5785:
5697:
5601:
5661:
5677:
5769:
5621:
5729:
5641:
5717:
5753:
3336:
2199:
2729:
20:
3990:
168:
913:
5433:
5535:
is a ruled surface (or a projective plane, if one uses the restrictive definition of ruled surface). Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2-dimensional vector bundle over some curve. The ruled surfaces with base curve of genus 0 are the
5493:
containing a straight line through any given point. This immediately implies that there is a projective line on the surface through any given point, and this condition is now often used as the definition of a ruled surface: ruled surfaces are defined to be abstract projective surfaces satisfying this
5440:
The determinant condition for developable surfaces is used to determine numerically developable connections between space curves (directrices). The diagram shows a developable connection between two ellipses contained in different planes (one horizontal, the other vertical) and its development.
3678:
4787:
3493:
3941:
2947:
5250:
4269:
4407:
892:
The geometric shape of the directrices and generators are of course essential to the shape of the ruled surface they produce. However, the specific parametric representations of them also influence the shape of the ruled surface.
5784:
3756:
4622:
5133:
3504:
4682:
704:
5415:
304:
5600:
1687:
3265:
5034:
5346:
4539:
4072:
5768:
888:
1274:
772:
1806:
3353:
4982:
4945:
2289:
1062:
2034:
1496:
1360:
4151:
2480:
5696:
2698:
1957:
1886:
3770:
2621:
2172:
4826:
4670:
3315:
820:
108:
A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a
499:
1151:
2375:
5676:
2774:
402:
5716:
2548:
1419:
604:
3176:
2099:
4908:
5620:
2763:
5502:
over a curve with fibers that are projective lines. This excludes the projective plane, which has a projective line though every point but cannot be written as such a fibration.
4864:
4310:
1569:
977:
3134:
222:
5728:
3081:
567:
3045:
446:
424:
5660:
2999:
5533:
5140:
2970:
532:
3973:
5640:
6057:
4158:
347:
327:
4318:
6251:
3687:
3673:{\displaystyle \mathbf {x} (u,v)=(1-v){\big (}(1-u)\mathbf {a} _{1}+u\mathbf {a} _{2}{\big )}+v{\big (}(1-u)\mathbf {b} _{1}+u\mathbf {b} _{2}{\big )}}
4782:{\displaystyle \mathbf {n} =\mathbf {x} _{u}\times \mathbf {x} _{v}=\mathbf {\dot {c}} \times \mathbf {r} +v(\mathbf {\dot {r}} \times \mathbf {r} ).}
4547:
5043:
5036:
lie in a plane, i.e. if they are linearly dependent. The linear dependency of three vectors can be checked using the determinant of these vectors:
619:
5353:
5498:
to the product of a curve and a projective line. Sometimes a ruled surface is defined to be one satisfying the stronger condition that it has a
5425:
The generators of any ruled surface coalesce with one family of its asymptotic lines. For developable surfaces they also form one family of its
234:
6175:
6138:
6024:
5986:
1580:
3181:
4987:
6104:
5922:
5277:
4470:
4003:
5960:
825:
86:
3488:{\displaystyle \mathbf {c} (u)=(1-u)\mathbf {a} _{1}+u\mathbf {a} _{2},\quad \mathbf {d} (u)=(1-u)\mathbf {b} _{1}+u\mathbf {b} _{2}}
1157:
712:
6112:
6087:
5841:
180:
32:
1695:
124:
are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points (
5739:
6246:
5429:. It can be shown that any developable surface is a cone, a cylinder, or a surface formed by all tangents of a space curve.
4950:
4913:
2212:
988:
6159:
3936:{\displaystyle \mathbf {a} _{1}=(0,0,0),\;\mathbf {a} _{2}=(1,0,0),\;\mathbf {b} _{1}=(0,1,0),\;\mathbf {b} _{2}=(1,1,1).}
1963:
1425:
1289:
184:
36:
4083:
2381:
6041:
2633:
1892:
1821:
2559:
2110:
6261:
6154:
4795:
4629:
3270:
777:
2942:{\displaystyle \mathbf {x} (u,v)=(1-v)(\cos(u-\varphi ),\sin(u-\varphi ),-1)+v(\cos(u+\varphi ),\sin(u+\varphi ),1)}
454:
6271:
6008:
1070:
159:
into affine or projective space, in which case "straight line" is understood to mean an affine or projective line.
2297:
6266:
6256:
352:
225:
121:
2495:
1366:
572:
3141:
2049:
5752:
5506:
907:
4869:
2735:
4831:
5722:
Village church in Selo, Slovenia: both the roof (conical) and the wall (cylindrical) are ruled surfaces.
5607:
5549:
5494:
condition that there is a projective line through any point. This is equivalent to saying that they are
4280:
3330:
1518:
926:
117:
3086:
198:
5973:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin,
5611:
5587:
3052:
2711:
610:
of the representation. The directrix may collapse to a point (in case of a cone, see example below).
541:
6149:
6126:
5888:
5707:
4458:
4446:
3004:
1507:
152:
136:
98:
71:
67:
429:
407:
6073:
5537:
5486:
5482:
5259:
5245:{\displaystyle \det(\mathbf {\dot {c}} (u_{0}),\mathbf {\dot {r}} (u_{0}),\mathbf {r} (u_{0}))=0}
4464:
4432:
192:
140:
2978:
6241:
6204:
6171:
6134:
6108:
6098:
6083:
6077:
6020:
5982:
5956:
5948:
5837:
5627:
5515:
5510:
5426:
2955:
504:
195:
of a differentiable one-parameter family of lines. Formally, a ruled surface is a surface in
116:
if through every one of its points there are two distinct lines that lie on the surface. The
6186:
6062:
6051:
6012:
5974:
5928:
5918:
5490:
148:
51:
6034:
5996:
6030:
5992:
5667:
5647:
5572:
4427:
4264:{\displaystyle \mathbf {r} (u)=(\cos u\cos 2u,\cos u\sin 2u,\sin u)\quad 0\leq u<\pi ,}
3949:
2040:
79:
75:
4402:{\displaystyle \det(\mathbf {\dot {c}} (0),\mathbf {\dot {r}} (0),\mathbf {r} (0))\neq 0}
3984:
172:
6045:
332:
312:
132:
6235:
6122:
5687:
5583:
5568:
5476:
59:
6226:
6066:
3751:{\displaystyle \mathbf {a} _{1},\mathbf {a} _{2},\mathbf {b} _{1},\mathbf {b} _{2}}
155:, but they are also sometimes considered as abstract algebraic surfaces without an
144:
90:
6207:
5631:
6221:
5903:
3761:
The surface is doubly ruled, because any point lies on two lines of the surface.
