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href="/atom.xml" title="Magicland" type="application/atom+xml"><link rel="alternate" href="/rss2.xml" title="Magicland" type="application/rss+xml"><title>LBS 球面距离公式 - Magicland</title><meta name="keywords" content="数学, Math,距离,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc name="description" content="详解LBS应用中的球面距离计算方案，推荐采用Haversine公式解决短距离计算时的舍入误差问题，提供Python实现代码：通过经纬度弧度转换、半正矢函数(hav)及地球半径(6371km)计算两点距离，附完整函数实现。 - MHuiG - Magicland"><meta property="og:type" content="article"><meta property="og:title" content="LBS 球面距离公式"><meta property="og:url" content="https://blog.mhuig.top/p/5e7ad437/"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="详解LBS应用中的球面距离计算方案，推荐采用Haversine公式解决短距离计算时的舍入误差问题，提供Python实现代码：通过经纬度弧度转换、半正矢函数(hav)及地球半径(6371km)计算两点距离，附完整函数实现。"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" 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color-about fa-fw"></i><p>关于</p></a><a href="https://www.travellings.cn/go-by-clouds.html" target="_blank" active-action="action-https:wwwtravellingscngo-by-cloudshtml" rel="external nofollow noopener noreferrer"><i class="fa-duotone fa-subway color-travellings fa-fw"></i><p>Travelling</p></a></div></div></div></div><div id="scroll-down" style="display:none"><i class="fa fa-chevron-down scroll-down-effects"></i></div></div></div><div id="safearea"><div class="body-wrapper"><div id="l_main" class><article itemscope itemtype="http://schema.org/Article" class="article post white-box reveal md shadow floatable blur article-type-post" id="post" itemprop="blogPost"><link itemprop="mainEntityOfPage" href="https://blog.mhuig.top/p/5e7ad437/"><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="Magicland"></span><span hidden itemprop="post" itemscope itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="LBS 球面距离公式"><meta itemprop="description" content="详解LBS应用中的球面距离计算方案，推荐采用Haversine公式解决短距离计算时的舍入误差问题，提供Python实现代码：通过经纬度弧度转换、半正矢函数(hav)及地球半径(6371km)计算两点距离，附完整函数实现。"></span><div class="article-meta" id="top"><h1 class="title allow-select" itemprop="name headline"> LBS 球面距离公式</h1><div class="new-meta-box"><div class="new-meta-item author" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a itemprop="url" class="author" target="_blank" href="https://mhuig.top/" rel="nofollow noopener" alt="author"><img itemprop="image" src="/lib/avatar/avatar-16.webp" class="lazyload" data-srcset="/lib/avatar/avatar-16.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="author"><p itemprop="name">MHuiG</p></a></div><div class="new-meta-item wordcount"><a class="notlink"><i class="fa-duotone fa-keyboard fa-fw" aria-hidden="true"></i><p>字数：323 字</p></a></div><div class="new-meta-item readtime"><a class="notlink"><i class="fa-duotone fa-hourglass-half fa-fw" aria-hidden="true"></i><p>时长：1 分钟</p></a></div></div></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p>维基百科推荐使用 Haversine 公式，理由是 Great-circle distance 公式用到了大量余弦函数， 而两点间距离很短时（比如地球表面上相距几百米的两点），余弦函数会得出 <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="7.425ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 3281.7 688"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="0.999…"><g data-mml-node="mn" data-latex="0.999"><path data-c="30" d="M249-22C390-22 460 92 460 320C460 473 428 575 365 625C330 652 291 666 250 666C109 666 39 551 39 320C39 136 88-22 249-22M361 524C368 489 371 425 371 332C371 240 367 172 360 128C347 48 310 8 249 8C226 8 203 17 182 34C155 57 139 104 132 176C129 201 128 253 128 332C128 419 131 480 136 513C145 