3236:
3235:
3230:
3195:
3194:
3173:
3148:
3146:
3145:
3140:
3108:
3106:
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3100:
3058:
3056:
3055:
3050:
2983:
2981:
2980:
2975:
2851:
2849:
2848:
2843:
2841:
2833:
2828:
2820:
2802:
2800:
2799:
2794:
2792:
2784:
2779:
2771:
2753:
2751:
2750:
2745:
2743:
2735:
2730:
2722:
2704:
2702:
2701:
2696:
2677:
2659:
2618:
2616:
2615:
2610:
2588:
2581:
2570:
2562:
2548:
2546:
2545:
2540:
2535:
2527:
2507:
2505:
2504:
2499:
2487:
2486:
2474:
2473:
2455:
2453:
2452:
2447:
2442:
2434:
2429:
2421:
2396:this shows that
2393:
2391:
2390:
2385:
2361:
2358:
2329:
2327:
2326:
2321:
2286:
2284:
2283:
2278:
2272:
2271:
2259:
2258:
2249:
2248:
2230:
2228:
2227:
2222:
2217:
2212:
2211:
2209:
2208:
2190:
2181:
2176:
2175:
2156:
2154:
2153:
2148:
2146:
2141:
2140:
2138:
2137:
2119:
2110:
2105:
2104:
2078:if we determine
2074:
2072:
2071:
2066:
2061:
2053:
2048:
2040:
2019:
2017:
2016:
2011:
1989:
1987:
1986:
1981:
1953:
1951:
1950:
1945:
1921:
1919:
1918:
1913:
1892:It follows that
1889:
1887:
1886:
1881:
1797:
1795:
1794:
1789:
1780:
1779:
1770:
1769:
1725:
1723:
1722:
1717:
1711:
1705:
1697:
1696:
1686:
1682:
1676:
1668:
1667:
1657:
1653:
1647:
1639:
1638:
1600:
1598:
1597:
1592:
1535:
1533:
1532:
1527:
1515:
1514:
1502:
1501:
1489:
1488:
1455:
1453:
1452:
1447:
1426:
1425:
1416:
1415:
1403:
1402:
1393:
1392:
1380:
1379:
1370:
1369:
1332:
1330:
1329:
1324:
1303:
1302:
1293:
1292:
1280:
1279:
1270:
1269:
1257:
1256:
1247:
1246:
1227:
1225:
1224:
1219:
1217:
1156:
1155:
1146:
1145:
1136:
1135:
1108:
1101:
1100:
1091:
1090:
1081:
1080:
1065:
1064:
1052:
1051:
1039:
1038:
1023:
1022:
1013:
1012:
1003:
1002:
990:
989:
980:
979:
970:
969:
957:
956:
947:
946:
937:
936:
920:
919:
910:
909:
897:
896:
887:
886:
874:
873:
864:
863:
837:
835:
834:
829:
728:
726:
725:
720:
708:
707:
695:
694:
682:
681:
637:Veronese surface
623:
619:
615:
608:
606:
605:
600:
592:
591:
564:
563:
540:
538:
537:
532:
503:
502:
479:
477:
476:
471:
442:
441:
416:
408:
397:
395:
394:
389:
371:
370:
358:
357:
348:
347:
335:
334:
325:
324:
312:
311:
302:
301:
278:
276:
275:
270:
136:
129:
125:
122:
116:
111:this article by
102:inline citations
89:
88:
81:
74:
67:
63:
60:
54:
27:
26:
19:
4694:
4693:
4689:
4688:
4687:
4685:
4684:
4683:
4669:
4668:
4667:
4658:
4622:
4599:Characteristics
4594:
4556:
4550:
4492:
4451:
4415:
4412:
4353:"Roman Surface"
4348:
4347:
4335:
4292:
4287:
4286:
4273:Coffman, Adam.