5759:
5580:
5561:
5553:
3335:
2723:
774:. To go back to the first description starting with two non intersecting curves
5969:
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004),
4617:{\displaystyle \mathbf {x} _{u}=\mathbf {\dot {c}} (u)+v\mathbf {\dot {r}} (u)}
2198:
6190:
5978:
5932:
5775:
5735:
5495:
5128:{\displaystyle \mathbf {x} (u_{0},v)=\mathbf {c} (u_{0})+v\mathbf {r} (u_{0})}
6016:
699:{\displaystyle \quad \mathbf {x} (u,v)=(1-v)\mathbf {c} (u)+v\mathbf {d} (u)}
6212:
5683:
5499:
5410:{\displaystyle \det(\mathbf {\dot {c}} ,\mathbf {\dot {r}} ,\mathbf {r} )=0}
4440:
156:
2728:
6007:, London Mathematical Society Student Texts, vol. 34 (2nd ed.),
5270:. The determinant condition can be used to prove the following statement:
19:
2193:
94:
28:
3989:
299:{\displaystyle \quad \mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)}
167:
5703:
2180:
For any cone one can choose the apex as the directrix. This shows that
912:
83:
1682:{\displaystyle \mathbf {x} (u,v)=(\cos u,\sin u,1)+v(\cos u,\sin u,1)}
5791:
5743:
5591:
4422:
4409:(see next section). Hence the given realization of a Möbius strip is
3260:{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}-{\frac {z^{2}}{c^{2}}}=1}
109:
5774:
Corrugated roof tiles ruled by parallel lines in one direction, and
5432:
5029:{\displaystyle \mathbf {\dot {c}} ,\mathbf {\dot {r}} ,\mathbf {r} }
4463:
For the determination of the normal vector at a point one needs the
2952:
has two horizontal circles as directrices. The additional parameter
5509:
of projective complex surfaces, because every algebraic surface of
5431:
5341:{\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)}
4534:{\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)}
4436:
4067:{\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)}
3988:
3684:
which is the hyperbolic paraboloid that interpolates the 4 points
3334:
2972:
allows to vary the parametric representations of the circles. For
2727:
2704:
2197:
911:
166:
102:
18:
5651:
5590:
that were laid out in a ruled surface to form the throat of the
883:{\displaystyle \mathbf {r} (u)=\mathbf {d} (u)-\mathbf {c} (u).}
1269:{\displaystyle =(1-v)(a\cos u,a\sin u,0)+v(a\cos u,a\sin u,1).}
767:{\displaystyle \mathbf {d} (u)=\mathbf {c} (u)+\mathbf {r} (u)}
131:
The properties of being ruled or doubly ruled are preserved by
6076:(2007), "16.5 There are no non-planar triply ruled surfaces",
5449:
1801:{\displaystyle =(1-v)(\cos u,\sin u,1)+v(2\cos u,2\sin u,2).}
6079:
Mathematical
Omnibus: Thirty Lectures on Classic Mathematics
143:, ruled surfaces are sometimes considered to be surfaces in
5436:
Developable connection of two ellipses and its development
4910:. The tangent planes along this line are all the same, if
2182:
the directrix of a ruled surface may degenerate to a point
613:
The ruled surface above may alternatively be described by
6168:
D-Forms: surprising new 3-D forms from flat curved shapes
5816:
5814:
23:
Definition of a ruled surface: every point lies on a line
6227:
Examples of developable surfaces on the Rhino3DE website
5834:
Curves and
Surfaces for Computer Aided Geometric Design
5908:, ACM Trans. Graph. (MONTH 2015), DOI: 10.1145/2832906
5444:
An impression of the usage of developable surfaces in
4977:{\displaystyle \mathbf {\dot {c}} \times \mathbf {r} }
4940:{\displaystyle \mathbf {\dot {r}} \times \mathbf {r} }
4828:(A mixed product with two equal vectors is always 0),
3321:
A hyperboloid of one sheet is a doubly ruled surface.
2284:{\displaystyle \mathbf {x} (u,v)=(v\cos u,v\sin u,ku)}
6222:
Ruled surface pictures from the
University of Arizona
5548:
Doubly ruled surfaces are the inspiration for curved
5518:
5356:
5280:
5143:
5046:
4990:
4953:
4916:
4872:
4834:
4798:
4685:
4632:
4550:
4473:
4321:
4283:
4161:
4086:
4006:
3952:
3773:
3690:
3507:
3356:
3273:
3184:
3144:
3089:
3055:
3007:
2981:
2958:
2777:
2738:
2636:
2562:
2498:
2384:
2300:
2215:
2113:
2052:
1966:
1895:
1824:
1698:
1583:
1521:
1428:
1369:
1292:
1160:
1073:
1057:{\displaystyle \mathbf {x} (u,v)=(a\cos u,a\sin u,v)}
991:
929:
828:
780:
715:
622:
575:
569:
describe the directions of the generators. The curve
544:
507:
457:
432:
410:
355:
335:
315:
237:
201:
2029:{\displaystyle \mathbf {d} (u)=(2\cos u,2\sin u,2).}
1491:{\displaystyle \mathbf {d} (u)=(a\cos u,a\sin u,1).}
1355:{\displaystyle \mathbf {c} (u)=(a\cos u,a\sin u,0),}
5924:
Developable
Surfaces: Their History and Application
5465:
Developable
Surfaces: Their History and Application
4146:{\displaystyle \mathbf {c} (u)=(\cos 2u,\sin 2u,0)}
2475:{\displaystyle =(1-v)(0,0,ku)+v(\cos u,\sin u,ku).}
1512:A right circular cylinder is given by the equation
920:A right circular cylinder is given by the equation
5927:, in  Nexus Network Journal 13(3) · October 2011,
5857:, Monatshefte fĂĽr Mathematik 66, 1962, S. 276-289.
5527:
5409:
5340:
5244:
5127:
5028:
4976:
4939:
4902:
4858:
4820:
4781:
4664:
4616:
4533:
4401:
4304:
4263:
4145:
4066:
3967:
3935:
3750:
3672:
3487:
3309:
3259:
3178:one gets a hyperboloid of one sheet with equation
3170:
3128:
3075:
3039:
2993:
2964:
2941:
2757:
2693:{\displaystyle \mathbf {d} (u)=(\cos u,\sin u,ku)}
2692:
2615:
2542:
2474:
2369:
2283:
2166:
2093:
2028:
1952:{\displaystyle \mathbf {r} (u)=(\cos u,\sin u,1),}
1951:
1881:{\displaystyle \mathbf {c} (u)=(\cos u,\sin u,1),}
1880:
1800:
1681:
1563:
1490:
1413:
1354:
1268:
1145:
1056:
971:
882:
814:
766:
698:
598:
561:
526:
493:
440:
418:
396:
341:
321:
298:
216:
2616:{\displaystyle \mathbf {r} (u)=(\cos u,\sin u,0)}
2167:{\displaystyle \mathbf {r} (u)=(\cos u,\sin u,1)}
5357:
5144:
4821:{\displaystyle \mathbf {n} \cdot \mathbf {r} =0}
4665:{\displaystyle \mathbf {x} _{v}=\mathbf {r} (u)}
4322:
125:
5463:survey on developable surfaces can be found in
3310:{\displaystyle a=\cos \varphi ,c=\cot \varphi }
815:{\displaystyle \mathbf {c} (u),\mathbf {d} (u)}
6082:, American Mathematical Society, p. 228,
494:{\displaystyle v\mapsto \mathbf {x} (u_{0},v)}
6058:Bulletin of the American Mathematical Society
4984:. This is possible only if the three vectors
4413:. But there exist developable Möbius strips.
3665:
3613:
3600:
3548:
1146:{\displaystyle =(a\cos u,a\sin u,0)+v(0,0,1)}
16:Surface containing a line through every point
8:
5953:Differential Geometry of Curves and Surfaces
5790:Construction of a planar surface by ruling (
5690:in Moscow, whose sections are doubly ruled.
