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Serpentine curve

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29: 490: 530: 178: 356: 284: 225: 388: 459: 422: 58: 571: 103: 80: 51: 295: 470: 564: 230: 516: 41: 45: 37: 595: 189: 557: 62: 590: 364: 427: 393: 545: 8: 474: 511: 17: 541: 584: 16:
This article is about the mathematical concept. For the design feature, see
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The curve has an inflection point at the origin. It has local extrema at
537: 489: 529: 430: 396: 367: 298: 233: 192: 106: 453: 416: 382: 350: 278: 219: 183:Equivalently, it has a parametric representation 173:{\displaystyle x^{2}y+a^{2}y-abx=0,\quad ab>0.} 172: 582: 50:but its sources remain unclear because it lacks 565: 351:{\displaystyle y={\frac {abx}{x^{2}+a^{2}}}.} 572: 558: 97:is a curve whose equation is of the form 81:Learn how and when to remove this message 488: 583: 524: 484: 22: 13: 469:Serpentine curves were studied by 279:{\displaystyle y=b\sin(t)\cos(t),} 14: 607: 505: 528: 27: 512:MathWorld – Serpentine Equation 157: 477:, and named and classified by 270: 264: 255: 249: 214: 208: 1: 289:or functional representation 544:. You can help Knowledge by 7: 10: 612: 523: 464: 390:, with a maximum value of 220:{\displaystyle x=a\cot(t)} 15: 493:The serpentine curve for 36:This article includes a 424:and a minimum value of 383:{\displaystyle x=\pm a} 65:more precise citations. 502: 455: 454:{\displaystyle y=-b/2} 418: 384: 352: 280: 221: 174: 492: 456: 419: 417:{\displaystyle y=b/2} 385: 353: 281: 222: 175: 428: 394: 365: 296: 231: 190: 104: 503: 451: 414: 380: 348: 276: 217: 170: 38:list of references 553: 552: 485:Visual appearance 343: 91: 90: 83: 603: 574: 567: 560: 538:geometry-related 532: 525: 460: 458: 457: 452: 447: 423: 421: 420: 415: 410: 389: 387: 386: 381: 357: 355: 354: 349: 344: 342: 341: 340: 328: 327: 317: 306: 285: 283: 282: 277: 226: 224: 223: 218: 179: 177: 176: 171: 132: 131: 116: 115: 95:serpentine curve 86: 79: 75: 72: 66: 61:this article by 52:inline citations 31: 30: 23: 18:Serpentine shape 611: 610: 606: 605: 604: 602: 601: 600: 581: 580: 579: 578: 521: 508: 501: = 1. 487: 467: 443: 429: 426: 425: 406: 395: 392: 391: 366: 363: 362: 336: 332: 323: 319: 318: 307: 305: 297: 294: 293: 232: 229: 228: 191: 188: 187: 127: 123: 111: 107: 105: 102: 101: 87: 76: 70: 67: 56: 42:related reading 32: 28: 21: 12: 11: 5: 609: 599: 598: 596:Geometry stubs 593: 577: 576: 569: 562: 554: 551: 550: 533: 519: 518: 514: 507: 506:External links 504: 486: 483: 466: 463: 450: 446: 442: 439: 436: 433: 413: 409: 405: 402: 399: 379: 376: 373: 370: 359: 358: 347: 339: 335: 331: 326: 322: 316: 313: 310: 304: 301: 287: 286: 275: 272: 269: 266: 263: 260: 257: 254: 251: 248: 245: 242: 239: 236: 216: 213: 210: 207: 204: 201: 198: 195: 181: 180: 169: 166: 163: 160: 156: 153: 150: 147: 144: 141: 138: 135: 130: 126: 122: 119: 114: 110: 89: 88: 46:external links 35: 33: 26: 9: 6: 4: 3: 2: 608: 597: 594: 592: 589: 588: 586: 575: 570: 568: 563: 561: 556: 555: 549: 547: 543: 540:article is a 539: 534: 531: 527: 526: 522: 517: 515: 513: 510: 509: 500: 497: =  496: 491: 482: 480: 476: 472: 462: 448: 444: 440: 437: 434: 431: 411: 407: 403: 400: 397: 377: 374: 371: 368: 345: 337: 333: 329: 324: 320: 314: 311: 308: 302: 299: 292: 291: 290: 273: 267: 261: 258: 252: 246: 243: 240: 237: 234: 211: 205: 202: 199: 196: 193: 186: 185: 184: 167: 164: 161: 158: 154: 151: 148: 145: 142: 139: 136: 133: 128: 124: 120: 117: 112: 108: 100: 99: 98: 96: 85: 82: 74: 64: 60: 54: 53: 47: 43: 39: 34: 25: 24: 19: 591:Cubic curves 546:expanding it 535: 520: 498: 494: 468: 360: 288: 182: 94: 92: 77: 68: 57:Please help 49: 63:introducing 585:Categories 471:L'HĂ´pital 438:− 375:± 262:⁡ 247:⁡ 206:⁡ 137:− 71:May 2024 475:Huygens 465:History 59:improve 479:Newton 536:This 44:, or 542:stub 473:and 165:> 259:cos 244:sin 203:cot 587:: 481:. 461:. 227:, 168:0. 93:A 48:, 40:, 573:e 566:t 559:v 548:. 499:b 495:a 449:2 445:/ 441:b 435:= 432:y 412:2 408:/ 404:b 401:= 398:y 378:a 372:= 369:x 346:. 338:2 334:a 330:+ 325:2 321:x 315:x 312:b 309:a 303:= 300:y 274:, 271:) 268:t 265:( 256:) 253:t 250:( 241:b 238:= 235:y 215:) 212:t 209:( 200:a 197:= 194:x 162:b 159:a 155:, 152:0 149:= 146:x 143:b 140:a 134:y 129:2 125:a 121:+ 118:y 113:2 109:x 84:) 78:( 73:) 69:( 55:. 20:.

Index

Serpentine shape
list of references
related reading
external links
inline citations
improve
introducing
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L'HĂ´pital
Huygens
Newton

MathWorld – Serpentine Equation

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geometry-related
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expanding it
v
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e
Categories
Cubic curves
Geometry stubs

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