@xtor/cga.js
Version:
Xtor Compute Geometry Algorithm Libary 计算几何算法库
601 lines • 151 kB
HTML
<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
<title>1.点与线性元素的距离</title><link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; }
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<div id='write' class = 'is-node'><h1><a name="第一章-点与线性元素的距离" class="md-header-anchor"></a><span>第一章 点与线性元素的距离</span></h1><p><span>基于 js 开源项目：计算机几何算法库 Compute Geometry Algorithm（CGA），开源地址：即将到来</span></p><h2><a name="点与线" class="md-header-anchor"></a><span>点与线</span></h2><blockquote><p><span>点由z,y,z三个坐标组成，可以表示为</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.346ex" height="2.577ex" viewBox="0 -806.1 5746.3 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E8-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="0" id="E8-MJMATHI-6F" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></path><path stroke-width="0" id="E8-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E8-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E8-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E8-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E8-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E8-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E8-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 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13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path stroke-width="0" id="E8-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E8-MJMATHI-50" x="0" y="0"></use><use xlink:href="#E8-MJMATHI-6F" x="751" y="0"></use><use xlink:href="#E8-MJMATHI-69" x="1236" y="0"></use><use xlink:href="#E8-MJMATHI-6E" x="1581" y="0"></use><use xlink:href="#E8-MJMATHI-74" x="2181" y="0"></use><use xlink:href="#E8-MJMAIN-28" x="2542" y="0"></use><use xlink:href="#E8-MJMATHI-78" x="2931" y="0"></use><use xlink:href="#E8-MJMAIN-2C" x="3503" y="0"></use><use xlink:href="#E8-MJMATHI-79" x="3947" y="0"></use><use xlink:href="#E8-MJMAIN-2C" x="4444" y="0"></use><use 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46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path stroke-width="0" id="E14-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E14-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E14-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E14-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E14-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 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98L298 109L352 326Z"></path><path stroke-width="0" id="E14-MJMAIN-20D7" d="M-123 694Q-123 702 -118 708T-103 714Q-93 714 -88 706T-80 687T-67 660T-40 633Q-29 626 -29 615Q-29 606 -36 600T-53 590T-83 571T-121 531Q-135 516 -143 516T-157 522T-163 536T-152 559T-129 584T-116 595H-287L-458 596Q-459 597 -461 599T-466 602T-469 607T-471 615Q-471 622 -458 635H-99Q-123 673 -123 694Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E14-MJMATHI-4C" x="0" y="0"></use><use xlink:href="#E14-MJMAIN-28" x="681" y="0"></use><use xlink:href="#E14-MJMATHI-74" x="1070" y="0"></use><use xlink:href="#E14-MJMAIN-29" x="1431" y="0"></use><use xlink:href="#E14-MJMAIN-3D" x="2097" y="0"></use><use xlink:href="#E14-MJMATHI-50" x="3153" y="0"></use><use xlink:href="#E14-MJMAIN-2B" x="4126" y="0"></use><use xlink:href="#E14-MJMATHI-74" x="5127" y="0"></use><g transform="translate(5488,0)"><use xlink:href="#E14-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E14-MJMAIN-20D7" x="665" y="253"></use></g></g></svg></span><script type="math/tex">L(t)=P+t\vec{d}</script><span>, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.744ex" height="1.877ex" viewBox="0 -755.9 751 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E18-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E18-MJMATHI-50" x="0" y="0"></use></g></svg></span><script type="math/tex">P</script><span>是直线上的一点，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.547ex" height="2.694ex" viewBox="0 -1057.4 666 1160" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E25-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E25-MJMAIN-20D7" d="M-123 694Q-123 702 -118 708T-103 714Q-93 714 -88 706T-80 687T-67 660T-40 633Q-29 626 -29 615Q-29 606 -36 600T-53 590T-83 571T-121 531Q-135 516 -143 516T-157 522T-163 536T-152 559T-129 584T-116 595H-287L-458 596Q-459 597 -461 599T-466 602T-469 607T-471 615Q-471 622 -458 635H-99Q-123 673 -123 694Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E25-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E25-MJMAIN-20D7" x="665" y="253"></use></g></svg></span><script type="math/tex">\vec{d}</script><span>是直线的方向。</span>
