82:
3205:
1658:
2938:
2582:
1649:
395:
2005:
2917:
2601:
2563:
25:
386:
3640:
3226:
1415:
1973:
3659:
2448:
758:
1406:
3190:
2902:
214:
2439:
3690:
205:
749:
3581:
3734:, particularly at low ratios; but there is very little additional gain for ratios increasing beyond 5:1. As the fineness ratio increases, the wetted area, and thus the skin friction component of drag, will also increase. Therefore, the minimum drag fineness ratio will ultimately be a trade-off between the decreasing wave drag and increasing friction drag.
1928:
3495:
1996:
1964:
2969:
is a member of the parabolic series (described above). The power series shape is characterized by its (usually) blunt tip, and by the fact that its base is not tangent to the body tube. There is always a discontinuity at the joint between nose cone and body that looks distinctly non-aerodynamic. The
3700:
This observation goes against the often-repeated conventional wisdom that a conical nose is optimum for "Mach-breaking". Fighter aircraft are probably good examples of nose shapes optimized for the transonic region, although their nose shapes are often distorted by other considerations of avionics
3650:
Many references on nose cone design contain empirical data comparing the drag characteristics of various nose shapes in different flight regimes. The chart shown here seems to be the most comprehensive and useful compilation of data for the flight regime of greatest interest. This chart generally
2633:. This construction is similar to that of the tangent ogive, except that a parabola is the defining shape rather than a circle. Just as it does on an ogive, this construction produces a nose shape with a sharp tip. For the blunt shape typically associated with a parabolic nose, see
1682:
3311:
2207:
2788:
232:. This shape is often chosen for its ease of manufacture. More optimal, streamlined shapes (described below) are often much more difficult to create. The sides of a conic profile are straight lines, so the diameter equation is simply:
2331:
2344:
and greater than half the length of the nose cone, then the result will be a secant ogive that bulges out to a maximum diameter that is greater than the base diameter. A classic example of this shape is the nose cone of the
511:
2035:(as it is for a tangent ogive), but rather is one of the factors to be chosen to define the nose shape. If the chosen ogive radius of a secant ogive is greater than the ogive radius of a tangent ogive with the same
1309:
3626:
region and beyond, where the pressure drag increases dramatically, the effect of nose shape on drag becomes highly significant. The factors influencing the pressure drag are the general shape of the nose cone, its
3723:, though that term is usually applied to wings and tails. Fineness ratio is often applied to the entire vehicle, considering the overall length and diameter. The length/diameter relation is also often called the
1923:{\displaystyle {\begin{aligned}x_{o}&=L-{\sqrt {\left(\rho -r_{n}\right)^{2}-(\rho -R)^{2}}}\\y_{t}&={\frac {r_{n}(\rho -R)}{\rho -r_{n}}}\\x_{t}&=x_{o}-{\sqrt {r_{n}^{2}-y_{t}^{2}}}\end{aligned}}}
1445:
to the curve of the nose cone at its base, and the base is on the radius of the circle. The popularity of this shape is largely due to the ease of constructing its profile, as it is simply a circular section.
1176:
2423:
3490:{\displaystyle {\begin{aligned}\theta (x)&=\arccos \left(1-{2x \over L}\right)\\y(\theta ,C)&={R \over {\sqrt {\pi }}}{\sqrt {\theta -{\sin(2\theta ) \over 2}+C\sin ^{3}(\theta )}}\end{aligned}}}
1100:
2889:
result in a slimmer shape, whose appearance is similar to that of the secant ogive. The shape is no longer tangent at the base, and the base is parallel to, but offset from, the axis of the parabola.
667:
3622:, the surface smoothness of that area, and the presence of any discontinuities in the shape. For example, in strictly subsonic rockets a short, blunt, smooth elliptical shape is usually best. In the
1618:
2474:) due to the blunt nose and tangent base. This is not a shape normally found in professional rocketry, which almost always flies at much higher velocities where other designs are more suitable. If
328:
3316:
1687:
2547:
1390:
3113:
2104:
1518:
2625:
This nose shape is not the blunt shape that is envisioned when people commonly refer to a "parabolic" nose cone. The parabolic series nose shape is generated by rotating a segment of a
2109:
564:
956:
1225:
2675:
2019:
The profile of this shape is also formed by a segment of a circle, but the base of the shape is not on the radius of the circle defined by the ogive radius. The rocket body will
997:
904:
733:
370:
271:
3062:
863:
2680:
191:. While the equations describe the "perfect" shape, practical nose cones are often blunted or truncated for manufacturing, aerodynamic, or thermodynamic reasons.
3291:
indicates minimum drag for a given length and volume. The Haack series nose cones are not perfectly tangent to the body at their base except for the case where
2466:, with the major axis being the centerline and the minor axis being the base of the nose cone. A rotation of a full ellipse about its major axis is called a
2238:
3757:
3305:, Haack nose cones bulge to a maximum diameter greater than the base diameter. Haack nose tips do not come to a sharp point, but are slightly rounded.
423:
3618:.8, the nose pressure drag is essentially zero for all shapes. The major significant factor is friction drag, which is largely dependent upon the
3812:
1231:
2470:
spheroid, so an elliptical nose shape would properly be known as a prolate hemispheroid. This shape is popular in subsonic flight (such as
3253:'s series shapes are not constructed from geometric figures. The shapes are instead mathematically derived for the purpose of minimizing
1113:
2355:
1002:
592:
3702:
1549:
286:
3678:
1.2. Although data are not available for many shapes in the transonic region, the table clearly suggests that either the
2492:
1314:
3067:
2049:
68:
46:
39:
3879:
2202:{\displaystyle \alpha =\arccos \left({{\sqrt {L^{2}+R^{2}}} \over 2\rho }\right)-\arctan \left({R \over L}\right)}
1463:
3789:
3602:
An aerospike can be used to reduce the forebody pressure acting on supersonic aircraft. The aerospike creates a
517:
3768:
909:
128:
geometrical shape for optimum performance. For many applications, such a task requires the definition of a
3783:
3204:
1181:
3743:
3670:
In many nose cone designs, the greatest concern is flight performance in the transonic region from Mach
2965:
includes the shape commonly referred to as a "parabolic" nose cone, but the shape correctly known as a
963:
870:
685:
333:
3779:
2783:{\displaystyle y=R\left({2\left({x \over L}\right)-K'\left({x \over L}\right)^{2} \over 2-K'}\right)}
1657:
2643:
238:
3819:
3651:
agrees with more detailed, but less comprehensive data found in other references (most notably the
3565:
3557:
The Haack series designs giving minimum drag for the given length and diameter, the LD-Haack where
3035:
2043:, then the resulting secant ogive appears as a tangent ogive with a portion of the base truncated.
