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Bertrand–Diguet–Puiseux theorem

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380: 290: 111: 421: 445: 440: 414: 450: 407: 362: 321: 35: 285:{\displaystyle K(p)=\lim _{r\to 0^{+}}3{\frac {2\pi r-C(r)}{\pi r^{3}}}=\lim _{r\to 0^{+}}12{\frac {\pi r^{2}-A(r)}{\pi r^{4}}}.} 296: 105:) denote the area of the disc contained within the circle. The Bertrand–Diguet–Puiseux theorem asserts that 387: 43: 17: 358: 317: 331: 55: 27:
Gives the Gaussian curvature of a surface from the length of a geodesic circle or its area
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circle, or the area of a geodesic disc. The theorem is named for
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A comprehensive introduction to differential geometry, Volume II
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is the set of all points whose geodesic distance from
114: 284: 205: 131: 97:) denote the circumference of this circle, and 330:Bertrand, J; Diguet, C.F.; Puiseux, V (1848), 415: 8: 422: 408: 270: 240: 230: 219: 208: 192: 156: 145: 134: 113: 314:A Panoramic View of Riemannian Geometry 332:"Démonstration d'un théorème de Gauss" 295:The theorem is closely related to the 7: 376: 374: 25: 446:Theorems in differential geometry 441:Differential geometry of surfaces 36:differential geometry of surfaces 378: 73:. The geodesic circle of radius 69:be a point on a smooth surface 62:, and Charles François Diguet. 40:Bertrand–Diguet–Puiseux theorem 18:Bertrand–Diquet–Puiseux theorem 258: 252: 212: 180: 174: 138: 124: 118: 1: 46:of a surface in terms of the 394:. You can help Knowledge by 451:Differential geometry stubs 357:, Publish or Perish Press, 467: 373: 339:Journal de Mathématiques 390:-related article is a 286: 388:differential geometry 287: 297:Gauss–Bonnet theorem 112: 316:, Springer-Verlag, 282: 226: 152: 44:Gaussian curvature 403: 402: 277: 204: 199: 130: 85:is equal to  16:(Redirected from 458: 424: 417: 410: 382: 375: 367: 346: 336: 326: 291: 289: 288: 283: 278: 276: 275: 274: 261: 245: 244: 231: 225: 224: 223: 200: 198: 197: 196: 183: 157: 151: 150: 149: 21: 466: 465: 461: 460: 459: 457: 456: 455: 431: 430: 429: 428: 371: 365: 351:Spivak, Michael 349: 334: 329: 324: 308: 305: 266: 262: 236: 232: 215: 188: 184: 158: 141: 110: 109: 56:Joseph Bertrand 28: 23: 22: 15: 12: 11: 5: 464: 462: 454: 453: 448: 443: 433: 432: 427: 426: 419: 412: 404: 401: 400: 383: 369: 368: 363: 347: 327: 322: 310:Berger, Marcel 304: 301: 293: 292: 281: 273: 269: 265: 260: 257: 254: 251: 248: 243: 239: 235: 229: 222: 218: 214: 211: 207: 203: 195: 191: 187: 182: 179: 176: 173: 170: 167: 164: 161: 155: 148: 144: 140: 137: 133: 129: 126: 123: 120: 117: 60:Victor Puiseux 42:expresses the 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 463: 452: 449: 447: 444: 442: 439: 438: 436: 425: 420: 418: 413: 411: 406: 405: 399: 397: 393: 389: 384: 381: 377: 372: 366: 364:0-914098-71-3 360: 356: 352: 348: 344: 340: 333: 328: 325: 323:3-540-65317-1 319: 315: 311: 307: 306: 302: 300: 298: 279: 271: 267: 263: 255: 249: 246: 241: 237: 233: 227: 220: 216: 209: 201: 193: 189: 185: 177: 171: 168: 165: 162: 159: 153: 146: 142: 135: 127: 121: 115: 108: 107: 106: 104: 100: 96: 92: 88: 84: 80: 76: 72: 68: 63: 61: 57: 53: 49: 48:circumference 45: 41: 37: 34:study of the 33: 19: 396:expanding it 385: 370: 354: 342: 338: 313: 294: 102: 98: 94: 90: 86: 82: 78: 77:centered at 74: 70: 66: 64: 39: 32:mathematical 29: 435:Categories 303:References 264:π 247:− 234:π 213:→ 186:π 169:− 163:π 139:→ 353:(1999), 312:(2004), 52:geodesic 345:: 80–90 30:In the 361:  320:  89:. Let 38:, the 386:This 335:(PDF) 50:of a 392:stub 359:ISBN 318:ISBN 65:Let 206:lim 132:lim 437:: 343:13 341:, 337:, 299:. 228:12 58:, 423:e 416:t 409:v 398:. 280:. 272:4 268:r 259:) 256:r 253:( 250:A 242:2 238:r 221:+ 217:0 210:r 202:= 194:3 190:r 181:) 178:r 175:( 172:C 166:r 160:2 154:3 147:+ 143:0 136:r 128:= 125:) 122:p 119:( 116:K 103:r 101:( 99:A 95:r 93:( 91:C 87:r 83:p 79:p 75:r 71:M 67:p 20:)

Index

Bertrand–Diquet–Puiseux theorem
mathematical
differential geometry of surfaces
Gaussian curvature
circumference
geodesic
Joseph Bertrand
Victor Puiseux
Gauss–Bonnet theorem
Berger, Marcel
ISBN
3-540-65317-1
"Démonstration d'un théorème de Gauss"
Spivak, Michael
ISBN
0-914098-71-3
Stub icon
differential geometry
stub
expanding it
v
t
e
Categories
Differential geometry of surfaces
Theorems in differential geometry
Differential geometry stubs

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