<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.763ex" height="1.877ex" viewBox="0 -755.9 759 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E12-MJMATHI-42" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E12-MJMATHI-42" x="0" y="0"></use></g></svg></span><script type="math/tex">B</script><span>是直线上任意的一点</span></p><p><img src="./img/直线的表达式.png" referrerpolicy="no-referrer" alt="avatar" title="直线的表达式"></p></blockquote><h2><a name="点到直线" class="md-header-anchor"></a><span>点到直线</span></h2><blockquote><p><span>假设我们有点</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.837ex" height="2.461ex" viewBox="0 -806.1 791 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E37-MJMATHI-51" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E37-MJMATHI-51" x="0" y="0"></use></g></svg></span><script type="math/tex">Q</script><span>和直线</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.293ex" height="3.161ex" viewBox="0 -1057.4 6154 1361" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E14-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path stroke-width="0" id="E14-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E14-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E14-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E14-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E14-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="0" id="E14-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E14-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E14-MJMAIN-20D7" d="M-123 694Q-123 702 -118 708T-103 714Q-93 714 -88 706T-80 687T-67 660T-40 633Q-29 626 -29 615Q-29 606 -36 600T-53 590T-83 571T-121 531Q-135 516 -143 516T-157 522T-163 536T-152 559T-129 584T-116 595H-287L-458 596Q-459 597 -461 599T-466 602T-469 607T-471 615Q-471 622 -458 635H-99Q-123 673 -123 694Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E14-MJMATHI-4C" x="0" y="0"></use><use xlink:href="#E14-MJMAIN-28" x="681" y="0"></use><use xlink:href="#E14-MJMATHI-74" x="1070" y="0"></use><use xlink:href="#E14-MJMAIN-29" x="1431" y="0"></use><use xlink:href="#E14-MJMAIN-3D" x="2097" y="0"></use><use xlink:href="#E14-MJMATHI-50" x="3153" y="0"></use><use xlink:href="#E14-MJMAIN-2B" x="4126" y="0"></use><use xlink:href="#E14-MJMATHI-74" x="5127" y="0"></use><g transform="translate(5488,0)"><use xlink:href="#E14-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E14-MJMAIN-20D7" x="665" y="253"></use></g></g></svg></span><script type="math/tex">L(t)=P+t\vec{d}</script><span>,我们要找出点</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.837ex" height="2.461ex" viewBox="0 -806.1 791 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E37-MJMATHI-51" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E37-MJMATHI-51" x="0" y="0"></use></g></svg></span><script type="math/tex">Q</script><span>与</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.582ex" height="1.994ex" viewBox="0 -755.9 681 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E38-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E38-MJMATHI-4C" x="0" y="0"></use></g></svg></span><script type="math/tex">L</script><span>的最短距离，如果</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.582ex" height="1.994ex" viewBox="0 -755.9 681 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E38-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E38-MJMATHI-4C" x="0" y="0"></use></g></svg></span><script type="math/tex">L</script><span>上离</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.744ex" height="1.877ex" viewBox="0 -755.9 751 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E18-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E18-MJMATHI-50" x="0" y="0"></use></g></svg></span><script type="math/tex">P</script><span>最近的点</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.521ex" height="2.577ex" viewBox="0 -856.4 1085.5 1109.7" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E20-MJMATHI-51" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path><path stroke-width="0" id="E20-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E20-MJMATHI-51" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E20-MJMAIN-2032" x="1118" y="513"></use></g></svg></span><script type="math/tex">Q^\prime</script><span> ，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.521ex" height="2.577ex" viewBox="0 -856.4 1085.5 1109.7" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E20-MJMATHI-51" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path><path stroke-width="0" id="E20-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E20-MJMATHI-51" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E20-MJMAIN-2032" x="1118" y="513"></use></g></svg></span><script type="math/tex">Q^\prime</script><span> 在直线上，此时</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.828ex" height="2.227ex" viewBox="0 -705.6 2509.1 958.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E21-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E21-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E21-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E21-MJMATHI-74" x="0" y="0"></use><use xlink:href="#E21-MJMAIN-3D" x="638" y="0"></use><g transform="translate(1694,0)"><use xlink:href="#E21-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E21-MJMAIN-30" x="510" y="-213"></use></g></g></svg></span><script type="math/tex">t=t_0</script></p><p><img src="img/点与直线.png" referrerpolicy="no-referrer" alt="avatar" title="点与直线图示"><span> </span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.892ex" height="2.227ex" viewBox="0 -705.6 814.6 958.