822:
33:
3597:
2937:
1457:, and it is related to the length and base radius of the nose cone as expressed by the formula:
50:
3917:
3693:
2581:
1648:
394:
3258:
8:
3845:
2971:
129:
2004:
3860:
2916:
174:
132:
shape that experiences minimal resistance to rapid motion through such a fluid medium.
117:
101:
2970:
shape can be modified at the base to smooth out this discontinuity. Both a flat-faced
16:
Geometry and construction of the foremost tip of airplanes, spacecraft and projectiles
3922:
3893:
3885:
3864:
3713:
The ratio of the length of a nose cone compared to its base diameter is known as the
3639:
2346:
2013:
Alternate secant ogive render and profile which show a bulge due to a smaller radius.
1666:
Spherically blunted tangent ogive nose cone render and profile with parameters shown.
2600:
2562:
385:
3585:
3225:
2326:{\displaystyle y={\sqrt {\rho ^{2}-(\rho \cos(\alpha )-x)^{2}}}-\rho \sin(\alpha )}
410:
3666:
to low-mach regions. Rankings are: superior (1), good (2), fair (3), inferior (4).
1676:. The tangency point where the sphere meets the tangent ogive can be found from:
1423:
Tangent ogive nose cone render and profile with parameters and ogive circle shown.
3679:
2975:
1414:
229:
3686:, would be preferable to the popular conical or ogive shapes, for this purpose.
3028:
decreases towards zero, the power series nose shape becomes increasingly blunt.
2878:
to the body at its base, and the base is on the axis of the parabola. Values of
1972:
506:{\displaystyle x_{t}={\frac {L^{2}}{R}}{\sqrt {\frac {r_{n}^{2}}{R^{2}+L^{2}}}}}
3730:
At supersonic speeds, the fineness ratio has a significant effect on nose cone
3715:
3652:
3628:
3603:
3254:
3250:
2637:
below. (The parabolic shape is also often confused with the elliptical shape.)
2471:
1434:
173:. The equations define the two-dimensional profile of the nose shape. The full
81:
3298:. However, the discontinuity is usually so slight as to be imperceptible. For
2447:
3911:
3897:
403:
Spherically blunted conic nose cone render and profile with parameters shown.
3658:
3261:. While the series is a continuous set of shapes determined by the value of
757:
2630:
1995:
1963:
1984:
ogive nose cone render and profile with parameters and ogive circle shown.
1405:
3619:
3615:
3189:
2901:
1981:
213:
177:
of the nose cone is formed by rotating the profile around the centerline
90:
1672:
A tangent ogive nose is often blunted by capping it with a segment of a
1304:{\displaystyle \phi _{2}=\arctan \left({R_{2}-R_{1} \over L_{2}}\right)}
417:. The tangency point where the sphere meets the cone can be found from:
3644:
2438:
3662:
Comparison of drag characteristics of various nose cone shapes in the
3731:
3663:
3623:
748:
125:
97:
413:, a conical nose is often blunted by capping it with a segment of a
204:
3580:
3280:
signifies minimum drag for the given length and diameter, and when
2626:
109:
3606:
ahead of the body, thus reducing the drag acting on the aircraft.
2023:
be tangent to the curve of the nose at its base. The ogive radius
1171:{\displaystyle \phi _{1}=\arctan \left({R_{1} \over L_{1}}\right)}
3689:
2875:
2467:
2463:
2418:{\displaystyle {\frac {L}{2}}<\rho <{R^{2}+L^{2} \over 2R}}
1442:
782:
113:
3889:
2483:
1673:
1632:
1438:
414:
121:
105:
93:
3765:
Bericht 139 der
Lilienthal-Gesellschaft für Luftfahrtforschung
2456:
Elliptical nose cone render and profile with parameters shown.
3705:
nose appears to be a very close match to a Von Kármán shape.
3573:
1430:
3853:
International
Research Journal of Engineering and Technology
3846:"A Review on Nose Cone Designs for Different Flight Regimes"
2806:, but the most common values used for nose cone shapes are:
1449:
The radius of the circle that forms the ogive is called the
1095:{\displaystyle y=R_{1}+{(x-L_{1})(R_{2}-R_{1}) \over L_{2}}}
766:
Bi-conic nose cone render and profile with parameters shown.
85:
General parameters used for constructing nose cone profiles.
807:
to the top radius of the smaller frustum with base radius
3778:
2981:
The power series nose shape is generated by rotating the
662:{\displaystyle x_{o}=x_{t}+{\sqrt {r_{n}^{2}-y_{t}^{2}}}}
222:
Conic nose cone render and profile with parameters shown.
100:
section of any vehicle or body meant to travel through a
1437:. The profile of this shape is formed by a segment of a
772:
A bi-conic nose cone shape is simply a cone with length
1613:{\displaystyle y={\sqrt {\rho ^{2}-(L-x)^{2}}}+R-\rho }
798:, where the base of the upper cone is equal in radius
323:{\displaystyle \phi =\arctan \left({R \over L}\right)}
3314:
3070:
3038:
2683:
2646:
2495:
2358:
2241:
2112:
2052:
1685:
1552:
1466:
1317:
1234:
1184:
1116:
1005:
966:
912:
873:
825:
688:
595:
520:
426:
336:
289:
241:
3257:; a related shape with similar derivation being the
1638:
124:), an important problem is the determination of the
3696:
with a nose cone very close to the Von Kármán shape
3016:controls the bluntness of the shape. For values of
145:In all of the following nose cone shape equations,
3708:
3489:
3107:
3056:
2782:
2669:
2542:{\displaystyle y=R{\sqrt {1-{x^{2} \over L^{2}}}}}
2541:
2417:
2325:
2201:
2098:
1922:
1612:
1512:
1385:{\displaystyle y=R_{1}+(x-L_{1})\tan(\phi _{2})\;}
1384:
1303:
1219:
1170:
1094:
991:
950:
898:
857:
727:
661:
558:
505:
364:
322:
265:
3785:Design of Aerodynamically Stabilized Free Rockets
3634:
3609:
3193:Graphs illustrating Haack series nose cone shapes
2905:Graphs illustrating power series nose cone shapes
276:Cones are sometimes defined by their half angle,
135:
3909:
3844:Iyer, Aditya Rajan; Pant, Anjali (August 2020).