5485:, ruled surfaces were originally defined as
2370:{\displaystyle =(0,0,ku)+v(\cos u,\sin u,0)}
349:ranging over the reals. It is required that
6100:Problems and Solutions in Mathematics, 3103
3946:The hyperbolic paraboloid has the equation
397:{\displaystyle \mathbf {r} (u)\neq (0,0,0)}
5906:Interactive design of developable surfaces
5454:Interactive design of developable surfaces
3893:
3853:
3813:
6050:, Cambridge University Press – via
5893:, lecture note, TU Darmstadt, p. 113
5702:A ruled helicoid spiral staircase inside
5517:
5393:
5379:
5378:
5364:
5363:
5355:
5324:
5304:
5281:
5279:
5224:
5212:
5200:
5182:
5181:
5169:
5151:
5150:
5142:
5116:
5104:
5089:
5077:
5059:
5047:
5045:
5021:
5007:
5006:
4992:
4991:
4989:
4969:
4955:
4954:
4952:
4932:
4918:
4917:
4915:
4885:
4873:
4871:
4847:
4835:
4833:
4807:
4799:
4797:
4768:
4754:
4753:
4739:
4725:
4724:
4715:
4710:
4700:
4695:
4686:
4684:
4648:
4639:
4634:
4631:
4594:
4593:
4567:
4566:
4557:
4552:
4549:
4517:
4497:
4474:
4472:
4376:
4353:
4352:
4329:
4328:
4320:
4282:
4162:
4160:
4087:
4085:
4050:
4030:
4007:
4005:
3951:
3900:
3895:
3860:
3855:
3820:
3815:
3780:
3775:
3772:
3742:
3737:
3727:
3722:
3712:
3707:
3697:
3692:
3689:
3664:
3663:
3657:
3652:
3639:
3634:
3612:
3611:
3599:
3598:
3592:
3587:
3574:
3569:
3547:
3546:
3508:
3506:
3479:
3474:
3461:
3456:
3423:
3413:
3408:
3395:
3390:
3357:
3355:
3272:
3243:
3233:
3227:
3216:
3205:
3192:
3185:
3183:
3160:
3143:
3120:
3107:
3094:
3088:
3065:
3054:
3025:
3012:
3006:
2980:
2957:
2778:
2776:
2749:
2737:
2637:
2635:
2563:
2561:
2499:
2497:
2383:
2299:
2216:
2214:
2114:
2112:
2053:
2051:
1967:
1965:
1896:
1894:
1825:
1823:
1697:
1584:
1582:
1552:
1539:
1526:
1520:
1429:
1427:
1370:
1368:
1293:
1291:
1159:
1072:
992:
990:
960:
947:
934:
928:
863:
846:
829:
827:
798:
781:
779:
750:
733:
716:
714:
682:
662:
624:
621:
582:
574:
545:
543:
518:
506:
476:
464:
456:
433:
431:
411:
409:
356:
354:
334:
314:
282:
262:
239:
236:
208:
204:
203:
200:
5820:
2543:{\displaystyle \mathbf {c} (u)=(0,0,ku)}
1414:{\displaystyle \mathbf {r} (u)=(0,0,1),}
599:{\displaystyle u\mapsto \mathbf {c} (u)}
163:Definition and parametric representation
5810:
5596:
4277:The diagram shows the Möbius strip for
3764:For the example shown in the diagram:
3171:{\displaystyle 0<\varphi <\pi /2}
2553:is the z-axis, the line directions are
2094:{\displaystyle \mathbf {c} (u)=(0,0,0)}
5614:, UK; the surface can be doubly ruled.
2710:The helicoid is a special case of the
4903:{\displaystyle \mathbf {x} (u_{0},v)}
2039:In this case one could have used the
7:
5471:Ruled surfaces in algebraic geometry
2758:{\displaystyle \varphi =63^{\circ }}
6183:Journal of Mathematics and the Arts
6181:. Review: SĂ©quin, Carlo H. (2009),
6133:(2nd ed.), New York: Chelsea,
6105:World Scientific Publishing Company
5567:Hyperboloids of one sheet, such as
4859:{\displaystyle \mathbf {r} (u_{0})}
2206:A helicoid can be parameterized as
5522:
5040:The tangent planes along the line
4305:{\displaystyle -0.3\leq v\leq 0.3}
1564:{\displaystyle x^{2}+y^{2}=z^{2}.}
972:{\displaystyle x^{2}+y^{2}=a^{2}.}
14:
6252:Differential geometry of surfaces
5855:Über ein abwickelbares Möbiusband
5666:Hyperboloid water tower, 1896 in
4866:is a tangent vector at any point
3129:{\displaystyle x^{2}+y^{2}=z^{2}}
5783:
5767:
5751:
5727:
5715:
5695:
5675:
5659:
5639:
5619:
5599:
5560:Hyperbolic paraboloids, such as
5394:
5384:
5381:
5369:
5366:
5325:
5305:
5282:
5213:
5187:
5184:
5156:
5153:
5105:
5078:
5048:
5022:
5012:
5009:
4997:
4994:
4970:
4960:
4957:
4933:
4923:
4920:
4874:
4836:
4808:
4800:
4769:
4759:
4756:
4740:
4730:
4727:
4711:
4696:
4687:
4649:
4635:
4599:
4596:
4572:
4569:
4553:
4518:
4498:
4475:
4377:
4358:
4355:
4334:
4331:
4163:
4088:
4051:
4031:
4008:
3896:
3856:
3816:
3776:
3738:
3723:
3708:
3693:
3653:
3635:
3588:
3570:
3509:
3475:
3457:
3424:
3409:
3391:
3358:
2779:
2638:
2564:
2500:
2217:
2115:
2054:
1968:
1897:
1826:
1585:
1430:
1371:
1294:
993:
864:
847:
830:
799:
782:
751:
734:
717:
683:
663:
625:
583:
546:
465:
434:
412:
357:
283:
263:
240:
217:{\displaystyle \mathbb {R} ^{3}}
135:, and therefore are concepts of
6067:10.1090/S0002-9904-1931-05248-4
5955:(1st ed.), Prentice-Hall,
5891:Geometry and Algorithms for CAD
5740:Warszawa Ochota railway station
4242:
3422:
3076:{\displaystyle \varphi =\pi /2}
2768:The parametric representation
2718:Cylinder, cone and hyperboloids
623:
562:{\displaystyle \mathbf {r} (u)}
238:
171:Ruled surface generated by two
5778:in the perpendicular direction
5654:, Japan, with a double ruling.
5626:Doubly ruled water tower with
5556:of straight elements, namely:
5544:Ruled surfaces in architecture
5398:
5360:
5348:is developable if and only if
5335:
5329:
5315:
5309:
5298:
5286:
5233:
5230:
5217:
5206:
5193:
5175:
5162:
5147:
5122:
5109:
5095:
5082:
5071:
5052:
4897:
4878:
4853:
4840:
4773:
4750:
4659:
4653:
4611:
4605:
4584:
4578:
4528:
4522:
4508:
4502:
4491:
4479:
4390:
4387:
4381:
4370:
4364:
4346:
4340:
4325:
4239:
4179:
4173:
4167:
4140:
4104:
4098:
4092:
4061:
4055:
4041:
4035:
4024:
4012:
3927:
3909:
3887:
3869:
3847:
3829:
3807:
3789:
3630:
3618:
3565:
3553:
3543:
3531:
3525:
3513:
3452:
3440:
3434:
3428:
3386:
3374:
3368:
3362:
2936:
2927:
2915:
2903:
2891:
2882:
2873:
2861:
2849:
2837:
2825:
2816:
2813:
2801:
2795:
2783:
2687:
2654:
2648:
2642:
2610:
2580:
2574:
2568:
2537:
2516:
2510:
2504:
2466:
2433:
2424:
2403:
2400:
2388:
2364:
2334:
2325:
2304:
2278:
2239:
2233:
2221:
2161:
2131:
2125:
2119:
2088:
2070:
2064:
2058:
2020:
1984:
1978:
1972:
1943:
1913:
1907:
1901:
1872:
1842:
1836:
1830:
1792:
1756:
1747:
1717:
1714:
1702:
1676:
1646:
1637:
1607:
1601:
1589:
1482:
1446:
1440:
1434:
1405:
1387:
1381:
1375:
1346:
1310:
1304:
1298:
1260:
1224:
1215:
1179:
1176:
1164:
1140:
1122:
1113:
1077:
1051:
1015:
1009:
997:
874:
868:
857:
851:
840:
834:
809:
803:
792:
786:
761:
755:
744:
738:
727:
721:
693:
687:
673:
667:
659:
647:
641:
629:
593:
587:
579:
556:
550:
488:
469:
461:
391:
373:
367:
361:
293:
287:
273:
267:
256:
244:
1:
5904:Tang, Bo, Wallner, Pottmann:
5505:Ruled surfaces appear in the
3040:{\displaystyle x^{2}+y^{2}=1}
2732:hyperboloid of one sheet for
329:varying over an interval and
185:3-dimensional Euclidean space
37:3-dimensional Euclidean space
6131:Geometry and the Imagination
6047:The Theory of Ruled Surfaces
5630:tank, by Jan Bogusławski in
441:{\displaystyle \mathbf {r} }
419:{\displaystyle \mathbf {c} }
126:Fuchs & Tabachnikov 2007
6155:Encyclopedia of Mathematics
5258:A smooth surface with zero
4676:Hence the normal vector is
4315:A simple calculation shows
2712:ruled generalized helicoids
1574:It can be parameterized as
982:It can be parameterized as
175:as directrices (red, green)
70:, the lateral surface of a
6288:
6148:Iskovskikh, V.A. (2001) ,
6097:Li, Ta-tsien, ed. (2011),
6009:Cambridge University Press
6005:Complex algebraic surfaces
6003:Beauville, Arnaud (1996),
5474:
4456:
3982:
3343:If the two directrices in
3328:
2994:{\displaystyle \varphi =0}
2721:
2191:
1505:
905:
709:with the second directrix
448:should be differentiable.