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E31-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E31-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E31-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E31-MJMAIN-30" x="510" y="-213"></use></g></svg></span><script type="math/tex">t_0</script><span>满足：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n14" cid="n14" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div 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x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E44-MJMAIN-30" x="510" y="-213"></use><use xlink:href="#E44-MJMAIN-3D" x="1092" y="0"></use><g transform="translate(1870,0)"><g transform="translate(397,0)"><rect stroke="none" width="5050" height="60" x="0" y="220"></rect><g transform="translate(60,715)"><use xlink:href="#E44-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E44-MJMAIN-20D7" x="665" y="253"></use><use xlink:href="#E44-MJMAIN-22C5" x="888" y="0"></use><use xlink:href="#E44-MJMAIN-28" x="1388" y="0"></use><use xlink:href="#E44-MJMATHI-51" x="1777" y="0"></use><use xlink:href="#E44-MJMAIN-2212" x="2790" y="0"></use><use xlink:href="#E44-MJMATHI-50" x="3790" y="0"></use><use xlink:href="#E44-MJMAIN-29" x="4541" y="0"></use></g><g transform="translate(1498,-934)"><use xlink:href="#E44-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E44-MJMAIN-20D7" x="665" y="253"></use><use xlink:href="#E44-MJMAIN-22C5" x="888" y="0"></use><g transform="translate(1388,0)"><use 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transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E45-MJMATHI-51" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E45-MJMAIN-2032" x="1118" y="583"></use><use xlink:href="#E45-MJMAIN-3D" x="1363" y="0"></use><use xlink:href="#E45-MJMATHI-50" x="2419" y="0"></use><use xlink:href="#E45-MJMAIN-2B" x="3392" y="0"></use><g transform="translate(4392,0)"><use xlink:href="#E45-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E45-MJMAIN-30" x="510" y="-213"></use></g><g transform="translate(5207,0)"><use xlink:href="#E45-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E45-MJMAIN-20D7" x="665" y="253"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">Q^\prime = P+t_0\vec{d}</script></div></div><p><span>那么点 Q 到直线 L 的距离为 d，说明:</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.651ex" height="2.577ex" viewBox="0 -806.1 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406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E24-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script><span>向量的长度(范数)</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n18" cid="n18" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span 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xlink:href="#E46-MJMATHI-51" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E46-MJMAIN-2032" x="1118" y="583"></use></g><use xlink:href="#E46-MJMAIN-2225" x="4932" y="0"></use></g><g transform="translate(0,-789)"><use xlink:href="#E46-MJMAIN-3D" x="277" y="0"></use><use xlink:href="#E46-MJMAIN-2225" x="1333" y="0"></use><use xlink:href="#E46-MJMATHI-51" x="1833" y="0"></use><use xlink:href="#E46-MJMAIN-2212" x="2846" y="0"></use><use xlink:href="#E46-MJMAIN-28" x="3847" y="0"></use><use xlink:href="#E46-MJMATHI-50" x="4236" y="0"></use><use xlink:href="#E46-MJMAIN-2B" x="5209" y="0"></use><g transform="translate(6209,0)"><use xlink:href="#E46-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E46-MJMAIN-30" x="510" y="-213"></use></g><g transform="translate(7023,0)"><use xlink:href="#E46-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E46-MJMAIN-20D7" x="665" y="253"></use></g><use xlink:href="#E46-MJMAIN-29" x="7689" y="0"></use><use xlink:href="#E46-MJMAIN-2225" x="8078" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">\begin{aligned}
d &= \|Q-Q^\prime\|\\
&=\|Q-(P+t_0\vec{d})\|
\end{aligned}</script></div></div><p><span>当</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.547ex" height="2.694ex" viewBox="0 -1057.4 666 1160" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E25-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 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536T-152 559T-129 584T-116 595H-287L-458 596Q-459 597 -461 599T-466 602T-469 607T-471 615Q-471 622 -458 635H-99Q-123 673 -123 694Z"></path><path stroke-width="0" id="E26-MJMAIN-2225" d="M133 736Q138 750 153 750Q164 750 170 739Q172 735 172 250T170 -239Q164 -250 152 -250Q144 -250 138 -244L137 -243Q133 -241 133 -179T132 250Q132 731 133 736ZM329 739Q334 750 346 750Q353 750 361 744L362 743Q366 741 366 679T367 250T367 -178T362 -243L361 -244Q355 -250 347 -250Q335 -250 329 -239Q327 -235 327 250T329 739Z"></path><path stroke-width="0" id="E26-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E26-MJMAIN-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path stroke-width="0" id="E26-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 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xlink:href="#E26-MJMAIN-3D" x="5387" y="0"></use><use xlink:href="#E26-MJMAIN-31" x="6443" y="0"></use></g></svg></span><script type="math/tex">||\vec{d}\|=\vec{d}\cdot\vec{d}=1</script><span>,那么</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n20" cid="n20" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-4-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.442ex" height="3.161ex" viewBox="0 -1057.4 7079 1361" role="img" focusable="false" style="vertical-align: -0.705ex; max-width: 100%;"><defs><path stroke-width="0" id="E47-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E47-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E47-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E47-MJMATHI-64" d="M366 683Q367 683 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144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path><path