4272:
4271:
4267:
4262:
4235:
4211:
4149:
4000:The paraboloid
3832:
3831:
3732:
3724:RP = S / (x~-x)
3540:
3539:
3459:
3458:
3429:
3359:
3348:
3328:
3323:
3322:
3267:
3247:
3242:
3241:
3186:
3166:
3161:
3160:
3112:
3111:
3062:
3061:
3012:
3011:
3003:, and latitude
2993:
2890:
2889:
2811:
2810:
2762:
2761:
2713:
2712:
2639:
2638:
2553:
2552:
2512:
2511:
2478:
2465:
2460:
2459:
2412:
2411:
2359: and
2334:
2333:
2299:
2298:
2263:
2250:
2240:
2235:
2234:
2200:
2182:
2167:
2162:
2161:
2129:
2111:
2096:
2091:
2090:
2031:
2030:
1992:
1991:
1956:
1955:
1926:
1925:
1894:
1893:
1847:
1846:
1771:
1761:
1756:
1755:
1698:
1669:
1640:
1626:
1625:
1540:
1539:
1506:
1493:
1480:
1475:
1474:
1417:
1407:
1394:
1384:
1371:
1361:
1356:
1355:
1294:
1284:
1271:
1261:
1248:
1238:
1233:
1232:
1215:
1214:
1147:
1137:
1127:
1106:
1105:
1092:
1082:
1072:
1056:
1043:
1030:
1014:
1004:
994:
981:
971:
961:
948:
938:
928:
921:
911:
901:
888:
878:
865:
855:
846:
845:
738:
737:
699:
686:
673:
668:
667:
645:
621:
617:
613:
583:
555:
544:
543:
494:
483:
482:
433:
422:
421:
414:
406:
362:
349:
339:
326:
316:
303:
293:
288:
287:
204:
203:
191:when he was in
185:singular points
161:Steiner surface
137:
126:
120:
117:
107:Please help to
106:
90:
86:
75:
64:
58:
55:
40:
28:
24:
17:
12:
11:
5:
4692:
4690:
4682:
4681:
4671:
4670:
4664:
4663:
4660:
4659:
4657:
4656:
4651:
4645:
4639:
4636:
4630:
4628:
4624:
4623:
4621:
4620:
4615:
4610:
4602:
4600:
4596:
4595:
4593:
4592:
4587:
4578:
4573:
4567:
4565:
4558:
4552:
4551:
4549:
4548:
4542:
4541:
4540:
4530:
4529:
4528:
4523:
4515:
4514:
4513:
4504:
4502:
4498:
4497:
4494:
4493:
4491:
4490:
4487:Dyck's surface
4484:
4478:
4477:
4476:
4471:
4459:
4457:
4456:Non-orientable
4453:
4452:
4450:
4449:
4446:
4443:
4437:
4430:
4428:
4421:
4417:
4416:
4413:
4411:
4410:
4403:
4396:
4388:
4382:
4381:
4376:
4368:Roman Surfaces
4364:
4345:
4334:
4333:External links
4331:
4330:
4329:
4318:, p. 30 (
4304:
4291:
4288:
4285:
4284:
4264:
4263:
4261:
4258:
4257:
4256:
4246:
4234:
4231:
4210:
4207:
4148:
4145:
4043:
4042:
4028:
3969:
3968:
3954:
3940:
3926:
3908:
3907:
3893:
3867:
3866:
3853:
3850:
3845:
3842:
3839:
3777:
3776:
3766:
3756:
3731:
3728:
3715:
3714:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3537:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3428:
3425:
3409:
3408:
3397:
3394:
3391:
3388:
3384:
3381:
3378:
3374:
3371:
3366:
3362:
3355:
3351:
3347:
3344:
3341:
3338:
3334:
3331:
3320:
3309:
3306:
3303:
3300:
3296:
3293:
3290:
3286:
3283:
3280:
3274:
3270:
3266:
3263:
3260:
3257:
3253:
3250:
3239:
3228:
3225:
3222:
3219:
3215:
3212:
3209:
3205:
3202:
3199:
3193:
3189:
3185:
3182:
3179:
3176:
3172:
3169:
3150:
3149:
3138:
3135:
3132:
3129:
3125:
3122:
3119:
3109:
3098:
3095:
3092:
3089:
3085:
3082:
3079:
3075:
3072:
3069:
3059:
3048:
3045:
3042:
3039:
3035:
3032:
3029:
3025:
3022:
3019:
2992:
2989:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2839:
2836:
2831:
2827:
2823:
2819:
2790:
2787:
2782:
2778:
2774:
2770:
2741:
2738:
2733:
2729:
2725:
2721:
2694:
2691:
2688:
2685:
2682:
2676:
2673:
2670:
2667:
2664:
2658:
2655:
2652:
2649:
2646:
2608:
2605:
2602:
2599:
2596:
2593:
2587:
2584:
2580:
2576:
2573:
2569:
2565:
2561:
2538:
2533:
2530:
2525:
2522:
2519:
2496:
2493:
2490:
2485:
2481:
2477:
2472:
2468:
2445:
2440:
2437:
2432:
2428:
2424:
2420:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2356:
2353:
2350:
2347:
2344:
2341:
2318:
2315:
2312:
2309:
2306:
2275:
2270:
2266:
2262:
2257:
2253:
2247:
2243:
2220:
2215:
2207:
2203:
2199:
2196:
2193:
2188:
2185:
2179:
2174:
2170:
2158:
2157:
2144:
2136:
2132:
2128:
2125:
2122:
2117:
2114:
2108:
2103:
2099:
2076:
2075:
2064:
2059:
2056:
2051:
2047:
2043:
2039:
2008:
2005:
2002:
1999:
1978:
1975:
1972:
1969:
1966:
1963:
1942:
1939:
1936:
1933:
1910:
1907:
1904:
1901:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1786:
1783:
1778:
1774:
1768:
1764:
1727:
1726:
1714:
1708:
1704:
1701:
1694:
1691:
1685:
1679:
1675:
1672:
1665:
1662:
1656:
1650:
1646:
1643:
1636:
1633:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1524:
1521:
1518:
1513:
1509:
1505:
1500:
1496:
1492:
1487:
1483:
1444:
1441:
1438:
1435:
1432:
1429:
1424:
1420:
1414:
1410:
1406:
1401:
1397:
1391:
1387:
1383:
1378:
1374:
1368:
1364:
1321:
1318:
1315:
1312:
1309:
1306:
1301:
1297:
1291:
1287:
1283:
1278:
1274:
1268:
1264:
1260:
1255:
1251:
1245:
1241:
1229:
1228:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1154:
1150:
1144:
1140:
1134:
1130:
1126:
1123:
1120:
1117:
1114:
1111:
1109:
1107:
1104:
1099:
1095:
1089:
1085:
1079:
1075:
1071:
1068:
1063:
1059:
1055:
1050:
1046:
1042:
1037:
1033:
1029:
1026:
1021:
1017:
1011:
1007:
1001:
997:
993:
988:
984:
978:
974:
968:
964:
960:
955:
951:
945:
941:
935:
931:
927:
924:
922:
918:
914:
908:
904:
900:
895:
891:
885:
881:
877:
872:
868:
862:
858:
854:
853:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
730:
729:
717:
714:
711:
706:
702:
698:
693:
689:
685:
680:
676:
644:
641:
610:
609:
598:
595:
590:
586:
582:
579:
576:
573:
570:
567:
562:
558:
554:
551:
541:
530:
527:
524:
521:
518:
515:
512:
509:
506:
501:
497:
493:
490:
480:
469:
466:
463:
460:
457:
454:
451:
448:
445:
440:
436:
432:
429:
399:
398:
386:
383:
380:
377:
374:
369:
365:
361:
356:
352:
346:
342:
338:
333:
329:
323:
319:
315:
310:
306:
300:
296:
268:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
139:
138:
93:
91:
84:
77:
76:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
4691:
4680:
4677:
4676:
4674:
4655:
4652:
4650:
4646:
4644:
4640:
4638:Making a hole
4637:
4635:
4634:Connected sum
4632:
4631:
4629:
4625:
4619:
4616:
4614:
4611:
4608:
4604:
4603:
4601:
4597:
4591:
4590:Orientability
4588:
4586:
4582:
4579:
4577:
4574:
4572:
4571:Connectedness
4569:
4568:
4566:
4562:
4559:
4553:
4546:
4543:
4539:
4536:
4535:
4534:
4531:
4527:
4524:
4522:
4519:
4518:
4516:
4511:
4510:
4509:
4506:
4505:
4503:
4501:With boundary
4499:
4489:(genus 3) ...