568 163 603 191 618C213 630 232 636 249 636C314 636 350 583 361 524Z"/><path data-c="2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z" transform="translate(500,0)"/><path data-c="39" d="M116 140C83 140 67 123 67 89C67 14 128-22 207-22C287-22 351 18 398 99C437 166 457 243 457 329C457 456 434 546 388 599C349 644 304 666 253 666C194 666 145 645 106 602C63 557 42 503 42 440C42 379 60 327 96 284C134 238 183 215 242 215C294 215 334 243 361 299L361 286C361 177 342 103 303 63C272 30 239 14 206 14C166 14 137 23 118 42C149 42 165 60 165 91C165 119 144 140 116 140M243 246C204 246 175 266 155 306C144 329 138 374 138 439C138 504 145 548 158 573C180 613 212 633 253 633C294 633 324 608 343 558C354 531 361 485 361 420C361 333 322 246 243 246Z" transform="translate(778,0)"/><path data-c="39" d="M116 140C83 140 67 123 67 89C67 14 128-22 207-22C287-22 351 18 398 99C437 166 457 243 457 329C457 456 434 546 388 599C349 644 304 666 253 666C194 666 145 645 106 602C63 557 42 503 42 440C42 379 60 327 96 284C134 238 183 215 242 215C294 215 334 243 361 299L361 286C361 177 342 103 303 63C272 30 239 14 206 14C166 14 137 23 118 42C149 42 165 60 165 91C165 119 144 140 116 140M243 246C204 246 175 266 155 306C144 329 138 374 138 439C138 504 145 548 158 573C180 613 212 633 253 633C294 633 324 608 343 558C354 531 361 485 361 420C361 333 322 246 243 246Z" transform="translate(1278,0)"/><path data-c="39" d="M116 140C83 140 67 123 67 89C67 14 128-22 207-22C287-22 351 18 398 99C437 166 457 243 457 329C457 456 434 546 388 599C349 644 304 666 253 666C194 666 145 645 106 602C63 557 42 503 42 440C42 379 60 327 96 284C134 238 183 215 242 215C294 215 334 243 361 299L361 286C361 177 342 103 303 63C272 30 239 14 206 14C166 14 137 23 118 42C149 42 165 60 165 91C165 119 144 140 116 140M243 246C204 246 175 266 155 306C144 329 138 374 138 439C138 504 145 548 158 573C180 613 212 633 253 633C294 633 324 608 343 558C354 531 361 485 361 420C361 333 322 246 243 246Z" transform="translate(1778,0)"/></g><g data-mml-node="mo" data-latex="…" transform="translate(2444.7,0)"><path data-c="2026" d="M751 53C751 83 725 106 695 106C664 106 638 83 638 53C638 23 664 0 695 0C725 0 751 23 751 53M475 53C475 83 449 106 419 106C388 106 363 83 363 53C363 23 388 0 419 0C449 0 475 23 475 53M200 53C200 83 174 106 143 106C113 106 87 83 87 53C87 23 113 0 143 0C173 0 200 23 200 53Z"/></g></g></g></svg></mjx-container> 的结果， 会导致较大的舍入误差。而 Haversine 公式采用了正弦函数，即使距离很小，也能保持足够的有效数字。 以前采用三角函数表计算时的确会有这个问题，但经过实际验证，采用计算机来计算时，两个公式的区别不大。 稳妥起见，这里还是采用 Haversine 公式。</p><div class="story post-story"><h2 id="Haversine公式"><a href="#Haversine公式" class="headerlink" title="Haversine公式"></a>Haversine公式</h2><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 73.729ex;"><svg style="vertical-align: -1.785ex; min-width: 73.729ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="4.701ex" role="img" focusable="false" data-mjx-viewbox="0 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\text{haversin}(\frac{d}{R}) = \text{haversin}(\varphi_2 - \varphi_1) + \cos(\varphi_1) \cos(\varphi_2) \text{haversin}(\Delta\lambda)
