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Limiting point (geometry)

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17: 320: 134:. The limiting points themselves can be found at this distance on either side of the intersection point, on the line through the two circle centers. From this fact it is straightforward to construct the limiting points algebraically or by 153:
produces the other limiting point. An inversion centered at one limiting point maps the other limiting point to the common center of the concentric circles.
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This follows from the pencil definition, together with the fact that every pencil has a unique orthogonal pencil; see
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The two points where the red circles cross are the limiting points of each pair of blue circles
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in the coordinates of the circle centers and their radii is given by Weisstein.
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crosses the line through their centers. This intersection point has equal
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that may be defined by any of the following equivalent properties:
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The midpoint of the two limiting points is the point where the
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contains a degenerate (radius zero) circle centered at 
263: 145:Inverting one of the two limiting points through 372: 288: 355: 209: 188: 126:to all the circles in the pencil containing 362: 348: 221: 169: 15: 173:A treatise on the circle and the sphere 373: 295:, University Press, Ex. 473, p. 109, 264: 314: 289:Godfrey, C.; Siddons, A. W. (1908), 176:, Oxford Clarendon Press, p. 97 13: 14: 402: 318: 170:Coolidge, Julian Lowell (1916), 231:Computer Aided Geometric Design 282: 257: 215: 203: 181: 163: 1: 212:, Example 2, p. 32. 156: 67:Every circle or line that is 334:. You can help Knowledge by 243:10.1016/0167-8396(93)90049-9 7: 222:Johnstone, John K. (1993), 195:Geometry of Complex Numbers 10: 407: 313: 391:Elementary geometry stubs 200:, Corollary, p. 31. 136:compass and straightedge 28:of two disjoint circles 330:-related article is a 21: 19: 210:Schwerdtfeger (1979) 328:elementary geometry 190:Schwerdtfeger, Hans 386:Inversive geometry 266:Weisstein, Eric W. 140:quadratic equation 22: 343: 342: 50:pencil of circles 24:In geometry, the 398: 364: 357: 350: 322: 315: 305: 303: 286: 280: 279: 278: 269:"Limiting Point" 261: 255: 253: 228: 219: 213: 207: 201: 199: 185: 179: 177: 167: 406: 405: 401: 400: 399: 397: 396: 395: 371: 370: 369: 368: 311: 309: 308: 292:Modern Geometry 287: 283: 262: 258: 226: 220: 216: 208: 204: 186: 182: 168: 164: 159: 79:passes through 38:Euclidean plane 26:limiting points 12: 11: 5: 404: 394: 393: 388: 383: 367: 366: 359: 352: 344: 341: 340: 323: 307: 306: 281: 256: 214: 202: 180: 161: 160: 158: 155: 124:power distance 108: 107: 84: 65: 9: 6: 4: 3: 2: 403: 392: 389: 387: 384: 382: 379: 378: 376: 365: 360: 358: 353: 351: 346: 345: 339: 337: 333: 329: 324: 321: 317: 316: 312: 302: 298: 294: 293: 285: 276: 275: 270: 267: 260: 252: 248: 244: 240: 236: 232: 225: 218: 211: 206: 197: 196: 191: 184: 175: 174: 166: 162: 154: 152: 148: 143: 141: 137: 133: 129: 125: 121: 117: 113: 105: 101: 97: 93: 89: 85: 82: 78: 74: 70: 69:perpendicular 66: 63: 59: 55: 51: 47: 46: 45: 43: 39: 35: 31: 27: 18: 336:expanding it 325: 310: 291: 284: 272: 259: 234: 230: 217: 205: 194: 183: 172: 165: 150: 146: 144: 131: 127: 119: 115: 112:radical axis 109: 99: 95: 91: 90:centered at 80: 76: 72: 61: 57: 53: 41: 33: 29: 25: 23: 237:(1): 1–24, 94:transforms 52:defined by 40:are points 375:Categories 157:References 104:concentric 274:MathWorld 88:inversion 301:6525169M 192:(1979), 106:circles. 71:to both 381:Circles 251:1202965 198:, Dover 36:in the 299:  249:  326:This 227:(PDF) 102:into 332:stub 130:and 118:and 98:and 75:and 56:and 48:The 32:and 239:doi 149:or 114:of 86:An 377:: 297:OL 271:. 247:MR 245:, 235:10 233:, 229:, 363:e 356:t 349:v 338:. 304:. 277:. 254:. 241:: 178:. 151:B 147:A 132:B 128:A 120:B 116:A 100:B 96:A 92:p 83:. 81:p 77:B 73:A 64:. 62:p 58:B 54:A 42:p 34:B 30:A

Index


Euclidean plane
pencil of circles
perpendicular
inversion
concentric
radical axis
power distance
compass and straightedge
quadratic equation
A treatise on the circle and the sphere
Schwerdtfeger, Hans
Geometry of Complex Numbers
Schwerdtfeger (1979)
"A new intersection algorithm for cyclides and swept surfaces using circle decomposition"
doi
10.1016/0167-8396(93)90049-9
MR
1202965
Weisstein, Eric W.
"Limiting Point"
MathWorld
Modern Geometry
OL
6525169M
Stub icon
elementary geometry
stub
expanding it
v

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