4488:
4485:
4482:
4479:
4475:
4474:Roman surface
4472:
4470:
4469:Boy's surface
4466:
4465:
4464:
4461:
4460:
4458:
4454:
4447:
4444:
4441:
4438:
4435:
4432:
4431:
4429:
4425:
4422:
4418:
4409:
4404:
4402:
4397:
4395:
4390:
4389:
4386:
4380:
4377:
4374:
4370:
4369:
4365:
4360:
4359:
4354:
4351:
4346:
4344:
4342:
4338:A. Coffman, "
4337:
4336:
4332:
4327:
4323:
4322:
4317:
4313:
4309:
4305:
4302:
4298:
4294:
4293:
4289:
4280:
4276:
4269:
4266:
4259:
4254:
4250:
4247:
4244:
4240:
4239:Boy's surface
4237:
4236:
4232:
4230:
4228:
4224:
4223:
4215:
4208:
4206:
4204:
4199:
4194:
4192:
4188:
4184:
4178:
4176:
4172:
4168:
4160:
4156:
4154:
4147:One-sidedness
4146:
4144:
4138:
4134:
4130:
4127:
4125:
4120:
4118:
4112:
4107:
4103:
4099:
4096:
4094:
4090:
4083:
4079:
4075:
4069:
4065:
4061:
4059:
4055:
4050:
4048:
4040:
4036:
4032:
4029:
4026:
4022:
4018:
4015:
4014:
4013:
4011:
4007:
4003:
3996:
3992:
3988:
3986:
3982:
3978:
3974:
3966:
3962:
3958:
3955:
3952:
3948:
3944:
3941:
3938:
3934:
3930:
3927:
3924:
3920:
3916:
3913:
3912:
3911:
3905:
3901:
3897:
3894:
3891:
3887:
3884:
3883:
3882:
3880:
3876:
3872:
3851:
3848:
3843:
3840:
3837:
3830:
3829:
3828:
3826:
3822:
3818:
3814:
3810:
3806:
3802:
3797:
3795:
3791:
3787:
3783:
3774:
3770:
3767:
3764:
3760:
3757:
3754:
3750:
3747:
3746:
3745:
3742:
3740:
3735:
3729:
3727:
3725:
3721:
3701:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3668:
3659:
3656:
3647:
3644:
3638:
3632:
3629:
3620:
3617:
3611:
3605:
3602:
3593:
3590:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3548:
3545:
3538:
3524:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3485:
3482:
3479:
3476:
3473:
3467:
3464:
3457:
3456:
3455:
3453:
3448:
3446:
3442:
3438:
3434:
3426:
3424:
3422:
3418:
3414:
3395:
3392:
3389:
3386:
3382:
3379:
3376:
3372:
3369:
3364:
3360:
3353:
3349:
3345:
3342:
3339:
3336:
3332:
3329:
3321:
3307:
3304:
3301:
3298:
3294:
3291:
3288:
3284:
3281:
3278:
3272:
3268:
3264:
3261:
3258:
3255:
3251:
3248:
3240:
3226:
3223:
3220:
3217:
3213:
3210:
3207:
3203:
3200:
3197:
3191:
3187:
3183:
3180:
3177:
3174:
3170:
3167:
3159:
3158:
3157:
3155:
3136:
3133:
3130:
3127:
3123:
3120:
3117:
3110:
3096:
3093:
3090:
3087:
3083:
3080:
3077:
3073:
3070:
3067:
3060:
3046:
3043:
3040:
3037:
3033:
3030:
3027:
3023:
3020:
3017:
3010:
3009:
3008:
3006:
3002:
2998:
2990:
2988:
2985:
2967:
2964:
2961:
2958:
2955:
2949:
2943:
2940:
2937:
2934:
2931:
2925:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2887:
2883:
2879:
2875:
2871:
2867:
2863:
2859:
2857:
2852:
2837:
2834:
2829:
2821:
2808:
2803:
2788:
2785:
2780:
2772:
2759:
2754:
2739:
2736:
2731:
2723:
2710:
2705:
2692:
2689:
2686:
2683:
2680:
2674:
2671:
2668:
2665:
2662:
2656:
2653:
2650:
2647:
2644:
2637:) satisfying
2636:
2632:
2628:
2624:
2619:
2606:
2603:
2600:
2597:
2594:
2591:
2585:
2582:
2578:
2574:
2571:
2563:
2549:
2536:
2531:
2528:
2523:
2520:
2517:
2508:
2494:
2491:
2488:
2483:
2479:
2475:
2470:
2466:
2456:
2443:
2438:
2435:
2430:
2422:
2409:
2405:
2401:
2399:
2394:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2354:
2351:
2348:
2345:
2342:
2339:
2330:
2316:
2313:
2310:
2307:
2304:
2296:
2292:
2287:
2273:
2268:
2264:
2260:
2255:
2251:
2245:
2241:
2231:
2218:
2213:
2205:
2201:
2197:
2194:
2191:
2186:
2183:
2177:
2172:
2168:
2142:
2134:
2130:
2126:
2123:
2120:
2115:
2112:
2106:
2101:
2097:
2089:
2088:
2087:
2085:
2081:
2062:
2057:
2054:
2049:
2041:
2029:
2028:
2027:
2025:
2021:
2006:
2003:
2000:
1997:
1976:
1973:
1970:
1967:
1964:
1961:
1954:implies that
1940:
1937:
1934:
1931:
1922:
1908:
1905:
1902:
1899:
1890:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1843:
1841:
1837:
1833:
1829:
1825:
1821:
1819:
1815:
1811:
1807:
1803:
1798:
1784:
1781:
1776:
1772:
1766:
1762:
1753:
1750:Suppose that
1749:
1745:
1743:
1739:
1735:
1730:
1712:
1706:
1702:
1699:
1692:
1689:
1683:
1677:
1673:
1670:
1663:
1660:
1654:
1648:
1644:
1641:
1634:
1631:
1624:
1623:
1622:
1620:
1616:
1612:
1608:
1604:
1601:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1536:
1522:
1519:
1516:
1511:
1507:
1503:
1498:
1494:
1490:
1485:
1481:
1471:
1469:
1465:
1461:
1456:
1442:
1439:
1436:
1433:
1430:
1427:
1422:
1418:
1412:
1408:
1404:
1399:
1395:
1389:
1385:
1381:
1376:
1372:
1366:
1362:
1352:
1351:) satisfying
1350:
1346:
1342:
1338:
1334:
1319:
1316:
1313:
1310:
1307:
1304:
1299:
1295:
1289:
1285:
1281:
1276:
1272:
1266:
1262:
1258:
1253:
1249:
1243:
1239:
1211:
1208:
1205:
1202:
1199:
1193:
1190:
1181:
1178:
1169:
1166:
1160:
1152:
1148:
1142:
1138:
1132:
1128:
1118:
1112:
1110:
1097:
1093:
1087:
1083:
1077:
1073:
1061:
1057:
1053:
1048:
1044:
1040:
1035:
1031:
1024:
1019:
1015:
1009:
1005:
999:
995:
991:
986:
982:
976:
972:
966:
962:
958:
953:
949:
943:
939:
933:
929:
925:
923:
916:
912:
906:
902:
898:
893:
889:
883:
879:
875:
870:
866:
860:
856:
844:
843:
842:
839:
824:
818:
815:
812:
809:
806:
800:
794:
791:
788:
785:
782:
779:
776:
773:
767:
761:
758:
755:
752:
749:
743:
735:
715:
712:
709:
704:
700:
696:
691:
687:
683:
678:
674:
666:
665:
664:
662:
658:
654:
650:
642:
640:
638:
634:
630:
627:
596:
593:
588:
584:
580:
577:
574:
571:
568:
565:
560:
556:
552:
549:
542:
528:
525:
522:
519:
516:
513:
510:
507:
504:
499:
495:
491:
488:
481:
467:
464:
461:
458:
455:
452:
449:
446:
443:
438:
434:
430:
427:
420:
419:
418:
412:
404:
384:
381:
378:
375:
372:
367:
363:
359:
354:
350:
344:
340:
336:
331:
327:
321:
317:
313:
308:
304:
298:
294:
286:
285:
284:
282:
266:
260:
257:
254:
251:
248:
245:
242:
239:
233:
227:
224:
221:
218:
215:
209:
201:
196:
194:
190:
189:Jakob Steiner
186:
182:
178:
174:
170:
166:
162:
158:
157:Roman surface
154:
145:
135:
132:
124:
114:
110:
104:
103:
97:
92:
83:
82:
73:
70:
62:
52:
49:and simplify
48:
44:
38:
37:
32:This article
30:
21:
20:
4533:Möbius strip
4481:Klein bottle
4473:
4372:
4367:
4356:
4339:
4326:Google Books
4324:, p. 30, at
4319:
4307:
4300:
4296:
4278:
4268:
4219:
4216:
4212:
4203:Klein bottle
4197:
4195:
4190:
4186:
4182:
4179:
4174:
4164:
4150:
4141:
4136:
4128:
4121:
4113:
4110:
4105:
4097:
4092:
4088:
4086:
4081:
4072:
4067:
4057:
4053:
4051:
4044:
4038:
4034:
4030:
4024:
4020:
4016:
4009:
4005:
4001:
3999:
3994:
3984:
3980:
3976:
3972:
3970:
3964:
3960:
3956:
3950:
3946:
3942:
3936:
3932:
3928:
3922:
3918:
3914:
3909:
3903:
3899:
3895:
3889:
3885:
3878:
3877:= 1 so that
3874:
3870:
3868:
3824:
3820:
3816:
3812:
3808:
3804:
3800:
3798:
3789:
3785:
3781:
3778:
3772:
3768:
3762:
3758:
3752:
3748:
3743:
3736:
3733:
3723:
3719:
3716:
3451:
3449:
3444:
3440:
3436:
3433:homeomorphic
3430:
3420:
3416:
3412:
3410:
3153:
3151:
3004:
3000:
2999:, longitude
2996:
2994:
2986:
2984:as desired.