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transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{1}" transform="translate(389,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>其中</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 44.905ex;"><svg style="vertical-align: -1.857ex; min-width: 44.905ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="4.846ex" role="img" focusable="false" data-mjx-viewbox="0 -1321 19848.1 2142"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1321)"><g data-mml-node="math" data-latex="\text{haversin}(\theta) = \sin(\frac{\theta}{2})^2 = \frac{1- \cos(\theta)}{2}"><g data-mml-node="mtable" data-latex="\text{haversin}(\theta) = \sin(\frac{\theta}{2})^2 = \frac{1- \cos(\theta)}{2}" 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59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.595ex" role="img" focusable="false" viewbox="0 -683 759 705"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="R"><g data-mml-node="mi" data-latex="R"><path data-c="1D445" d="M739 531C739 582 713 621 662 649C621 672 572 683 517 683L235 683C212 683 202 682 202 659C202 652 205 647 211 646C221 645 229 644 234 644C264 643 281 641 286 640C291 639 294 636 294 631C294 629 293 623 290 613L158 82C153 60 143 46 128 41C121 39 103 38 72 38C50 38 41 37 41 15C41 4 47-1 59 0L183 3L309 0C325-1 333 8 333 23C333 33 322 38 301 38C261 38 241 43 241 52C241 52 242 54 244 68L308 327L423 327C492 327 527 298 527 241C527 232 522 210 513 174C502 132 497 104 497 89C497 13 556-22 632-22C660-22 687-9 714 16C741 41 755 68 755 96C755 106 750 111 739 111C732 111 726 106 723 95C712 64 698 41 682 28C666 15 651 8 636 8C613 8 601 27 601 64C601 88 604 125 611 176C614 197 615 212 615 223C615 276 587 315 531 339C625 362 739 429 739 531M609 616C629 603 639 581 639 550C639 530 635 507 626 480C599 398 531 357 422 357L316 357L379 610C384 631 392 642 403 643C408 644 428 644 463 644C528 644 566 642 609 616Z"/></g></g></g></svg></mjx-container>为地球半径，可取平均值 6371.137km</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.493ex;" xmlns="http://www.w3.org/2000/svg" width="2.467ex" height="1.493ex" role="img" focusable="false" viewbox="0 -442 1090.6 660"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\varphi_1"><g data-mml-node="msub" data-latex="\varphi_1"><g data-mml-node="mi" data-latex="\varphi"><path data-c="1D711" d="M503 442C486 442 471 439 456 434C393 413 349 351 324 294C282 201 270 163 239 58C136 79 84 126 84 200C84 267 131 364 164 402C170 409 173 414 173 417C173 426 168 431 157 431C140 431 118 402 91 344C64 286 50 234 50 189C50 85 129 19 220-3L169-165C165-176 163-183 163-186C163-207 174-218 196-218C215-218 228-208 236-189L270-12C281-13 290-13 299-13C381-13 454 19 520 83C586 147 619 220 619 302C619 380 578 442 503 442M583 280C583 217 553 162 492 115C435 73 374 52 308 52C302 52 296 52 291 53C285 53 282 53 282 52C297 131 306 178 310 192C333 271 401 377 493 377C549 377 583 337 583 280Z"/></g><g data-mml-node="mn" transform="translate(687,-150) scale(0.707)" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g></g></g></g></svg></mjx-container>, <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.493ex;" xmlns="http://www.w3.org/2000/svg" width="2.467ex" height="1.493ex" role="img" focusable="false" viewbox="0 -442 1090.6 660"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\varphi_2"><g data-mml-node="msub" data-latex="\varphi_2"><g data-mml-node="mi" data-latex="\varphi"><path data-c="1D711" d="M503 442C486 442 471 439 456 434C393 413 349 351 324 294C282 201 270 163 239 58C136 79 84 126 84 200C84 267 131 364 164 402C170 409 173 414 173 417C173 426 168 431 157 431C140 431 118 402 91 344C64 286 50 234 50 189C50 85 129 19 220-3L169-165C165-176 163-183 163-186C163-207 174-218 196-218C215-218 228-208 236-189L270-12C281-13 290-13 299-13C381-13 454 19 520 83C586 147 619 220 619 302C619 380 578 442 503 442M583 280C583 217 553 162 492 115C435 73 374 52 308 52C302 52 296 52 291 53C285 53 282 53 282 52C297 131 306 178 