3108:{\displaystyle y=R\left({x \over L}\right)^{n}}
2099:{\displaystyle \rho >{R^{2}+L^{2} \over 2R}}
2619:Renders of common parabolic nose cone shapes.
3758:"Geschoßformen kleinsten Wellenwiderstandes"
2462:The profile of this shape is one-half of an
1513:{\displaystyle \rho ={R^{2}+L^{2} \over 2R}}
375:
153:is the radius of the base of the nose cone.
3810:
149:is the overall length of the nose cone and
3249:Unlike all of the nose cone shapes above,
1381:
1216:
361:
228:A very common nose-cone shape is a simple
1947:is the center of the spherical nose cap.
1631:. If they are equal, then the shape is a
576:is the radius of the spherical nose cap.
559:{\displaystyle y_{t}={\frac {x_{t}R}{L}}}
69:Learn how and when to remove this message
3843:
3688:
3657:
3638:
3579:
3188:
2900:
80:
32:This article includes a list of general
3910:
3813:The Descriptive Geometry of Nose Cones
3804:
3265:in the equations below, two values of
951:{\displaystyle y={xR_{1} \over L_{1}}}
579:The center of the spherical nose cap,
3755:
140:
3877:
3871:
3837:
3719:. This is sometimes also called the
1220:{\displaystyle y=x\tan(\phi _{1})\;}
18:
3269:have particular significance: when
2867:For the case of the full parabola (
1429:Next to a simple cone, the tangent
13:
3749:
3682:shape, or power series shape with
409:In practical applications such as
169:, at the tip of the nose cone, to
38:it lacks sufficient corresponding
14:
3934:
3643:Closeup view of a nose cone on a
2978:are members of the power series.
2634:
1639:Spherically blunted tangent ogive
992:{\displaystyle L_{1}\leq x\leq L}
899:{\displaystyle 0\leq x\leq L_{1}}
728:{\displaystyle x_{a}=x_{o}-r_{n}}
3614:For aircraft and rockets, below
3224:
3203:
2936:
2915:
2599:
2580:
2561:
2446:
2437:
2003:
1994:
1971:
1962:
1656:
1647:
1627:, must be less than or equal to
1413:
1404:
1395:
756:
747:
393:
384:
365:{\displaystyle y=x\tan(\phi )\;}
212:
203:
23:
3790:U.S. Government Printing Office
3709:Influence of the fineness ratio
3180:
2892:
2340:is less than the tangent ogive
1953:
785:of a cone (commonly known as a
3884:. New York City: McGraw-Hill.
3635:Influence of the general shape
3610:Nose cone drag characteristics
3504:(as described above) include:
3478:
3472:
3444:
3435:
3394:
3382:
3328:
3322:
3024:, the tip is fairly sharp. As
2670:{\displaystyle 0\leq K'\leq 1}
2629:around a line parallel to its
2320:
2314:
2291:
2281:
2275:
2263:
1821:
1809:
1764:
1751:
1587:
1574:
1433:shape is the most familiar in
1378:
1365:
1356:
1337:
1213:
1200:
1076:
1050:
1047:
1028:
358:
352:
266:{\displaystyle y={xR \over L}}
136:Nose cone shapes and equations
1:
3811:Crowell Sr., Gary A. (1996).
3797:
3552:
3057:{\displaystyle 0\leq n\leq 1}
2428:
1441:such that the rocket body is
858:{\displaystyle L=L_{1}+L_{2}}
3881:Missile Configuration Design
3701:and inlets. For example, an
3591:
2552:
7:
3737:
3631:, and its bluffness ratio.
738:
157:is the radius at any point
10:
3939:
3744:Index of aviation articles
3595:
2798:can vary anywhere between
787:conical transition section
3780:U.S. Army Missile Command
3564:, is commonly called the
376:Spherically blunted conic
89:Given the problem of the
3818:(Report). Archived from
3756:Haack, Wolfgang (1941).
194:
3767:: 14–28. Archived from
3598:Drag-reducing aerospike
3210:LD-Haack (Von Kármán) (
53:more precise citations.
3697:
3667:
3647:
3588:
3521:LD-Haack (Von Kármán)
3491:
3194:
3109:
3058:
2906:
2784:
2671:
2543:
2419:
2327:
2203:
2100:
1924:
1623:The nose cone length,
1614:
1514:
1386:
1305:
1221:
1172:
1096:
993:
952:
900:
859:
729:
663:
560:
507:
366:
324:
267:
86:
3694:General Dynamics F-16
3692:
3661:
3642:
3583:
3492:
3192:
3110:
3059:
2904:
2785:
2672:
2544:
2420:
2328:
2204:
2101:
2027:is not determined by
1925:
1615:
1515:
1387:
1306:
1222:
1173:
1097:
994:
953:
901:
860:
730:
664:
586:, can be found from:
561:
508:
367:
325:
268:
84:
3878:Chin, S. S. (1961).
3703:F-16 Fighting Falcon
3584:An aerospike on the
3312:
3068:
3036:
3004:-axis for values of
2681:
2644:
2493:
2356:
2239:
2110:
2050:
1683:
1550:
1464:
1315:
1232:
1182:
1114:
1003:
964:
910:
871:
823:
781:stacked on top of a
686:
672:And the apex point,
593:
518:
424:
334:
287:
239:
3792:. MIL-HDBK-762(MI).
2967:parabolic nose cone
1913:
1895:
789:shape) with length
679:can be found from:
656:
638:
474:
130:solid of revolution
3698:
3668:
3648:
3589:
3510:Haack series type
3500:Special values of
3487:
3485:
3195:
3105:
3054:
2907:
2780:
2667:
2539:
2415:
2323:
2199:
2096:
1940:is the radius and
1920:
1918:
1899:
1881:
1610:
1510:
1382:
1301:
1217:
1168:
1092:
989:
948:
896:
855:
725:
659:
642:
624:
556:
503:
460:
362:
320:
263:
175:body of revolution
141:General dimensions
104:medium (such as a
102:compressible fluid
87:
3550:
3549:
3481:
3451:
3416:
3414:
3368:
3247:
3246:
3178:
3177:
3118:Common values of
3093:
2959:
2958:
2865:
2864:
2774:
2745:
2715:
2537:
2535:
2413:
2367:
2300:
2193:
2166:
2156:
2094:
1914:
1843:
1773:
1596:
1508:
1295:
1162:
1090:
946:
657:
554:
501:
500:
455:
411:re-entry vehicles
314:
261:
79:
78:
71:
3930:
3902:
3901:
3875:
3869:
3868:
3859:(8): 3546–3554.