2885:
2881:
2877:
2873:
2869:
2865:
2861:
2860:
2855:
2853:
2806:
2804:
2757:
2755:
2708:
2706:
2634:
2630:
2626:
2622:
2620:
2550:
2509:
2457:
2407:
2403:
2402:
2398:this subcase
2397:
2395:
2331:
2294:
2290:
2288:
2232:
2159:
2083:
2079:
2077:
2023:
2022:
1923:
1891:
1844:
1835:
1831:
1827:
1823:
1822:
1817:
1813:
1809:
1805:
1801:
1799:
1751:
1747:
1746:
1741:
1737:
1733:
1731:
1728:
1618:
1614:
1610:
1606:
1605:
1602:
1537:
1472:
1470:) such that
1467:
1463:
1459:
1457:
1353:
1348:
1344:
1340:
1336:
1335:
1333:as desired.
1230:
840:
733:
731:
663:) such that
660:
656:
652:
648:
646:
611:
400:
197:
160:
156:
150:
127:
118:
99:
65:
56:
33:
4576:Compactness
4222:singularity
4165:Imagine an
3869:and either
2805:and (0, 0,
2760:, 0) with
2332:Since also
736:defined by
626:tetrahedral
153:mathematics
113:introducing
4627:Operations
4609:components
4605:Number of
4585:smoothness
4564:Properties
4512:Semisphere
4427:Orientable
4260:References
4253:polyhedron
4153:orientable
3445:(-x,-y,-z)
2410:, we have
1990:and hence
1842:we assume
1538:for which
1337:Conversely
121:March 2018
96:references
59:March 2018
4654:Immersion
4649:cross-cap
4647:Gluing a
4641:Gluing a
4538:Cross-cap
4483:(genus 2)
4467:genus 1;
4442:(genus 1)
4436:(genus 0)
4358:MathWorld
4243:immersion
4106:Figure 4.
4068:Figure 2.
3995:Figure 1.
3788:= 0, and
3657:−
3645:−
3630:−
3618:−
3603:−
3591:−
3582:→
3573:−
3564:−
3555:−
3492:→
3393:ϕ
3390:
3383:ϕ
3380:
3373:θ
3370:
3305:ϕ
3302:
3295:θ
3292:
3285:θ
3282:
3224:ϕ
3221:
3214:θ
3211:
3204:θ
3201:
3134:θ
3131:
3094:ϕ
3091:
3084:θ
3081:
3044:ϕ
3041:
3034:θ
3031:
2621:there is
2524:≤
2195:−
2187:−
2124:−
2050:≤
1935:≠
1856:≠
1820:to be 0.
1428:−
1305:−
597:φ
594:
581:θ
578:
572:θ
569:
529:φ
526:
520:φ
517:
511:θ
508:
468:φ
465:
459:φ
456:
450:θ
447:
403:longitude
360:−
195:in 1844.
181:immersion
4679:Surfaces
4673:Category
4607:boundary
4526:Cylinder
4233:See also
4220:Whitney
3421:π/2
3333:′
3252:′
3171:′
629:symmetry
411:latitude
177:symmetry
4557:notions
4555:Related
4521:Annulus
4517:Ribbon
4371:at the
4227:tangent
4175:z = x y
4124:bulbous
4047:orchids
4006:x = y z
4002:y = x z
3873:= 0 or
3827:. Then
3441:(x,y,z)
2809:) with
1924:(since
1838:are 0.
1231:and so
635:of the
620:-, and
281:formula
167:of the
165:mapping
109:improve
41:Please
4643:handle
4434:Sphere
4314:
4033:= −1,
3794:origin
3417:θ
3413:φ
3005:θ
3001:φ
2678:
2660:
2589:
2458:Since
1687:
1658:
409:) and
200:sphere
155:, the
98:, but
4613:Genus
4440:Torus
4299:. In
4241:– an
4198:x = 0
4191:z = 0
4183:z = 0
4019:= 1,
4010:z = 0
3967:= −1.
3939:= −1;
3906:= −1.
3886:x = y
3784:= 0,
1845:(***)
1473:(**)
838:say.
171:into
4508:Disk
4312:ISBN
4251:– a
3979:and
3953:= 1;
3925:= 1;
3892:= 1;
3807:and
2864:If (
2830:>
2781:>
2732:>
2572:>
2431:>
2293:and
2160:and
2082:and
1354:(*)
193:Rome
4583:or
4547:...
4205:.)