6191:10.1080/17513470903332913
5979:10.1007/978-3-642-57739-0
5933:10.1007/s00004-011-0087-z
5552:that can be built with a
4274:contains a Möbius strip.
2627:and the second directrix
226:parametric representation
6017:10.1017/CBO9780511623936
5971:Compact Complex Surfaces
5836:, Academic Press, 1990,
5528:{\displaystyle -\infty }
5264:developable into a plane
2965:{\displaystyle \varphi }
2177:as the line directions.
122:hyperboloid of one sheet
5507:Enriques classification
2043:as the directrix, i.e.
908:Right circular cylinder
902:Right circular cylinder
527:{\displaystyle u=u_{0}}
66:. Examples include the
5550:hyperboloid structures
5529:
5437:
5411:
5342:
5246:
5129:
5030:
4978:
4941:
4904:
4860:
4822:
4783:
4666:
4618:
4535:
4467:of the representation
4403:
4306:
4265:
4153:(circle as directrix),
4147:
4068:
3994:
3969:
3937:
3752:
3674:
3489:
3340:
3311:
3261:
3172:
3130:
3077:
3041:
3001:one gets the cylinder
2995:
2966:
2943:
2765:
2759:
2694:
2617:
2544:
2476:
2371:
2285:
2203:
2168:
2095:
2030:
1953:
1882:
1802:
1683:
1565:
1492:
1415:
1356:
1270:
1147:
1058:
973:
917:
884:
816:
768:
700:
600:
563:
528:
495:
442:
420:
398:
343:
323:
300:
218:
176:
24:
6247:Differential geometry
5949:do Carmo, Manfredo P.
5868:Differentialgeometrie
5736:hyperbolic paraboloid
5530:
5446:Computer Aided Design
5435:
5412:
5343:
5247:
5130:
5031:
4979:
4942:
4905:
4861:
4823:
4784:
4667:
4619:
4536:
4457:Further information:
4404:
4307:
4266:
4148:
4069:
3992:
3983:Further information:
3970:
3938:
3753:
3675:
3490:
3339:Hyperbolic paraboloid
3338:
3331:Hyperbolic paraboloid
3329:Further information:
3325:Hyperbolic paraboloid
3312:
3262:
3173:
3131:
3078:
3042:
2996:
2967:
2944:
2760:
2731:
2722:Further information:
2695:
2618:
2545:
2477:
2372:
2286:
2201:
2192:Further information:
2169:
2096:
2031:
1954:
1883:
1803:
1684:
1566:
1506:Further information:
1493:
1416:
1357:
1271:
1148:
1059:
974:
915:
906:Further information:
885:
817:
769:
701:
601:
564:
529:
501:with fixed parameter
496:
443:
421:
399:
344:
324:
301:
219:
170:
118:hyperbolic paraboloid
22:
6166:Sharp, John (2008),
6127:Cohn-Vossen, Stephan
6061:37 (1931), 791-793,
5612:Didcot Power Station
5516:
5354:
5278:
5141:
5044:
4988:
4951:
4914:
4870:
4832:
4796:
4683:
4630:
4548:
4471:
4453:Developable surfaces
4319:
4281:
4159:
4084:
4004:
3968:{\displaystyle z=xy}
3950:
3771:
3688:
3505:
3354:
3271:
3182:
3142:
3087:
3053:
3005:
2979:
2956:
2775:
2736:
2634:
2560:
2496:
2382:
2298:
2213:
2111:
2050:
1964:
1893:
1822:
1696:
1581:
1519:
1426:
1367:
1290:
1158:
1071:
989:
927:
826:
822:as directrices, set
778:
713:
620:
573:
542:
505:
455:
430:
408:
353:
333:
313:
235:
199:
5538:Hirzebruch surfaces
5487:projective surfaces
4465:partial derivatives
4459:Developable surface
4447:Tangent developable
4433:Developable rollers
1508:Right circular cone
1502:Right circular cone
137:projective geometry
99:tangent developable
50:) if through every
6262:Algebraic surfaces
6205:Weisstein, Eric W.
6074:Tabachnikov, Serge
5586:employed straight
5525:
5483:algebraic geometry
5438:
5427:lines of curvature
5407:
5338:
5260:Gaussian curvature
5242:
5125:
5026:
4974:
4937:
4900:
4856:
4818:
4779:
4662:
4614:
4531:
4399:
4302:
4261:
4143:
4064:
3997:The ruled surface
3995:
3965:
3933:
3748:
3670:
3485:
3341:
3307:
3267:and the semi axes
3257:
3168:
3126:
3083:one gets the cone
3073:
3037:
2991:
2962:
2939:
2766:
2755:
2690:
2613:
2540:
2472:
2367:
2281:
2204:
2164:
2091:
2026:
1949:
1878:
1798:
1679:
1561:
1488:
1411:
1352:
1266:
1143:
1054:
969:
918:
880:
812:
764:
696:
596:
559:
524:
491:
451:Any straight line
438:
416:
394:
339:
319:
296:
224:is described by a
214:
177:
141:algebraic geometry
25:
6272:Analytic geometry
6177:978-1-899618-87-3
6140:978-0-8284-1087-8
6026:978-0-521-49510-3
5988:978-3-540-00832-3
5608:hyperbolic towers
5511:Kodaira dimension
5387:
5372:
5190:
5159:
5015:
5000:
4963:
4947:is a multiple of
4926:
4762:
4733:
4602:
4575:
4361:
4337:
3249:
3222:
342:{\displaystyle v}
322:{\displaystyle u}
6279:
6267:Geometric shapes
6257:Complex surfaces
6218:
6217:
6180:
6162:
6143:
6117:
6103:(2nd ed.),
6092:
6054:
6052:Internet Archive
6037:
5999:
5965:
5935:
5919:Snezana Lawrence
5916:
5910:
5901:
5895:
5886:
5880:
5879:G. Farin: p. 380
5877:
5871:
5864:
5858:
5851:
5845:
5830:
5824:
5818:
5787:
5771:
5755:
5731:
5719:
5699:
5679:
5663:
5643:
5623:
5603:
5588:cooling channels
5534:
5532:
5531:
5526:
5491:projective space
5416:
5414:
5413:
5408:
5397:
5389:
5388:
5380:
5374:
5373:
5365:
5347:
5345:
5344:
5339:
5328:
5308:
5285:
5274:A ruled surface
5251:
5249:
5248:
5243:
5229:
5228:
5216:
5205:
5204:
5192:
5191:
5183:
5174:
5173:
5161:
5160:
5152:
5134:
5132:
5131:
5126:
5121:
5120:
5108:
5094:
5093:
5081:
5064:
5063:
5051:
5035:
5033:
5032:
5027:
5025:
5017:
5016:
5008:
5002:
5001:
4993:
4983:
4981:
4980:
4975:
4973:
4965:
4964:
4956:
4946:
4944:
4943:
4938:
4936:
4928:
4927:
4919:
4909:
4907:
4906:
4901:
4890:
4889:
4877:
4865:
4863:
4862:
4857:
4852:
4851:
4839:
4827:
4825:
4824:
4819:
4811:
4803:
4788:
4786:
4785:
4780:
4772:
4764:
4763:
4755:
4743:
4735:
4734:
4726:
4720:
4719:
4714:
4705:
4704:
4699:
4690:
4671:
4669:
4668:
4663:
4652:
4644:
4643:
4638:
4623:
4621:
4620:
4615:
4604:
4603:
4595:
4577:
4576:
4568:
4562:
4561:
4556:
4540:
4538:
4537:
4532:
4521:
4501:
4478:
4417:Further examples
4408:
4406:
4405:
4400:
4380:
4363:
4362:
4354:
4339:
4338:
4330:
4311:
4309:
4308:
4303:
4270:
4268:
4267:
4262:
4166:
4152:
4150:
4149:
4144:
4091:
4073:
4071:
4070:
4065:
4054:
4034:
4011:
3974:
3972:
3971:
3966:
3942:
3940:
3939:
3934:
3905:
3904:
3899:
3865:
3864:
3859:
3825:
3824:
3819:
3785:
3784:
3779:
3757:
3755:
3754:
3749:
3747:
3746:
3741:
3732:
3731:
3726:
3717:
3716:
3711:
3702:
3701:
3696:
3679:
3677:
3676:
3671:
3669:
3668:
3662:
3661:
3656:
3644:
3643:
3638:
3617:
3616:
3604:
3603:
3597:
3596:
3591:
3579:
3578:
3573:
3552:
3551:
3512:
3494:
3492:
3491:
3486:
3484:
3483:
3478:
3466:
3465:
3460:
3427:
3418:
3417:
3412:
3400:
3399:
3394:
3361:
3316:
3314:
3313:
3308:
3266:
3264:
3263:
3258:
3250:
3248:
3247:
3238:
3237:
3228:
3223:
3221:
3220:
3211:
3210:
3209:
3197:
3196:
3186:
3177:
3175:
3174:
3169:
3164:
3135:
3133:
3132:
3127:
3125:
3124:
3112:
3111:
3099:
3098:
3082:
3080:
3079:
3074:
3069:
3046:
3044:
3043:
3038:
3030:
3029:
3017:
3016:
3000:
2998:
2997:
2992:
2971:
2969:
2968:
2963:
2948:
2946:
2945:
2940:
2782:
2764:
2762:
2761:
2756:
2754:
2753:
2699:
2697:
2696:
2691:
2641:
2622:
2620:
2619:
2614:
2567:
2549:
2547:
2546:
2541:
2503:
2481:
2479:
2478:
2473:
2376:
2374:
2373:
2368:
2290:
2288:
2287:
2282:
2220:
2173:
2171:
2170:
2165:
2118:
2100:
2098:
2097:
2092:
2057:
2035:
2033:
2032:
2027:
1971:
1958:
1956:
1955:
1950:
1900:
1887:
1885:
1884:
1879:
1829:
1807:
1805:
1804:
1799:
1688:
1686:
1685:
1680:
1588:
1570:
1568:
1567:
1562:
1557:
1556:
1544:
1543:
1531:
1530:
1497:
1495:
1494:
1489:
1433:
1420:
1418:
1417:
1412:
1374:
1361:
1359:
1358:
1353:
1297:
1275:
1273:
1272:
1267:
1152:
1150:
1149:
1144:
1063:
1061:
1060:
1055:
996:
978:
976:
975:
970:
965:
964:
952:
951:
939:
938:
889:
887:
886:
881:
867:
850:
833:
821:
819:
818:
813:
802:
785:
773:
771:
770:
765:
754:
737:
720:
705:
703:
702:
697:
686:
666:
628:
605:
603:
602:
597:
586:
568:
566:
565:
560:
549:
533:
531:
530:
525:
523:
522:
500:
498:
497:
492:
481:
480:
468:
447:
445:
444:
439:
437:
425:
423:
422:
417:
415:
403:
401:
400:
395:
360:
348:
346:
345:
340:
328:
326:
325:
320:
305:
303:
302:
297:
286:
266:
243:
223:
221:
220:
215:
213:
212:
207:
149:projective space
65:
57:
41:
6287:
6286:
6282:
6281:
6280:
6278:
6277:
6276:
6232:
6231:
6208:"Ruled Surface"
6203:
6202:
6199:
6178:
6165:
6150:"Ruled surface"
6147:
6141:
6121:
6115:
6096:
6090:
6071:
6040:
6027:
6002:
5989:
5968:
5963:
5947:
5944:
5939:
5938:
5917:
5913:
5902:
5898:
5887:
5883:
5878:
5874:
5865:
5861:
5853:W. Wunderlich:
5852:
5848:
5831:
5827:
5819:
5812:
5807:
5802:
5795:
5788:
5779:
5772:
5763:
5756:
5747:
5732:
5723:
5720:
5711:
5700:
5691:
5680:
5671:
5668:Nizhny Novgorod
5664:
5655:
5648:Kobe Port Tower
5644:
5635:
5624:
5615:
5604:
5546:
5514:
5513:
5479:
5473:
5421:at every point.
5352:
5351:
5276:
5275:
5220:
5196:
5165:
5139:
5138:
5112:
5085:
5055:
5042:
5041:
4986:
4985:
4949:
4948:
4912:
4911:
4881:
4868:
4867:
4843:
4830:
4829:
4794:
4793:
4709:
4694:
4681:
4680:
4633:
4628:
4627:
4551:
4546:
4545:
4469:
4468:
4461:
4455:
4428:Catalan surface
4419:
4411:not developable
4317:
4316:
4279:
4278:
4157:
4156:
4082:
4081:
4002:
4001:
3987:
3981:
3948:
3947:
3894:
3854:
3814:
3774:
3769:
3768:
3736:
3721:
3706:
3691:
3686:
3685:
3651:
3633:
3586:
3568:
3503:
3502:
3473:
3455:
3407:
3389:
3352:
3351:
3347:are the lines
3333:
3327:
3269:
3268:
3239:
3229:
3212:
3201:
3188:
3187:
3180:
3179:
3140:
3139:
3116:
3103:
3090:
3085:
3084:
3051:
3050:
3021:
3008:
3003:
3002:
2977:
2976:
2954:
2953:
2773:
2772:
2745:
2734:
2733:
2726:
2720:
2632:
2631:
2558:
2557:
2494:
2493:
2380:
2379:
2296:
2295:
2211:
2210:
2196:
2190:
2109:
2108:
2048:
2047:
1962:
1961:
1891:
1890:
1820:
1819:
1694:
1693:
1579:
1578:
1548:
1535:
1522:
1517:
1516:
1510:
1504:
1424:
1423:
1365:
1364:
1288:
1287:
1156:
1155:
1069:
1068:
987:
986:
956:
943:
930:
925:
924:
910:
904:
899:
824:
823:
776:
775:
711:
710:
618:
617:
571:
570:
540:
539:
514:
503:
502:
472:
453:
452:
428:
427:
406:
405:
351:
350:
331:
330:
311:
310:
233:
232:
202:
197:
196:
165:
133:projective maps
112:. A surface is
80:conical surface
63:
55:
46:(also called a
39:
17:
12:
11:
5:
6285:
6283:
6275:
6274:
6269:
6264:
6259:
6254:
6249:
6244:
6234:
6233:
6230:
6229:
6224:
6219:
6198:
6197:External links
6195:
6194:
6193:
6176:
6163:
6145:
6139:
6123:Hilbert, David
6119:
6113:
6094:
6088:
6069:
6038:
6025:
6000:
5987:
5966:
5962:978-0132125895
5961:
5943:
5940:
5937:
5936:
5911:
5896:
5881:
5872:
5859:
5846:
5825:
5823:, p. 188.