3850:
3841:
3835:
3834:
3832:
3830:
3825:on 11 April 2011
3824:
3817:
3808:
3793:
3782:(17 July 1990).
3775:
3773:
3762:
3727:of a nose cone.
3685:
3677:
3673:
3586:UGM-96 Trident I
3563:
3546:
3536:
3526:
3515:
3507:
3506:
3503:
3496:
3494:
3493:
3488:
3486:
3482:
3468:
3467:
3452:
3447:
3427:
3419:
3417:
3415:
3410:
3405:
3374:
3370:
3369:
3364:
3356:
3304:
3297:
3290:
3286:
3279:
3275:
3268:
3264:
3259:Sears–Haack body
3237:
3228:
3216:
3207:
3185:
3184:
3174:
3164:
3154:
3149:Half (parabola)
3144:
3133:
3125:
3124:
3121:
3114:
3112:
3111:
3106:
3104:
3103:
3098:
3094:
3086:
3063:
3061:
3060:
3055:
3027:
3023:
3019:
3015:
3011:
3007:
3003:
3000:curve about the
2999:
2949:
2940:
2928:
2919:
2897:
2896:
2888:
2884:
2873:
2861:
2851:
2841:
2831:
2820:
2809:
2808:
2805:
2801:
2797:
2789:
2787:
2786:
2781:
2779:
2775:
2773:
2772:
2757:
2756:
2755:
2750:
2746:
2738:
2731:
2720:
2716:
2708:
2698:
2676:
2674:
2673:
2668:
2660:
2612:
2603:
2593:
2584:
2574:
2565:
2548:
2546:
2545:
2540:
2538:
2536:
2534:
2533:
2524:
2523:
2514:
2506:
2481:
2477:
2450:
2441:
2424:
2422:
2421:
2416:
2414:
2412:
2404:
2403:
2402:
2390:
2389:
2379:
2368:
2360:
2343:
2339:
2332:
2330:
2329:
2324:
2301:
2299:
2298:
2259:
2258:
2249:
2231:
2227:
2223:
2219:
2215:
2212:Then the radius
2208:
2206:
2205:
2200:
2198:
2194:
2186:
2171:
2167:
2165:
2157:
2155:
2154:
2142:
2141:
2132:
2130:
2105:
2103:
2102:
2097:
2095:
2093:
2085:
2084:
2083:
2071:
2070:
2060:
2042:
2038:
2034:
2030:
2026:
2007:
1998:
1975:
1966:
1946:
1939:
1929:
1927:
1926:
1921:
1919:
1915:
1912:
1907:
1894:
1889:
1880:
1875:
1874:
1858:
1857:
1844:
1842:
1841:
1840:
1824:
1808:
1807:
1797:
1788:
1787:
1774:
1772:
1771:
1747:
1746:
1741:
1737:
1736:
1735:
1714:
1699:
1698:
1660:
1651:
1630:
1626:
1619:
1617:
1616:
1611:
1597:
1595:
1594:
1570:
1569:
1560:
1542:
1538:
1534:
1530:
1526:
1519:
1517:
1516:
1511:
1509:
1507:
1499:
1498:
1497:
1485:
1484:
1474:
1456:
1417:
1408:
1391:
1389:
1388:
1383:
1377:
1376:
1355:
1354:
1333:
1332:
1310:
1308:
1307:
1302:
1300:
1296:
1294:
1293:
1284:
1283:
1282:
1270:
1269:
1259:
1244:
1243:
1226:
1224:
1223:
1218:
1212:
1211:
1177:
1175:
1174:
1169:
1167:
1163:
1161:
1160:
1151:
1150:
1141:
1126:
1125:
1101:
1099:
1098:
1093:
1091:
1089:
1088:
1079:
1075:
1074:
1062:
1061:
1046:
1045:
1026:
1021:
1020:
998:
996:
995:
990:
976:
975:
957:
955:
954:
949:
947:
945:
944:
935:
934:
933:
920:
905:
903:
902:
897:
895:
894:
864:
862:
861:
856:
854:
853:
841:
840:
815:
806:
797:
780:
760:
751:
734:
732:
731:
726:
724:
723:
711:
710:
698:
697:
678:
668:
666:
665:
660:
658:
655:
650:
637:
632:
623:
618:
617:
605:
604:
585:
575:
565:
563:
562:
557:
555:
550:
546:
545:
535:
530:
529:
512:
510:
509:
504:
502:
499:
498:
497:
485:
484:
473:
468:
459:
458:
456:
451:
450:
441:
436:
435:
397:
388:
371:
369:
368:
363:
329:
327:
326:
321:
319:
315:
307:
279:
272:
270:
269:
264:
262:
257:
249:
216:
207:
190:
189:
188:
183:
182:
172:
168:
164:
160:
156:
152:
148:
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
3938:
3937:
3933:
3932:
3931:
3929:
3928:
3927:
3908:
3907:
3906:
3905:
3876:
3872:
3848:
3842:
3838:
3828:
3826:
3822:
3815:
3809:
3805:
3800:
3771:
3760:
3752:
3750:Further reading
3740:
3711:
3683:
3675:
3671:
3637:
3612:
3600:
3594:
3558:
3555:
3544:
3534:
3524:
3513:
3501:
3484:
3483:
3463:
3459:
3428:
3426:
3418:
3409:
3404:
3397:
3376:
3375:
3357:
3355:
3348:
3344:
3331:
3315:
3313:
3310:
3309:
3299:
3292:
3288:
3281:
3277:
3276:, the notation
3270:
3266:
3262:
3243:
3242:
3241:
3240:
3239:
3232:
3229:
3220:
3219:
3218:
3211:
3208:
3183:
3172:
3162:
3152:
3142:
3131:
3119:
3099:
3085:
3081:
3080:
3069:
3066:
3065:
3037:
3034:
3033:
3025:
3021:
3017:
3013:
3009:
3005:
3001:
2982:
2955:
2954:
2953:
2952:
2951:
2944:
2943:Three-quarter (
2941:
2932:
2931:
2930:
2923:
2920:
2895:
2886:
2879:
2874:) the shape is
2868:
2859:
2849:
2839:
2829:
2815:
2803:
2799:
2792:
2765:
2758:
2751:
2737:
2733:
2732:
2724:
2707:
2703:
2699:
2697:
2693:
2682:
2679:
2678:
2653:
2645:
2642:
2641:
2623:
2622:
2621:
2620:
2616:
2615:
2614:
2607:
2604:
2596:
2595:
2588:
2587:Three-quarter (
2585:
2577:
2576:
2569:
2566:
2555:
2529:
2525:
2519:
2515:
2513:
2505:
2494:
2491:
2490:
2479:
2475:
2460:
2459:
2458:
2457:
2453:
2452:
2451:
2443:
2442:
2431:
2405:
2398:
2394:
2385:
2381:
2380:
2378:
2359:
2357:
2354:
2353:
2341:
2337:
2294:
2290:
2254:
2250:
2248:
2240:
2237:
2236:
2229:
2225:
2221:
2217:
2213:
2185:
2181:
2158:
2150:
2146:
2137:
2133:
2131:
2129:
2125:
2111:
2108:
2107:
2086:
2079:
2075:
2066:
2062:
2061:
2059:
2051:
2048:
2047:
2040:
2036:
2032:
2028:
2024:
2017:
2016:
2015:
2014:
2010:
2009:
2008:
2000:
1999:
1988:
1987:
1986:
1985:
1978:
1977:
1976:
1968:
1967:
1956:
1950:
1945:
1941:
1938:
1934:
1917:
1916:
1908:
1903:
1890:
1885:
1879:
1870:
1866:
1859:
1853:
1849:
1846:
1845:
1836:
1832:
1825:
1803:
1799:
1798:
1796:
1789:
1783:
1779:
1776:
1775:
1767:
1763:
1742:
1731:
1727:
1720:
1716:
1715:
1713:
1700:
1694:
1690:
1686:
1684:
1681:
1680:
1670:
1669:
1668:
1667:
1663:
1662:
1661:
1653:
1652:
1641:
1628:
1624:
1590:
1586:
1565:
1561:
1559:
1551:
1548:
1547:
1540:
1536:
1532:
1528:
1524:
1500:
1493:
1489:
1480:
1476:
1475:
1473:
1465:
1462:
1461:
1454:
1427:
1426:
1425:
1424:
1420:
1419:
1418:
1410:
1409:
1398:
1372:
1368:
1350:
1346:
1328:
1324:
1316:
1313:
1312:
1289:
1285:
1278:
1274:
1265:
1261:
1260:
1258:
1254:
1239:
1235:
1233:
1230:
1229:
1207:
1203:
1183:
1180:
1179:
1156:
1152:
1146:
1142:
1140:
1136:
1121:
1117:
1115:
1112:
1111:
1084:
1080:
1070:
1066:
1057:
1053:
1041:
1037:
1027:
1025:
1016:
1012:
1004:
1001:
1000:
971:
967:
965:
962:
961:
940:
936:
929:
925:
921:
919:
911:
908:
907:
890:
886:
872:
869:
868:
849:
845:
836:
832:
824:
821:
820:
814:
808:
805:
799:
796:
790:
779:
773:
770:
769:
768:
767:
763:
762:
761:
753:
752:
741:
719:
715:
706:
702:
693:
689:
687:
684:
683:
677:
673:
651:
646:
633:
628:
622:
613:
609:
600:
596:
594:
591:
590:
584:
580:
574:
570:
541:
537:
536:
534:
525:
521:
519:
516:
515:
493:
489:
480:
476:
475:
469:
464:
457:
446:
442:
440:
431:
427:
425:
422:
421:
407:
406:
405:
404:
400:
399:
398:
390:
389:
378:
335:
332:
331:
306:
302:
288:
285:
284:
277:
250:
248:
240:
237:
236:
226:
225:
224:
223:
219:
218:
217:
209:
208:
197:
186:
185:
180:
179:
178:
170:
166:
162:
158:
154:
150:
146:
143:
138:
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
3936:
3926:
3925:
3920:
3904:
3903:
3870:
3836:
3802:
3801:
3799:
3796:
3795:
3794:
3776:
3774:on 2007-09-27.
3751:
3748:
3747:
3746:
3739:
3736:
3716:fineness ratio
3710:
3707:
3636:
3633:
3629:fineness ratio
3611:
3608:
3604:detached shock
3596:Main article:
3593:
3590:
3554:
3551:
3548:
3547:
3542:
3538:
3537:
3532:
3528:
3527:
3522:
3518:
3517:
3511:
3498:
3497:
3480:
3477:
3474:
3471:
3466:
3462:
3458:
3455:
3450:
3446:
3443:
3440:
3437:
3434:
3431:
3425:
3422:
3413:
3408:
3403:
3400:
3398:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3377:
3373:
3367:
3363:
3360:
3354:
3351:
3347:
3343:
3340:
3337:
3334:
3332:
3330:
3327:
3324:
3321:
3318:
3317:
3251:Wolfgang Haack
3245:
3244:
3230:
3223:
3222:
3221:
3209:
3202:
3201:
3200:
3199:
3198:
3196:
3182:
3179:
3176:
3175:
3170:
3166:
3165:
3160:
3159:Three quarter
3156:
3155:
3150:
3146:
3145:
3140:
3136:
3135:
3129:
3116:
3115:
3102:
3097:
3092:
3089:
3084:
3079:
3076:
3073:
3053:
3050:
3047:
3044:
3041:
2957:
2956:
2942:
2935:
2934:
2933:
2921:
2914:
2913:
2912:
2911:
2910:
2908:
2894:
2891:
2863:
2862:
2857:
2853:
2852:
2847:
2846:Three quarter
2843:
2842:
2837:
2833:
2832:
2827:
2823:
2822:
2813:
2812:Parabola type
2778:
2771:
2768:
2764:
2761:
2754:
2749:
2744:
2741:
2736:
2730:
2727:
2723:
2719:
2714:
2711:
2706:
2702:
2696:
2692:
2689:
2686:
2666:
2663:
2659:
2656:
2652:
2649:
2618:
2617:
2605:
2598:
2597:
2586:
2579:
2578:
2567:
2560:
2559:
2558:
2557:
2556:
2554:
2551:
2550:
2549:
2532:
2528:
2522:
2518:
2512:
2509:
2504:
2501:
2498:
2472:model rocketry
2455:
2454:
2445:
2444:
2436:
2435:
2434:
2433:
2432:
2430:
2427:
2426:
2425:
2411:
2408:
2401:
2397:
2393:
2388:
2384:
2377:
2374:
2371:
2366:
2363:
2336:If the chosen
2334:
2333:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2297:
2293:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2257:
2253:
2247:
2244:
2210:
2209:
2197:
2192:
2189:
2184:
2180:
2177:
2174:
2170:
2164:
2161:
2153:
2149:
2145:
2140:
2136:
2128:
2124:
2121:
2118:
2115:
2092:
2089:
2082:
2078:
2074:
2069:
2065:
2058:
2055:
2012:
2011:
2002:
2001:
1993:
1992:
1991:
1990:
1989:
1980:
1979:
1970:
1969:
1961:
1960:
1959:
1958:
1957:
1955:
1952:
1943:
1936:
1931:
1930:
1911:
1906:
1902:
1898:
1893:
1888:
1884:
1878:
1873:
1869:
1865:
1862:
1860:
1856:
1852:
1848:
1847:
1839:
1835:
1831:
1828:
1823:
1820:
1817:
1814:
1811:
1806:
1802:
1795:
1792:
1790:
1786:
1782:
1778:
1777:
1770:
1766:
1762:
1759:
1756:
1753:
1750:
1745:
1740:
1734:
1730:
1726:
1723:
1719:
1712:
1709:
1706:
1703:
1701:
1697:
1693:
1689:
1688:
1665:
1664:
1655:
1654:
1646:
1645:
1644:
1643:
1642:
1640:
1637:
1621:
1620:
1609:
1606:
1603:
1600:
1593:
1589:
1585:
1582:
1579:
1576:
1573:
1568:
1564:
1558:
1555:
1521:
1520:
1506:
1503:
1496:
1492:
1488:
1483:
1479:
1472:
1469:
1435:hobby rocketry
1422:
1421:
1412:
1411:
1403:
1402:
1401:
1400:
1399:
1397:
1394:
1393:
1392:
1380:
1375:
1371:
1367:
1364:
1361:
1358:
1353:
1349:
1345:
1342:
1339:
1336:
1331:
1327:
1323:
1320:
1299:
1292:
1288:
1281:
1277:
1273:
1268:
1264:
1257:
1253:
1250:
1247:
1242:
1238:
1227:
1215:
1210:
1206:
1202:
1199:
1196:
1193:
1190:
1187:
1166:
1159:
1155:
1149:
1145:
1139:
1135:
1132:
1129:
1124:
1120:
1103:
1102:
1087:
1083:
1078:
1073:
1069:
1065:
1060:
1056:
1052:
1049:
1044:
1040:
1036:
1033:
1030:
1024:
1019:
1015:
1011:
1008:
988:
985:
982:
979:
974:
970:
958:
943:
939:
932:
928:
924:
918:
915:
893:
889:
885:
882:
879:
876:
865:
852:
848:
844:
839:
835:
831:
828:
812:
803:
794:
777:
765:
764:
755:
754:
746:
745:
744:
743:
742:
740:
737:
736:
735:
722:
718:
714:
709:
705:
701:
696:
692:
675:
670:
669:
654:
649:
645:
641:
636:
631:
627:
621:
616:
612:
608:
603:
599:
582:
572:
567:
566:
553:
549:
544:
540:
533:
528:
524:
513:
496:
492:
488:
483:
479:
472:
467:
463:
454:
449:
445:
439:
434:
430:
402:
401:
392:
391:
383:
382:
381:
380:
379:
377:
374:
373:
372:
360:
357:
354:
351:
348:
345:
342:
339:
318:
313:
310:
305:
301:
298:
295:
292:
274:
273:
260:
256:
253:
247:
244:
221:
220:
211:
210:
202:
201:
200:
199:
198:
196:
193:
142:
139:
137:
134:
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3935:
3924:
3921:
3919:
3916:
3915:
3913:
3899:
3895:
3891:
3887:
3883:
3882:
3874:
3866:
3862:
3858:
3854:
3847:
3840:
3821:
3814:
3807:
3803:
3791:
3787:
3786:
3781:
3777:
3770:
3766:
3759:
3754:
3753:
3745:
3742:
3741:
3735:
3733:
3728:
3726:
3722:
3718:
3717:
3706:
3704:
3695:
3691:
3687:
3681:
3665:
3660:
3656:
3654:
3646:
3641:
3632:
3630:
3625:
3621:
3617:
3607:
3605:
3599:
3587:
3582:
3578:
3576:
3575:
3569:
3568:
3561:
3543:
3540:
3539:
3533:
3530:
3529:
3523:
3520:
3519:
3512:
3509:
3508:
3505:
3475:
3469:
3464:
3460:
3456:
3453:
3448:
3441:
3438:
3432:
3429:
3423:
3420:
3411:
3406:
3401:
3399:
3391:
3388:
3385:
3379:
3371:
3365:
3361:
3358:
3352:
3349:
3345:
3341:
3338:
3335:
3333:
3325:
3319:
3308:
3307:
3306:
3302:
3295:
3284:
3273:
3260:
3256:
3252:
3235:
3227:
3214:
3206:
3197:
3191:
3187:
3186:
3171:
3168:
3167:
3161:
3158:
3157:
3151:
3148:
3147:
3141:
3138:
3137:
3130:
3127:
3126:
3123:
3100:
3095:
3090:
3087:
3082:
3077:
3074:
3071:
3051:
3048:
3045:
3042:
3039:
3031:
3030:
3029:
3012:. The factor
2997:
2993:
2989:
2985:
2979:
2977:
2973:
2968:
2964:
2947:
2939:
2926:
2918:
2909:
2903:
2899:
2898:
2890:
2882:
2877:
2871:
2858:
2855:
2854:
2848:
2845:
2844:
2838:
2835:
2834:
2828:
2825:
2824:
2818:
2814:
2811:
2810:
2807:
2795:
2790:
2776:
2769:
2766:
2762:
2759:
2752:
2747:
2742:
2739:
2734:
2728:
2725:
2721:
2717:
2712:
2709:
2704:
2700:
2694:
2690:
2687:
2684:
2664:
2661:
2657:
2654:
2650:
2647:
2638:
2636:
2632:
2628:
2610:
2602:
2591:
2583:
2572:
2564:
2530:
2526:
2520:
2516:
2510:
2507:
2502:
2499:
2496:
2489:
2488:
2487:
2485:
2473:
2469:
2465:
2449:
2440:
2409:
2406:
2399:
2395:
2391:
2386:
2382:
2375:
2372:
2369:
2364:
2361:
2352:
2351:
2350:
2348:
2317:
2311:
2308:
2305:
2302:
2295:
2287:
2284:
2278:
2272:
2269:
2266:
2260:
2255:
2251:
2245:
2242:
2235:
2234:
2233:
2216:at any point
2195:
2190:
2187:
2182:
2178:
2175:
2172:
2168:
2162:
2159:
2151:
2147:
2143:
2138:
2134:
2126:
2122:
2119:
2116:
2113:
2090:
2087:
2080:
2076:
2072:
2067:
2063:
2056:
2053:
2046:
2045:
2044:
2022:
2006:
1997:
1983:
1974:
1965:
1951:
1948:
1909:
1904:
1900:
1896:
1891:
1886:
1882:
1876:
1871:
1867:
1863:
1861:
1854:
1850:
1837:
1833:
1829:
1826:
1818:
1815:
1812:
1804:
1800:
1793:
1791:
1784:
1780:
1768:
1760:
1757:
1754:
1748:
1743:
1738:
1732:
1728:
1724:
1721:
1717:
1710:
1707:
1704:
1702:
1695:
1691:
1679:
1678:
1677:
1675:
1659:
1650:
1636:
1634:
1607:
1604:
1601:
1598:
1591:
1583:
1580:
1577:
1571:
1566:
1562:
1556:
1553:
1546:
1545:
1544:
1527:at any point
1504:
1501:
1494:
1490:
1486:
1481:
1477:
1470:
1467:
1460:
1459:
1458:
1452:
1447:
1444:
1440:
1436:
1432:
1416:
1407:
1396:Tangent ogive
1373:
1369:
1362:
1359:
1351:
1347:
1343:
1340:
1334:
1329:
1325:
1321:
1318:
1297:
1290:
1286:
1279:
1275:
1271:
1266:
1262:
1255:
1251:
1248:
1245:
1240:
1236:
1228:
1208:
1204:
1197:
1194:
1191:
1188:
1185:
1164:
1157:
1153:
1147:
1143:
1137:
1133:
1130:
1127:
1122:
1118:
1110:
1109:
1108:
1107:
1085:
1081:
1071:
1067:
1063:
1058:
1054:
1042:
1038:
1034:
1031:
1022:
1017:
1013:
1009:
1006:
986:
983:
980:
977:
972:
968:
959:
941:
937:
930:
926:
922:
916:
913:
891:
887:
883:
880:
877:
874:
866:
850:
846:
842:
837:
833:
829:
826:
819:
818:
817:
811:
802:
793:
788:
784:
776:
759:
750:
720:
716:
712:
707:
703:
699:
694:
690:
682:
681:
680:
652:
647:
643:
639:
634:
629:
625:
619:
614:
610:
606:
601:
597:
589:
588:
587:
577:
551:
547:
542:
538:
531:
526:
522:
514:
494:
490:
486:
481:
477:
470:
465:
461:
452:
447:
443:
437:
432:
428:
420:
419:
418:
416:
412:
396:
387:
355:
349:
346:
343:
340:
337:
316:
311:
308:
303:
299:
296:
293:
290:
283:
282:
281:
258:
254:
251:
245:
242:
235:
234:
233:
231:
215:
206:
192:
176:
133:
131:
127:
123:
119:
115:
111:
107:
103:
99:
95:
92:
83:
73:
70:
62:
52:
48:
42:
41:
35:
30:
21:
20:
3918:Aerodynamics
3880:
3873:
3856:
3852:
3839:
3827:. Retrieved
3820:the original
3806:
3784:
3769:the original
3764:
3729:
3724:
3721:aspect ratio
3720:
3714:
3712:
3699:
3669:
3649:
3613:
3601:
3571:
3566:
3559:
3556:
3499:
3300:
3293:
3282:
3271:
3248:
3233:
3212:
3181:Haack series
3117:
3020:above about
2995:
2991:
2987:
2983:
2980:
2966:
2963:power series
2962:
2960:
2945:
2924:
2893:Power series
2880:
2869:
2866:
2816:
2793:
2791:
2639:
2635:power series
2631:latus rectum
2624:
2608:
2589:
2570:
2482:, this is a
2461:
2335:
2224:varies from
2211:
2020:
2018:
1954:Secant ogive
1949:
1932:
1671:
1622:
1535:varies from
1522:
1451:ogive radius
1450:
1448:
1428:
1106:Half angles:
1105:
1104:
809:
800:
791:
786:
774:
771:
671:
578:
568:
408:
275:
227:
165:varies from
144:
88:
65:
56:
37:
3674:0.8 to Mach
3653:USAF Datcom
3620:wetted area
3572:Von Kármán
3128:Power type
2347:Honest John
1523:The radius
91:aerodynamic
51:introducing
3912:Categories
3798:References
3680:Von Kármán
3645:Boeing 737
3567:Von Kármán
3553:Von Kármán
3231:LV-Haack (
3008:less than
2885:less than
2484:hemisphere
2429:Elliptical
1633:hemisphere
34:references
3898:253099252
3865:221684654
3732:wave drag
3664:transonic
3624:transonic
3592:Aerospike
3531:LV-Haack
3476:θ
3470:
3442:θ
3433:
3424:−
3421:θ
3412:π
3386:θ
3353:−
3342:
3320:θ
3139:Cylinder
3122:include:
3049:≤
3043:≤
2763:−
2722:−
2662:≤
2651:≤
2553:Parabolic
2511:−
2373:ρ
2318:α
2312:
2306:ρ
2303:−
2285:−
2279:α
2273:
2267:ρ
2261:−
2252:ρ
2179:
2173:−
2163:ρ
2123:
2114:α
2054:ρ
1897:−
1877:−
1830:−
1827:ρ
1816:−
1813:ρ
1758:−
1755:ρ
1749:−
1725:−
1722:ρ
1711:−
1608:ρ
1605:−
1581:−
1572:−
1563:ρ
1468:ρ
1370:ϕ
1363:
1344:−
1272:−
1252:
1237:ϕ
1205:ϕ
1198:
1134:
1119:ϕ
1064:−
1035:−
984:≤
978:≤
884:≤
878:≤
713:−
640:−
356:ϕ
350:
300:
291:ϕ
126:nose cone
98:nose cone
59:July 2018
3923:Rocketry
3890:60-15518
3829:11 April
3738:See also
3541:Tangent
3303:> 2/3
2972:cylinder
2770:′
2729:′
2677: :
2658:′
2627:parabola
999: :
906: :
739:Bi-conic
110:aircraft
3725:caliber
3684:n = 1/2
2876:tangent
2592:′ = 3/4
2573:′ = 1/2
2478:equals
2468:prolate
2464:ellipse
1443:tangent
783:frustum
184:⁄
114:missile
96:of the
47:improve
3896:
3888:
3863:
3676:
3672:
3516:value
3339:arccos
3134:value
2974:and a
2922:Half (
2821:value
2606:Full (
2568:Half (
2342:ρ
2338:ρ
2176:arctan
2120:arccos
2025:ρ
1982:Secant
1933:where
1674:sphere
1629:ρ
1455:ρ
1439:circle
1249:arctan
1131:arctan
569:where
415:sphere
297:arctan
278:φ
122:bullet
106:rocket
94:design
36:, but
3861:S2CID
3849:(PDF)
3823:(PDF)
3816:(PDF)
3772:(PDF)
3761:(PDF)
3574:ogive
3296:= 2/3
3285:= 1/3
3236:= 1/3
3169:Cone
3032:For
2948:= 3/4
2927:= 1/2
2872:′ = 1
2856:Full
2836:Half
2826:Cone
2611:′ = 1
1531:, as
1431:ogive
195:Conic
161:, as
118:shell
3894:OCLC
3886:LCCN
3831:2011
3616:Mach
3255:drag
2976:cone
2961:The
2802:and
2640:For
2376:<
2370:<
2232:is:
2106:and
2057:>
2039:and
2031:and
1543:is:
1311:and
1178:and
960:For
867:For
330:and
230:cone
3655:).