4187:x-y
4167:ant
4037:= −
3959:= −
3931:= −
3898:= −
3387:sin
3377:cos
3361:cos
3299:cos
3289:sin
3279:cos
3218:sin
3208:sin
3198:cos
3128:sin
3088:sin
3078:cos
3038:cos
3028:cos
2086:by
616:-,
585:cos
575:sin
566:cos
523:sin
514:cos
505:sin
462:sin
453:cos
444:cos
283:of
159:or
151:In
4675::
4355:.
4277:.
4173:,
4093:xy
4091:=
4058:xy
4056:=
4023:=
3985:yz
3983:=
3977:xz
3975:=
3963:,
3949:,
3945:=
3935:,
3921:,
3917:=
3902:,
3888:,
3819:=
3813:zx
3811:=
3805:yz
3803:=
3796:.
3773:xy
3771:=
3763:zx
3761:=
3753:yz
3751:=
3720:RP
3437:RP
3423:.
2884:,
2880:,
2872:,
2868:,
2862:4.
2858:.
2633:,
2629:,
2623:no
2408:3.
2404:b.
2024:a.
1877:0.
1834:,
1830:,
1824:3.
1816:,
1812:,
1804:,
1748:2.
1740:,
1736:,
1617:,
1613:,
1607:1.
1443:0.
1347:,
1343:,
659:,
655:,
639:.
622:xz
618:yz
614:xy
385:0.
4407:e
4400:t
4393:v
4361:.
4343:"
4328:)
4089:z
4054:z
4041:.
4039:x
4035:y
4031:z
4027:;
4025:x
4021:y
4017:z
3981:x
3973:y
3965:x
3961:z
3957:y
3951:x
3947:z
3943:y
3937:y
3933:z
3929:x
3923:y
3919:z
3915:x
3904:z
3900:y
3896:x
3890:z
3879:z
3875:z
3871:y
3852:z
3849:y
3844:=
3841:z
3838:y
3825:z
3823:/
3821:y
3817:x
3809:y
3801:x
3790:z
3786:y
3782:x
3775:.
3769:z
3765:,
3759:y
3755:,
3749:x
3702:.
3699:)
3696:y
3693:x
3690:,
3687:x
3684:z
3681:,
3678:z
3675:y
3672:(
3669:=
3666:)
3663:)
3660:y
3654:(
3651:)
3648:x
3642:(
3639:,
3636:)
3633:x
3627:(
3624:)
3621:z
3615:(
3612:,
3609:)
3606:z
3600:(
3597:)
3594:y
3588:(
3585:(
3579:)
3576:z
3570:,
3567:y
3561:,
3558:x
3552:(
3549::
3546:T
3525:,
3522:)
3519:y
3516:x
3513:,
3510:x
3507:z
3504:,
3501:z
3498:y
3495:(
3489:)
3486:z
3483:,
3480:y
3477:,
3474:x
3471:(
3468::
3465:T
3452:T
3396:,
3365:2
3354:2
3350:r
3346:=
3343:y
3340:x
3337:=
3330:z
3308:,
3273:2
3269:r
3265:=
3262:x
3259:z
3256:=
3249:y
3227:,
3192:2
3188:r
3184:=
3181:z
3178:y
3175:=
3168:x
3154:T
3137:.
3124:r
3121:=
3118:z
3097:,
3074:r
3071:=
3068:y
3047:,
3024:r
3021:=
3018:x
2997:r
2971:)
2968:W
2965:,
2962:V
2959:,
2956:U
2953:(
2950:=
2947:)
2944:0
2941:,
2938:0
2935:,
2932:0
2929:(
2926:=
2923:)
2920:x
2917:z
2914:,
2911:z
2908:y
2905:,
2902:y
2899:x
2896:(
2886:z
2882:y
2878:x
2874:W
2870:V
2866:U
2838:2
2835:1
2826:|
2822:W
2818:|
2807:W
2789:2
2786:1
2777:|
2773:V
2769:|
2758:V
2740:2
2737:1
2728:|
2724:U
2720:|
2709:U
2693:.
2690:x
2687:z
2684:=
2681:W
2675:,
2672:z
2669:y
2666:=
2663:V
2657:,
2654:y
2651:x
2648:=
2645:U
2635:z
2631:y
2627:x
2625:(
2607:,
2604:0
2601:=
2598:W
2595:=
2592:V
2586:,
2583:2
2579:/
2575:1
2568:|
2564:U
2560:|
2537:,
2532:2
2529:1
2521:y
2518:x
2495:,
2492:1
2489:=
2484:2
2480:y
2476:+
2471:2
2467:x
2444:.