5809:
5808:
5806:
5803:
5801:
5798:
5797:
5796:
5789:
5782:
5780:
5773:
5766:
5764:
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5750:
5748:
5733:
5726:
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5721:
5714:
5712:
5701:
5694:
5692:
5681:
5674:
5672:
5665:
5658:
5656:
5646:A hyperboloid
5645:
5638:
5636:
5625:
5618:
5616:
5605:
5598:
5577:
5576:
5569:cooling towers
5565:
5545:
5542:
5524:
5521:
5475:Main article:
5472:
5469:
5452:) is given in
5423:
5422:
5419:
5418:
5417:
5406:
5403:
5400:
5396:
5392:
5386:
5383:
5377:
5371:
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5362:
5359:
5337:
5334:
5331:
5327:
5323:
5320:
5317:
5314:
5311:
5307:
5303:
5300:
5297:
5294:
5291:
5288:
5284:
5256:
5255:
5254:
5253:
5241:
5238:
5235:
5232:
5227:
5223:
5219:
5215:
5211:
5208:
5203:
5199:
5195:
5189:
5186:
5180:
5177:
5172:
5168:
5164:
5158:
5155:
5149:
5146:
5135:are equal, if
5124:
5119:
5115:
5111:
5107:
5103:
5100:
5097:
5092:
5088:
5084:
5080:
5076:
5073:
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5067:
5062:
5058:
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5024:
5020:
5014:
5011:
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4996:
4972:
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4888:
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4876:
4855:
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4771:
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4752:
4749:
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4729:
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4708:
4703:
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2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2502:
2489:The directrix
2487:
2486:
2485:
2484:
2483:
2482:
2471:
2468:
2465:
2462:
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2456:
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2280:
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2250:
2247:
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2226:
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2189:
2186:
2175:
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2163:
2160:
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2145:
2142:
2139:
2136:
2133:
2130:
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2090:
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2078:
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2072:
2069:
2066:
2063:
2060:
2056:
2037:
2036:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1970:
1959:
1948:
1945:
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1939:
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1933:
1930:
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1924:
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1918:
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1909:
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1460:
1457:
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1442:
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1436:
1432:
1421:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1373:
1362:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1296:
1281:
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1278:
1277:
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1262:
1259:
1256:
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1250:
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1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1153:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
995:
980:
979:
968:
963:
959:
955:
950:
946:
942:
937:
933:
916:cylinder, cone
903:
900:
898:
895:
879:
876:
873:
870:
866:
862:
859:
856:
853:
849:
845:
842:
839:
836:
832:
811:
808:
805:
801:
797:
794:
791:
788:
784:
763:
760:
757:
753:
749:
746:
743:
740:
736:
732:
729:
726:
723:
719:
707:
706:
695:
692:
689:
685:
681:
678:
675:
672:
669:
665:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
627:
606:is called the
595:
592:
589:
585:
581:
578:
558:
555:
552:
548:
538:. The vectors
521:
517:
513:
510:
490:
487:
484:
479:
475:
471:
467:
463:
460:
436:
414:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
359:
338:
318:
307:
306:
295:
292:
289:
285:
281:
278:
275:
272:
269:
265:
261:
258:
255:
252:
249:
246:
242:
211:
206:
164:
161:
15:
13:
10:
9:
6:
4:
3:
2:
6284:
6273:
6270:
6268:
6265:
6263:
6260:
6258:
6255:
6253:
6250:
6248:
6245:
6243:
6240:
6239:
6237:
6228:
6225:
6223:
6220:
6215:
6214:
6209:
6206:
6201:
6200:
6196:
6192:
6188:
6184:
6179:
6173:
6169:
6164:
6161:
6157:
6156:
6151:
6146:
6142:
6136:
6132:
6128:
6124:
6120:
6116:
6114:9789810234805
6110:
6106:
6102:
6101:
6095:
6091:
6089:9780821843161
6085:
6081:
6080:
6075:
6070:
6068:
6064:
6060:
6059:
6053:
6049:
6048:
6043:
6039:
6036:
6032:
6028:
6022:
6018:
6014:
6010:
6006:
6001:
5998:
5994:
5990:
5984:
5980:
5976:
5972:
5967:
5964:
5958:
5954:
5950:
5946:
5945:
5941:
5934:
5930:
5926:
5925:
5920:
5915:
5912:
5909:
5907:
5900:
5897:
5894:
5892:
5889:E. Hartmann:
5885:
5882:
5876:
5873:
5869:
5863:
5860:
5856:
5850:
5847:
5843:
5842:0-12-249051-7
5839:
5835:
5829:
5826:
5822:
5821:do Carmo 1976
5817:
5815:
5811:
5804:
5799:
5793:
5786:
5781:
5777:
5770:
5765:
5761:
5754:
5749:
5745:
5741:
5737:
5730:
5725:
5718:
5713:
5709:
5705:
5698:
5693:
5689:
5688:Shukhov Tower
5685:
5678:
5673:
5669:
5662:
5657:
5653:
5649:
5642:
5637:
5633:
5629:
5622:
5617:
5613:
5609:
5602:
5597:
5595:
5593:
5589:
5585:
5584:rocket engine
5582:
5574:
5570:
5566:
5563:
5559:
5558:
5557:
5555:
5551:
5543:
5541:
5539:
5519:
5512:
5508:
5503:
5501:
5497:
5492:
5488:
5484:
5478:
5477:Ruled variety
5470:
5468:
5466:
5462:
5457:
5455:
5451:
5447:
5442:
5434:
5430:
5428:
5420:
5404:
5401:
5390:
5375:
5350:
5349:
5332:
5321:
5318:
5312:
5301:
5295:
5292:
5289:
5273:
5272:
5271:
5269:
5265:
5261:
5239:
5236:
5225:
5221:
5209:
5201:
5197:
5178:
5170:
5166:
5137:
5136:
5117:
5113:
5101:
5098:
5090:
5086:
5074:
5068:
5065:
5060:
5056:
5039:
5038:
5037:
5018:
5003:
4966:
4929:
4894:
4891:
4886:
4882:
4848:
4844:
4815:
4812:
4804:
4776:
4765:
4747:
4744:
4736:
4721:
4716:
4706:
4701:
4691:
4679:
4678:
4677:
4656:
4645:
4640:
4626:
4608:
4590:
4587:
4581:
4563:
4558:
4544:
4543:
4542:
4525:
4514:
4511:
4505:
4494:
4488:
4485:
4482:
4466:
4460:
4452:
4448:
4445:
4442:
4438:
4434:
4431:
4429:
4426:
4424:
4421:
4420:
4416:
4414:
4412:
4396:
4393:
4384:
4373:
4367:
4349:
4343:
4313:
4299:
4296:
4293:
4290:
4287:
4284:
4275:
4258:
4255:
4252:
4249:
4246:
4243:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4176:
4170:
4155:
4137:
4134:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4101:
4095:
4080:
4079:
4078:
4058:
4047:
4044:
4038:
4027:
4021:
4018:
4015:
4000:
3999:
3998:
3991:
3986:
3978:
3976:
3962:
3959:
3956:
3953:
3930:
3924:
3921:
3918:
3915:
3912:
3906:
3901:
3890:
3884:
3881:
3878:
3875:
3872:
3866:
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3467:
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3414:
3404:
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3396:
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3277:
3274:
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3165:
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3138:
3121:
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2988:
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2717:
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2713:
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2684:
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2678:
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2645:
2630:
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2628:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
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2577:
2571:
2556:
2555:
2554:
2534:
2531:
2528:
2525:
2522:
2519:
2513:
2507:
2492:
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2236:
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2227:
2224:
2209:
2208:
2207:
2200:
2195:
2187:
2185:
2183:
2178:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2128:
2122:
2107:
2106:
2105:
2085:
2082:
2079:
2076:
2073:
2067:
2061:
2046:
2045:
2044:
2042:
2023:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1981:
1975:
1960:
1946:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
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1916:
1910:
1904:
1889:
1875:
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1197:
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1188:
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1128:
1125:
1119:
1116:
1110:
1107:
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1101:
1098:
1095:
1092:
1089:
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1080:
1074:
1067:
1066:
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1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1012:
1006:
1003:
1000:
985:
984:
983:
966:
961:
957:
953:
948:
944:
940:
935:
931:
923:
922:
921:
914:
909:
901:
896:
894:
890:
877:
871:
860:
854:
843:
837:
806:
795:
789:
758:
747:
741:
730:
724:
690:
679:
676:
670:
656:
653:
650:
644:
638:
635:
632:
616:
615:
614:
611:
609:
590:
576:
553:
537:
519:
515:
511:
508:
485:
482:
477:
473:
458:
449:
388:
385:
382:
379:
376:
370:
364:
336:
316:
290:
279:
276:
270:
259:
253:
250:
247:
231:
230:
229:
227:
209:
194:
191:if it is the
190:
189:ruled surface
186:
182:
174:
173:BĂ©zier curves
169:
162:
160:
158:
154:
150:
146:
142:
138:
134:
129:
127:
123:
119:
115:
111:
106:
104:
100:
96:
92:
88:
85:
81:
77:
73:
69:
62:that lies on
61:
60:straight line
58:, there is a
53:
49:
45:
38:
34:
30:
21:
6211:
6185:3: 229–230,
6182:
6167:
6153:
6130:
6099:
6078:
6056:
6046:
6004:
5970:
5952:
5923:
5914:
5905:
5899:
5890:
5884:
5875:
5867:
5862:
5854:
5849:
5833:
5828:
5578:
5562:saddle roofs
5547:
5504:
5480:
5464:
5460:
5458:
5453:
5445:
5443:
5439:
5424:
5267:
5263:
5257:
4791:
4675:
4462:
4410:
4314:
4276:
4273:
4076:
3996:
3993:Möbius strip
3985:Möbius strip
3979:Möbius strip
3945:
3763:
3760:
3758:bilinearly.