stroke-width="0" id="E47-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E47-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="0" id="E47-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E47-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E47-MJMAIN-30" x="510" y="-213"></use><use xlink:href="#E47-MJMAIN-3D" x="1092" y="0"></use><g transform="translate(2148,0)"><use xlink:href="#E47-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E47-MJMAIN-20D7" x="665" y="253"></use></g><use xlink:href="#E47-MJMAIN-22C5" x="3036" y="0"></use><use xlink:href="#E47-MJMAIN-28" x="3536" y="0"></use><use xlink:href="#E47-MJMATHI-51" x="3925" y="0"></use><use xlink:href="#E47-MJMAIN-2212" x="4938" y="0"></use><use xlink:href="#E47-MJMATHI-50" x="5938" y="0"></use><use xlink:href="#E47-MJMAIN-29" x="6689" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-4">t_0 =\vec{d}\cdot(Q-P)</script></div></div><p><span>伪代码如下，JavaScript：</span></p></blockquote><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang="javascript" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="javascript"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 8px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><span><span></span>x</span></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">/**</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* 点到直线的距离</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* @param {Point} point</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* @param {Line} line</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* @returns</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* {</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* lineParameter 最近点的参数</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* lineCloset 最近点</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* distanceSqr //到最近点距离的平方</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* distance//到最近点距离</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">* }</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">*/</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">function</span> <span class="cm-def">distancePointLine</span>(<span class="cm-def">point，line</span>) {</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">var</span> <span class="cm-def">result</span> <span class="cm-operator">=</span> {};</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">var</span> <span class="cm-def">diff</span> <span class="cm-operator">=</span> <span class="cm-variable">point</span>.<span class="cm-property">clone</span>().<span class="cm-property">sub</span>(<span class="cm-variable">line</span>.<span class="cm-property">origin</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable-2">result</span>.<span class="cm-property">lineParameter</span> <span class="cm-operator">=</span> <span class="cm-variable">line</span>.<span class="cm-property">direction</span>.<span class="cm-property">dot</span>(<span class="cm-variable-2">diff</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable-2">result</span>.<span class="cm-property">lineCloset</span> <span class="cm-operator">=</span> <span class="cm-variable">line</span>.<span class="cm-property">direction</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> .<span class="cm-property">clone</span>()</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> .<span class="cm-property">multiplyScalar</span>(<span class="cm-variable-2">result</span>.<span class="cm-property">lineParameter</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> .<span class="cm-property">add</span>(<span class="cm-variable">line</span>.<span class="cm-property">origin</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable-2">diff</span> <span class="cm-operator">=</span> <span class="cm-variable">point</span>.<span class="cm-property">clone</span>().<span class="cm-property">sub</span>(<span class="cm-variable-2">result</span>.<span class="cm-property">lineCloset</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable-2">result</span>.<span class="cm-property">distanceSqr</span> <span class="cm-operator">=</span> <span class="cm-variable-2">diff</span>.<span class="cm-property">dot</span>(<span class="cm-variable-2">diff</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable-2">result</span>.<span class="cm-property">distance</span> <span class="cm-operator">=</span> <span class="cm-variable">Math</span>.<span class="cm-property">sqrt</span>(<span class="cm-variable-2">result</span>.<span class="cm-property">distanceSqr</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable-2">result</span>;</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> }</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 598px;"></div><div class="CodeMirror-gutters" style="display: none; height: 598px;"></div></div></div></pre><h2><a name="点到射线和线段" class="md-header-anchor"></a><span>点到射线和线段</span></h2><ol start='' ><li><p><span>点到射线</span></p><blockquote><p><span>如果</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.582ex" height="1.994ex" viewBox="0 -755.9 681 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E38-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E38-MJMATHI-4C" x="0" y="0"></use></g></svg></span><script type="math/tex">L</script><span>是直线，上文已经详细说过. 