3570:or
3562:= 0
3545:2/3
3535:1/3
3461:sin
3430:sin
3274:= 0
3215:= 0
3163:3/4
3153:1/2
3022:0.7
2850:3/4
2840:1/2
2309:sin
2270:cos
2228:to
2220:as
2021:not
1539:to
1360:tan
1195:tan
347:tan
120:or
108:or
3914::
3892:.
3855:.
3851:.
3788:.
3763:.
3577:.
3289:LV
3287:,
3278:LD
3064::
2986:=
2486:.
2349:.
1635:.
1453:,
816:.
280::
116:,
112:,
3900:.
3867:.
3857:7
3833:.
3560:C
3525:0
3514:C
3502:C
3479:)
3473:(
3465:3
3457:C
3454:+
3449:2
3445:)
3439:2
3436:(
3407:R
3402:=
3395:)
3392:C
3389:,
3383:(
3380:y
3372:)
3366:L
3362:x
3359:2
3350:1
3346:(
3336:=
3329:)
3326:x
3323:(
3301:C
3294:C
3283:C
3272:C
3267:C
3263:C
3238:)
3234:C
3217:)
3213:C
3173:1
3143:0
3132:n
3120:n
3101:n
3096:)
3091:L
3088:x
3083:(
3078:R
3075:=
3072:y
3052:1
3046:n
3040:0
3026:n
3018:n
3014:n
3010:1
3006:n
3002:x
2998:)
2996:L
2994:/
2992:x
2990:(
2988:R
2984:y
2950:)
2946:n
2929:)
2925:n
2887:1
2883:′
2881:K
2870:K
2860:1
2830:0
2819:′
2817:K
2804:1
2800:0
2796:′
2794:K
2777:)
2767:K
2760:2
2753:2
2748:)
2743:L
2740:x
2735:(
2726:K
2718:)
2713:L
2710:x
2705:(
2701:2
2695:(
2691:R
2688:=
2685:y
2665:1
2655:K
2648:0
2613:)
2609:K
2594:)
2590:K
2575:)
2571:K
2531:2
2527:L
2521:2
2517:x
2508:1
2503:R
2500:=
2497:y
2480:L
2476:R
2410:R
2407:2
2400:2
2396:L
2392:+
2387:2
2383:R
2365:2
2362:L
2321:)
2315:(
2296:2
2292:)
2288:x
2282:)
2276:(
2264:(
2256:2
2246:=
2243:y
2230:L
2226:0
2222:x
2218:x
2214:y
2196:)
2191:L
2188:R
2183:(
2169:)
2160:2
2152:2
2148:R
2144:+
2139:2
2135:L
2127:(
2117:=
2091:R
2088:2
2081:2
2077:L
2073:+
2068:2
2064:R
2041:L
2037:R
2033:L
2029:R
1944:o
1942:x
1937:n
1935:r
1910:2
1905:t
1901:y
1892:2
1887:n
1883:r
1872:o
1868:x
1864:=
1855:t
1851:x
1838:n
1834:r
1822:)
1819:R
1810:(
1805:n
1801:r
1794:=
1785:t
1781:y
1769:2
1765:)
1761:R
1752:(
1744:2
1739:)
1733:n
1729:r
1718:(
1708:L
1705:=
1696:o
1692:x
1625:L
1602:R
1599:+
1592:2
1588:)
1584:x
1578:L
1575:(
1567:2
1557:=
1554:y
1541:L
1537:0
1533:x
1529:x
1525:y
1505:R
1502:2
1495:2
1491:L
1487:+
1482:2
1478:R
1471:=
1379:)
1374:2
1366:(
1357:)
1352:1
1348:L
1341:x
1338:(
1335:+
1330:1
1326:R
1322:=
1319:y
1298:)
1291:2
1287:L
1280:1
1276:R
1267:2
1263:R
1256:(
1246:=
1241:2
1214:)
1209:1
1201:(
1192:x
1189:=
1186:y
1165:)
1158:1
1154:L
1148:1
1144:R
1138:(
1128:=
1123:1
1086:2
1082:L
1077:)
1072:1
1068:R
1059:2
1055:R
1051:(
1048:)
1043:1
1039:L
1032:x
1029:(
1023:+
1018:1
1014:R
1010:=
1007:y
987:L
981:x
973:1
969:L
942:1
938:L
931:1
927:R
923:x
917:=
914:y
892:1
888:L
881:x
875:0
851:2
847:L
843:+
838:1
834:L
830:=
827:L
813:2
810:R
804:1
801:R
795:2
792:L
778:1
775:L
721:n
717:r
708:o
704:x
700:=
695:a
691:x
676:a
674:x
653:2
648:t
644:y
635:2
630:n
626:r
620:+
615:t
611:x
607:=
602:o
598:x
583:o
581:x
573:n
571:r
552:L
548:R
543:t
539:x
532:=
527:t
523:y
495:2
491:L
487:+
482:2
478:R
471:2
466:n
462:r
453:R
448:2
444:L
438:=
433:t
429:x
359:)
353:(
344:x
341:=
338:y
317:)
312:L
309:R
304:(
294:=
259:L
255:R
252:x
246:=
243:y
187:L
181:C
171:L
167:0
163:x
159:x
155:y
151:R
147:L
72:)
66:(
61:)
57:(
43:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