3683:
3497:
3344:
3342:
3320:
2951:
2767:
2709:
2702:
2626:
2552:
2488:
2205:
2181:
2179:
2176:
2103:
2038:
1814:
1573:
1511:
1282:
981:
919:
891:
708:
612:
607:
535:
534:is called a
450:
308:
228:of the form
188:
187:is called a
178:
130:
114:doubly ruled
113:
107:
101:of a smooth
91:right conoid
47:
43:
26:
6170:, Tarquin,
6072:Fuchs, D.;
6042:Edge, W. L.
5866:W. KĂĽhnel:
5760:conical hat
5581:RM-81 Agena
5554:latticework
5268:developable
2724:Hyperboloid
404:, and both
6236:Categories
6055:. Review:
5870:, p. 58–60
5832:G. Farin:
5800:References
5794:) concrete
5776:sinusoidal
5573:trash bins
5496:birational
5461:historical
5266:, or just
5262:is called
3498:one gets
105:in space.
97:, and the
84:elliptical
6213:MathWorld
6160:EMS Press
5792:screeding
5746:, Poland.
5684:gridshell
5632:CiechanĂłw
5594:section.
5571:and some
5523:∞
5520:−
5500:fibration
5385:˙
5370:˙
5188:˙
5157:˙
5013:˙
4998:˙
4967:×
4961:˙
4930:×
4924:˙
4805:⋅
4766:×
4760:˙
4737:×
4731:˙
4707:×
4600:˙
4573:˙
4441:sphericon
4394:≠
4359:˙
4335:˙
4297:≤
4291:≤
4285:−
4256:π
4247:≤
4234:
4219:
4210:
4195:
4186:
4126:
4111:
3625:−
3560:−
3538:−
3447:−
3381:−
3305:φ
3302:
3287:φ
3284:
3225:−
3158:π
3152:φ
3063:π
3057:φ
2983:φ
2960:φ
2925:φ
2913:
2901:φ
2889:
2868:−
2859:φ
2856:−
2847:
2835:φ
2832:−
2823:
2808:−
2751:∘
2740:φ
2673:
2661:
2599:
2587:
2452:
2440:
2395:−
2353:
2341:
2264:
2249:
2150:
2138:
2009:
1994:
1932:
1920:
1861:
1849:
1781:
1766:
1736:
1724:
1709:−
1665:
1653:
1626:
1614:
1471:
1456:
1335:
1320:
1249:
1234:
1204:
1189:
1171:−
1102:
1087:
1040:
1025:
861:−
654:−
608:directrix
580:↦
536:generator
462:↦
371:≠
157:embedding
87:directrix
6242:Surfaces
6129:(1952),
6044:(1931),
5951:(1976),
5844:, p. 250
5758:A ruled
5738:roof of
5708:Torrazzo
5634:, Poland
5628:toroidal
5606:Cooling
2202:helicoid
2194:Helicoid
2188:Helicoid
897:Examples
120:and the
95:helicoid
72:cylinder
29:geometry
6035:1406314
5997:2030225
5942:Sources
5704:Cremona
181:surface
151:over a
33:surface
6174:
6137:
6111:
6086:
6033:
6023:
5995:
5985:
5959:
5840:
5744:Warsaw
5592:nozzle
4792:Since
4423:Conoid
1283:with
145:affine
110:circle
93:, the
89:, the
48:scroll
5805:Notes
4437:oloid
4077:with
2705:helix
2703:is a
1815:with
193:union
153:field
139:. In
103:curve
82:with
68:plane
52:point
44:ruled
6172:ISBN
6135:ISBN
6109:ISBN
6084:ISBN
6021:ISBN
5983:ISBN
5957:ISBN
5838:ISBN
5682:The
5652:Kobe
5579:The
4253:<
3345:(CD)
3155:<
3149:<
2104:and
2041:apex
426:and
309:for
78:, a
76:cone
31:, a
6187:doi
6063:doi
6013:doi
5975:doi
5929:doi
5742:in
5706:'s
5686:of
5610:at
5489:in
5481:In
5450:CAD
5358:det
5145:det
4323:det
4300:0.3
4288:0.3
4231:sin
4216:sin
4207:cos
4192:cos
4183:cos
4123:sin
4108:cos
3299:cot
3281:cos
2910:sin
2886:cos
2844:sin
2820:cos
2670:sin
2658:cos
2596:sin
2584:cos
2449:sin
2437:cos
2350:sin
2338:cos
2261:sin
2246:cos
2147:sin
2135:cos
2006:sin
1991:cos
1929:sin
1917:cos
1858:sin
1846:cos
1778:sin
1763:cos
1733:sin
1721:cos
1662:sin
1650:cos
1623:sin
1611:cos
1468:sin
1453:cos
1332:sin
1317:cos
1246:sin
1231:cos
1201:sin
1186:cos
1099:sin
1084:cos
1037:sin
1022:cos
183:in
147:or
128:).
74:or
54:of
42:is
35:in
27:In
6238::
6210:.
6158:,
6152:,
6125:;
6107:,
6031:MR
6029:,
6019:,
6011:,
5993:MR
5991:,
5981:,
5921::
5813:^
5734:A
5650:,
5540:.
5467:.
5459:A
5456:.
4541::
4439:,
4312:.
3975:.
2747:63
2714:.
2707:.
2184:.
179:A
6216:.
6189::
6144:.
6118:.
6093:.
6065::
6015::
5977::
5931::
5762:.
5710:.
5670:.
5575:.
5564:.
5448:(
5405:0
5402:=
5399:)
5395:r
5391:,
5382:r
5376:,
5367:c
5361:(
5336:)
5333:u
5330:(
5326:r
5322:v
5319:+
5316:)
5313:u
5310:(
5306:c
5302:=
5299:)
5296:v
5293:,
5290:u
5287:(
5283:x
5252:.
5240:0
5237:=
5234:)
5231:)
5226:0
5222:u
5218:(
5214:r
5210:,
5207:)
5202:0
5198:u
5194:(
5185:r
5179:,
5176:)
5171:0
5167:u
5163:(
5154:c
5148:(
5123:)
5118:0
5114:u
5110:(
5106:r
5102:v
5099:+
5096:)
5091:0
5087:u
5083:(
5079:c
5075:=
5072:)
5069:v
5066:,
5061:0
5057:u
5053:(
5049:x
5023:r
5019:,
5010:r
5004:,
4995:c
4971:r
4958:c
4934:r
4921:r
4898:)
4895:v
4892:,
4887:0
4883:u
4879:(
4875:x
4854:)
4849:0
4845:u
4841:(
4837:r
4816:0
4813:=
4809:r
4801:n
4777:.
4774:)
4770:r
4757:r
4751:(
4748:v
4745:+
4741:r
4728:c
4722:=
4717:v
4712:x
4702:u
4697:x
4692:=
4688:n
4672:.
4660:)
4657:u
4654:(
4650:r
4646:=
4641:v
4636:x
4624:,
4612:)
4609:u
4606:(
4597:r
4591:v
4588:+
4585:)
4582:u
4579:(
4570:c
4564:=
4559:u
4554:x
4529:)
4526:u
4523:(
4519:r
4515:v
4512:+
4509:)
4506:u
4503:(
4499:c
4495:=
4492:)
4489:v
4486:,
4483:u
4480:(
4476:x
4443:)
4435:(
4397:0
4391:)
4388:)
4385:0
4382:(
4378:r
4374:,
4371:)
4368:0
4365:(
4356:r
4350:,
4347:)
4344:0
4341:(
4332:c
4326:(
4294:v
4259:,
4250:u
4244:0
4240:)
4237:u
4228:,
4225:u
4222:2
4213:u
4204:,
4201:u
4198:2
4189:u
4180:(
4177:=
4174:)
4171:u
4168:(
4164:r
4141:)
4138:0
4135:,
4132:u
4129:2
4120:,
4117:u
4114:2
4105:(
4102:=
4099:)
4096:u
4093:(
4089:c
4062:)
4059:u
4056:(
4052:r
4048:v
4045:+
4042:)
4039:u
4036:(
4032:c
4028:=
4025:)
4022:v
4019:,
4016:u
4013:(
4009:x
3963:y
3960:x
3957:=
3954:z
3931:.