如果</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.582ex" height="1.994ex" viewBox="0 -755.9 681 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E38-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E38-MJMATHI-4C" x="0" y="0"></use></g></svg></span><script type="math/tex">L</script><span>是射线，那么我们限制</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.838ex" height="1.877ex" viewBox="0 -705.6 361 808.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E29-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E29-MJMATHI-74" x="0" y="0"></use></g></svg></span><script type="math/tex">t</script><span>为非负数</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.904ex" height="1.994ex" viewBox="0 -755.9 2972.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E30-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E30-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E30-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E30-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E30-MJMATHI-74" x="0" y="0"></use><g transform="translate(638,0)"><use xlink:href="#E30-MJMAIN-3E"></use><use xlink:href="#E30-MJMAIN-3D" x="778" y="0"></use></g><use xlink:href="#E30-MJMAIN-30" x="2472" y="0"></use></g></svg></span><script type="math/tex">t>=0</script><span>, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.892ex" height="2.227ex" viewBox="0 -705.6 814.6 958.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E31-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E31-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E31-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E31-MJMAIN-30" x="510" y="-213"></use></g></svg></span><script type="math/tex">t_0</script><span> 是点</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.837ex" height="2.461ex" viewBox="0 -806.1 791 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E37-MJMATHI-51" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E37-MJMATHI-51" x="0" y="0"></use></g></svg></span><script type="math/tex">Q</script><span>到射线的最近点的系数，如果</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.15ex" height="2.344ex" viewBox="0 -755.9 2648.1 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E33-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E33-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E33-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E33-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E33-MJMAIN-30" x="510" y="-213"></use><use xlink:href="#E33-MJMAIN-3E" x="1092" y="0"></use><use xlink:href="#E33-MJMAIN-30" x="2148" y="0"></use></g></svg></span><script type="math/tex">t_0>0</script><span>,那么点</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.837ex" height="2.461ex" viewBox="0 -806.1 791 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E37-MJMATHI-51" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E37-MJMATHI-51" x="0" y="0"></use></g></svg></span><script type="math/tex">Q</script><span>到最近点的的线是与</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.582ex" height="1.994ex" viewBox="0 -755.9 681 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E38-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E38-MJMATHI-4C" x="0" y="0"></use></g></svg></span><script type="math/tex">L</script><span>垂直的，但是如果 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.15ex" height="2.344ex" viewBox="0 -755.9 2648.1 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E36-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E36-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E36-MJMAIN-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E36-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E36-MJMAIN-30" x="510" y="-213"></use><use xlink:href="#E36-MJMAIN-3C" x="1092" y="0"></use><use xlink:href="#E36-MJMAIN-30" x="2148" y="0"></use></g></svg></span><script type="math/tex">t_0<0</script><span>是，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.837ex" height="2.461ex" viewBox="0 -806.1 791 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E37-MJMATHI-51" d="M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E37-MJMATHI-51" x="0" y="0"></use></g></svg></span><script type="math/tex">Q</script><span>到射线所在的直线上的最近点是在射线之外，那么最近点，就是到</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.582ex" height="1.994ex" viewBox="0 -755.9 681 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E38-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E38-MJMATHI-4C" x="0" y="0"></use></g></svg></span><script type="math/tex">L</script><span>的原点</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n29" cid="n29" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" 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xlink:href="#E48-MJMAIN-30" x="510" y="-213"></use><use xlink:href="#E48-MJMAIN-2264" x="1092" y="0"></use><use xlink:href="#E48-MJMAIN-30" x="2148" y="0"></use></g><g transform="translate(0,-784)"><use xlink:href="#E48-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E48-MJMAIN-30" x="510" y="-213"></use><use xlink:href="#E48-MJMAIN-3E" x="1092" y="0"></use><use xlink:href="#E48-MJMAIN-30" x="2148" y="0"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">d=\left\{
\begin{aligned}
&\|Q-P\| & t_0\leq0 \\
&\|Q-(P+t_0\vec{d})\| &t_0>0
\end{aligned}
\right .</script></div></div><p><span>图示如下：</span>