2439:2
2436:1
2427:|
2423:U
2419:|
2381:,
2378:W
2375:=
2372:0
2369:=
2366:x
2363:z
2355:V
2352:=
2349:0
2346:=
2343:z
2340:y
2317:.
2314:U
2311:=
2308:y
2305:x
2295:y
2291:x
2274:,
2269:2
2265:U
2261:=
2256:2
2252:y
2246:2
2242:x
2219:,
2214:2
2206:2
2202:U
2198:4
2192:1
2184:1
2178:=
2173:2
2169:y
2143:2
2135:2
2131:U
2127:4
2121:1
2116:+
2113:1
2107:=
2102:2
2098:x
2084:y
2080:x
2063:,
2058:2
2055:1
2046:|
2042:U
2038:|
2007:,
2004:0
2001:=
1998:U
1977:,
1974:0
1971:=
1968:y
1965:=
1962:x
1941:,
1938:0
1932:z
1909:,
1906:0
1903:=
1900:z
1874:=
1871:W
1868:=
1865:V
1862:,
1859:0
1853:U
1836:W
1832:V
1828:U
1818:W
1814:V
1810:U
1806:V
1802:U
1785:0
1782:=
1777:2
1773:V
1767:2
1763:U
1752:W
1742:z
1738:y
1734:x
1713:.
1707:U
1703:W
1700:V
1693:=
1690:z
1684:,
1678:W
1674:V
1671:U
1664:=
1661:y
1655:,
1649:V
1645:U
1642:W
1635:=
1632:x
1619:W
1615:V
1611:U
1588:,
1585:x
1582:z
1579:=
1576:W
1573:,
1570:z
1567:y
1564:=
1561:V
1558:,
1555:y
1552:x
1549:=
1546:U
1523:,
1520:1
1517:=
1512:2
1508:z
1504:+
1499:2
1495:y
1491:+
1486:2
1482:x
1468:z
1466:,
1464:y
1462:,
1460:x
1440:=
1437:W
1434:V
1431:U
1423:2
1419:U
1413:2
1409:W
1405:+
1400:2
1396:W
1390:2
1386:V
1382:+
1377:2
1373:V
1367:2
1363:U
1349:W
1345:V
1341:U
1320:0
1317:=
1314:W
1311:V
1308:U
1300:2
1296:U
1290:2
1286:W
1282:+
1277:2
1273:W
1267:2
1263:V
1259:+
1254:2
1250:V
1244:2
1240:U
1212:,
1209:W
1206:V
1203:U
1200:=
1197:)
1194:x
1191:z
1188:(
1185:)
1182:z
1179:y
1176:(
1173:)
1170:y
1167:x
1164:(
1161:=
1158:)
1153:2
1149:z
1143:2
1139:y
1133:2
1129:x
1125:(
1122:)
1119:1
1116:(
1113:=
1103:)
1098:2
1094:z
1088:2
1084:y
1078:2
1074:x
1070:(
1067:)
1062:2
1058:z
1054:+
1049:2
1045:y
1041:+
1036:2
1032:x
1028:(
1025:=
1020:4
1016:x
1010:2
1006:z
1000:2
996:y
992:+
987:4
983:z
977:2
973:y
967:2
963:x
959:+
954:4
950:y
944:2
940:x
934:2
930:z
926:=
917:2
913:U
907:2
903:W
899:+
894:2
890:W
884:2
880:V
876:+
871:2
867:V
861:2
857:U
825:,
822:)
819:W
816:,
813:V
810:,
807:U
804:(
801:=
798:)
795:y
792:x
789:,
786:x
783:z
780:,
777:z
774:y
771:(
768:=
765:)
762:z
759:,
756:y
753:,
750:x
747:(
744:T
734:T
716:,
713:1
710:=
705:2
701:z
697:+
692:2
688:y
684:+
679:2
675:x
661:z
657:y
653:x
649:r
589:2
561:2
557:r
553:=
550:z
500:2
496:r
492:=
489:y
439:2
435:r
431:=
428:x
415:φ
413:(
407:θ
405:(
382:=
379:z
376:y
373:x
368:2
364:r
355:2
351:x
345:2
341:z
337:+
332:2
328:z
322:2
318:y
314:+
309:2
305:y
299:2
295:x
267:.
264:)
261:y
258:x
255:,
252:z
249:x
246:,
243:z
240:y
237:(
234:=
231:)
228:z
225:,
222:y
219:,
216:x
213:(
210:f
134:)
128:(
123:)
119:(
105:.
72:)
66:(
61:)
57:(
53:.
39:.
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