3928:)
3925:1
3922:,
3919:1
3916:,
3913:1
3910:(
3907:=
3902:2
3897:b
3891:,
3888:)
3885:0
3882:,
3879:1
3876:,
3873:0
3870:(
3867:=
3862:1
3857:b
3851:,
3848:)
3845:0
3842:,
3839:0
3836:,
3833:1
3830:(
3827:=
3822:2
3817:a
3811:,
3808:)
3805:0
3802:,
3799:0
3796:,
3793:0
3790:(
3787:=
3782:1
3777:a
3744:2
3739:b
3734:,
3729:1
3724:b
3719:,
3714:2
3709:a
3704:,
3699:1
3694:a
3680:,
3666:)
3659:2
3654:b
3649:u
3646:+
3641:1
3636:b
3631:)
3628:u
3622:1
3619:(
3614:(
3609:v
3606:+
3601:)
3594:2
3589:a
3584:u
3581:+
3576:1
3571:a
3566:)
3563:u
3557:1
3554:(
3549:(
3544:)
3541:v
3535:1
3532:(
3529:=
3526:)
3523:v
3520:,
3517:u
3514:(
3510:x
3481:2
3476:b
3471:u
3468:+
3463:1
3458:b
3453:)
3450:u
3444:1
3441:(
3438:=
3435:)
3432:u
3429:(
3425:d
3420:,
3415:2
3410:a
3405:u
3402:+
3397:1
3392:a
3387:)
3384:u
3378:1
3375:(
3372:=
3369:)
3366:u
3363:(
3359:c
3317:.
3296:=
3293:c
3290:,
3278:=
3275:a
3255:1
3252:=
3245:2
3241:c
3235:2
3231:z
3218:2
3214:a
3207:2
3203:y
3199:+
3194:2
3190:x
3166:2
3162:/
3146:0
3136:,
3122:2
3118:z
3114:=
3109:2
3105:y
3101:+
3096:2
3092:x
3071:2
3067:/
3060:=
3047:,
3035:1
3032:=
3027:2
3023:y
3019:+
3014:2
3010:x
2989:0
2986:=
2937:)
2934:1
2931:,
2928:)
2922:+
2919:u
2916:(
2907:,
2904:)
2898:+
2895:u
2892:(
2883:(
2880:v
2877:+
2874:)
2871:1
2865:,
2862:)
2853:u
2850:(
2841:,
2838:)
2829:u
2826:(
2817:(
2814:)
2811:v
2805:1
2802:(
2799:=
2796:)
2793:v
2790:,
2787:u
2784:(
2780:x
2743:=
2688:)
2685:u
2682:k
2679:,
2676:u
2667:,
2664:u
2655:(
2652:=
2649:)
2646:u
2643:(
2639:d
2623:,
2611:)
2608:0
2605:,
2602:u
2593:,
2590:u
2581:(
2578:=
2575:)
2572:u
2569:(
2565:r
2538:)
2535:u
2532:k
2529:,
2526:0
2523:,
2520:0
2517:(
2514:=
2511:)
2508:u
2505:(
2501:c
2470:.
2467:)
2464:u
2461:k
2458:,
2455:u
2446:,
2443:u
2434:(
2431:v
2428:+
2425:)
2422:u
2419:k
2416:,
2413:0
2410:,
2407:0
2404:(
2401:)
2398:v
2392:1
2389:(
2386:=
2365:)
2362:0
2359:,
2356:u
2347:,
2344:u
2335:(
2332:v
2329:+
2326:)
2323:u
2320:k
2317:,
2314:0
2311:,
2308:0
2305:(
2302:=
2279:)
2276:u
2273:k
2270:,
2267:u
2258:v
2255:,
2252:u
2243:v
2240:(
2237:=
2234:)
2231:v
2228:,
2225:u
2222:(
2218:x
2162:)
2159:1
2156:,
2153:u
2144:,
2141:u
2132:(
2129:=
2126:)
2123:u
2120:(
2116:r
2089:)
2086:0
2083:,
2080:0
2077:,
2074:0
2071:(
2068:=
2065:)
2062:u
2059:(
2055:c
2024:.
2021:)
2018:2
2015:,
2012:u
2003:2
2000:,
1997:u
1988:2
1985:(
1982:=
1979:)
1976:u
1973:(
1969:d
1947:,
1944:)
1941:1
1938:,
1935:u
1926:,
1923:u
1914:(
1911:=
1908:)
1905:u
1902:(
1898:r
1876:,
1873:)
1870:1
1867:,
1864:u
1855:,
1852:u
1843:(
1840:=
1837:)
1834:u
1831:(
1827:c
1796:.
1793:)
1790:2
1787:,
1784:u
1775:2
1772:,
1769:u
1760:2
1757:(
1754:v
1751:+
1748:)
1745:1
1742:,
1739:u
1730:,
1727:u
1718:(
1715:)
1712:v
1706:1
1703:(
1700:=
1677:)
1674:1
1671:,
1668:u
1659:,
1656:u
1647:(
1644:v
1641:+
1638:)
1635:1
1632:,
1629:u
1620:,
1617:u
1608:(
1605:=
1602:)
1599:v
1596:,
1593:u
1590:(
1586:x
1559:.
1554:2
1550:z
1546:=
1541:2
1537:y
1533:+
1528:2
1524:x
1486:.
1483:)
1480:1
1477:,
1474:u
1465:a
1462:,
1459:u
1450:a
1447:(
1444:=
1441:)
1438:u
1435:(
1431:d
1409:,
1406:)
1403:1
1400:,
1397:0
1394:,
1391:0
1388:(
1385:=
1382:)
1379:u
1376:(
1372:r
1350:,
1347:)
1344:0
1341:,
1338:u
1329:a
1326:,
1323:u
1314:a
1311:(
1308:=
1305:)
1302:u
1299:(
1295:c
1264:.
1261:)
1258:1
1255:,
1252:u
1243:a
1240:,
1237:u
1228:a
1225:(
1222:v
1219:+
1216:)
1213:0
1210:,
1207:u
1198:a
1195:,
1192:u
1183:a
1180:(
1177:)
1174:v
1168:1
1165:(
1162:=
1141:)
1138:1
1135:,
1132:0
1129:,
1126:0
1123:(
1120:v
1117:+
1114:)
1111:0
1108:,
1105:u
1096:a
1093:,
1090:u
1081:a
1078:(
1075:=
1052:)
1049:v
1046:,
1043:u
1034:a
1031:,
1028:u
1019:a
1016:(
1013:=
1010:)
1007:v
1004:,
1001:u
998:(
994:x
967:.
962:2
958:a
954:=
949:2
945:y
941:+
936:2
932:x
878:.
875:)
872:u
869:(
865:c
858:)
855:u
852:(
848:d
844:=
841:)
838:u
835:(
831:r
810:)
807:u
804:(
800:d
796:,
793:)
790:u
787:(
783:c
762:)
759:u
756:(
752:r
748:+
745:)
742:u
739:(
735:c
731:=
728:)
725:u
722:(
718:d
694:)
691:u
688:(
684:d
680:v
677:+
674:)
671:u
668:(
664:c
660:)
657:v
651:1
648:(
645:=
642:)
639:v
636:,
633:u
630:(
626:x
594:)
591:u
588:(
584:c
577:u
557:)
554:u
551:(
547:r
520:0
516:u
512:=
509:u
489:)
486:v
483:,
478:0
474:u
470:(
466:x
459:v
435:r
413:c
392:)
389:0
386:,
383:0
380:,
377:0
374:(
368:)
365:u
362:(
358:r
337:v
317:u
294:)
291:u
288:(
284:r
280:v
277:+
274:)
271:u
268:(
264:c
260:=
257:)
254:v
251:,
248:u
245:(
241:x
210:3
205:R
64:S
56:S
40:S
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