<img src="./img/点与射线.png" referrerpolicy="no-referrer" alt="avatar" title="点与射线"></p></blockquote></li><li><p><span>点到射线</span></p><blockquote><p><span>线段可以定义成两个端点</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.122ex" height="2.344ex" viewBox="0 -755.9 2635.8 1009.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E39-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="0" id="E39-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E39-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E39-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E39-MJMATHI-50" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E39-MJMAIN-30" x="907" y="-213"></use><use xlink:href="#E39-MJMAIN-2C" x="1095" y="0"></use><g transform="translate(1540,0)"><use xlink:href="#E39-MJMATHI-50" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E39-MJMAIN-31" x="907" y="-213"></use></g></g></svg></span><script type="math/tex">P_0,P_1</script><span>,方向可以定义为</span>
<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.572ex" height="3.044ex" viewBox="0 -1057.4 5413.1 1310.7" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E40-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E40-MJMAIN-20D7" d="M-123 694Q-123 702 -118 708T-103 714Q-93 714 -88 706T-80 687T-67 660T-40 633Q-29 626 -29 615Q-29 606 -36 600T-53 590T-83 571T-121 531Q-135 516 -143 516T-157 522T-163 536T-152 559T-129 584T-116 595H-287L-458 596Q-459 597 -461 599T-466 602T-469 607T-471 615Q-471 622 -458 635H-99Q-123 673 -123 694Z"></path><path stroke-width="0" id="E40-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E40-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="0" id="E40-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E40-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E40-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E40-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E40-MJMAIN-20D7" x="665" y="253"></use><use xlink:href="#E40-MJMAIN-3D" x="943" y="0"></use><g transform="translate(1999,0)"><use xlink:href="#E40-MJMATHI-50" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E40-MJMAIN-31" x="907" y="-213"></use></g><use xlink:href="#E40-MJMAIN-2212" x="3317" y="0"></use><g transform="translate(4317,0)"><use xlink:href="#E40-MJMATHI-50" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E40-MJMAIN-30" x="907" y="-213"></use></g></g></svg></span><script type="math/tex">\vec{d}=P_1-P_0</script>
<span>根据</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.227ex" height="2.577ex" viewBox="0 -806.1 1820 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E41-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path stroke-width="0" id="E41-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E41-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E41-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E41-MJMATHI-4C" x="0" y="0"></use><use xlink:href="#E41-MJMAIN-28" x="681" y="0"></use><use xlink:href="#E41-MJMATHI-74" x="1070" y="0"></use><use xlink:href="#E41-MJMAIN-29" x="1431" y="0"></use></g></svg></span><script type="math/tex">L(t)</script><span>的定义，两个端点可以表示为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.416ex" height="2.577ex" viewBox="0 -806.1 9220.9 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E42-MJMATHI-50" d="M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z"></path><path stroke-width="0" id="E42-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E42-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" 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12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E42-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E42-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E42-MJMATHI-50" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E42-MJMAIN-30" x="907" y="-213"></use><use xlink:href="#E42-MJMAIN-3D" x="1373" y="0"></use><use xlink:href="#E42-MJMATHI-4C" x="2429" y="0"></use><use xlink:href="#E42-MJMAIN-28" x="3110" y="0"></use><use xlink:href="#E42-MJMAIN-30" x="3499" y="0"></use><use xlink:href="#E42-MJMAIN-29" x="3999" y="0"></use><use xlink:href="#E42-MJMAIN-2C" x="4388" y="0"></use><g transform="translate(4832,0)"><use xlink:href="#E42-MJMATHI-50" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E42-MJMAIN-31" x="907" y="-213"></use></g><use xlink:href="#E42-MJMAIN-3D" x="6206" y="0"></use><use xlink:href="#E42-MJMATHI-4C" x="7261" y="0"></use><use xlink:href="#E42-MJMAIN-28" x="7942" y="0"></use><use xlink:href="#E42-MJMAIN-31" x="8331" y="0"></use><use xlink:href="#E42-MJMAIN-29" x="8831" y="0"></use></g></svg></span><script type="math/tex">P_0=L(0),P_1 = L(1)</script><span>,我们就可以得到点到线段的最短距离</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" 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y="0"></use></g></svg></span><script type="math/tex">d</script><span>的方程:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n35" cid="n35" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="39ex" height="9.815ex" viewBox="0 -2364.2 16791.7 4225.8" role="img" focusable="false" style="vertical-align: -4.324ex; max-width: 100%;"><defs><path stroke-width="0" id="E49-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 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\begin{aligned}
&\|Q-P\| &t_0\leq0 \\
&\|Q-(P+t_0\vec{d})\| & 0 <t_0<1\\
&\|Q-(P+\vec{d})\| & t_0\geq 1
\end{aligned}
\right .</script></div></div><p><span>图是如下:</span>
<img src="./img/点与线段.png" referrerpolicy="no-referrer" alt="avatar" title="点与线"></p></blockquote></li></ol></div>
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