310 192C333 271 401 377 493 377C549 377 583 337 583 280Z"/></g><g data-mml-node="mn" transform="translate(687,-150) scale(0.707)" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g></g></g></g></svg></mjx-container> 表示两点的纬度；</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.029ex;" xmlns="http://www.w3.org/2000/svg" width="3.204ex" height="1.649ex" role="img" focusable="false" viewbox="0 -716 1416 729"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Delta\lambda"><g data-mml-node="mi" data-latex="\Delta"><path data-c="394" d="M759 0C776 0 785 4 785 13L445 696C438 709 429 716 416 716C403 716 394 709 387 696L52 25C49 18 47 14 47 12C47 4 56 0 73 0M384 604L644 83L124 83Z"/></g><g data-mml-node="mi" data-latex="\lambda" transform="translate(833,0)"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 589L342 322L72 53C59 40 53 29 53 20C53 2 68-13 86-13Z"/></g></g></g></svg></mjx-container> 表示两点经度的差值。</p></div><div class="story post-story"><h2 id="距离计算函数"><a href="#距离计算函数" class="headerlink" title="距离计算函数"></a>距离计算函数</h2><p>下面就是计算球面间两点 (lat0, lng)-(lat1, lng1) 之间距离的函数。</p><figure class="highlight python"><table><tbody><tr><td class="code"><pre><span class="line"><span class="keyword">from</span> math <span class="keyword">import</span> sin, asin, cos, radians, fabs, sqrt</span><br><span class="line"> </span><br><span class="line">EARTH_RADIUS=<span class="number">6371</span> <span class="comment"># 地球平均半径，6371km</span></span><br><span class="line"> </span><br><span class="line"><span class="keyword">def</span> <span class="title function_">hav</span>(<span class="params">theta</span>):</span><br><span class="line"> s = sin(theta / <span class="number">2</span>)</span><br><span class="line"> <span class="keyword">return</span> s * s</span><br><span class="line"> </span><br><span class="line"><span class="keyword">def</span> <span class="title function_">get_distance_hav</span>(<span class="params">lat0, lng0, lat1, lng1</span>):</span><br><span class="line"> <span class="string">"用haversine公式计算球面两点间的距离。"</span></span><br><span class="line"> <span class="comment"># 经纬度转换成弧度</span></span><br><span class="line"> lat0 = radians(lat0)</span><br><span class="line"> lat1 = radians(lat1)</span><br><span class="line"> lng0 = radians(lng0)</span><br><span class="line"> lng1 = radians(lng1)</span><br><span class="line"> </span><br><span class="line"> dlng = fabs(lng0 - lng1)</span><br><span class="line"> dlat = fabs(lat0 - lat1)</span><br><span class="line"> h = hav(dlat) + cos(lat0) * cos(lat1) * hav(dlng)</span><br><span class="line"> distance = <span class="number">2</span> * EARTH_RADIUS * asin(sqrt(h))</span><br><span class="line"> </span><br><span class="line"> <span class="keyword">return</span> distance</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="相关文献"><a href="#相关文献" class="headerlink" title="相关文献"></a>相关文献</h2><p><a target="_blank" rel="external nofollow noopener noreferrer" href="/go.html?u=aHR0cDovL29yYWNsZS1hYmMud2lraWRvdC5jb20vemgtYmxvZzoyMA">LBS 球面距离公式</a></p></div></div><div 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class="article-meta" id="bottom"><div class="new-meta-box"><div class="new-meta-item category"><i class="fa-duotone fa-folder-open fa-fw" aria-hidden="true"></i> <a class="category-link" href="/categories/%E6%95%B0%E5%AD%A6/">数学</a> <span hidden itemprop="about" itemscope itemtype="http://schema.org/Thing"><a href="/categories/%E6%95%B0%E5%AD%A6/" itemprop="url"><span itemprop="name">数学</span></a></span></div><div class="new-meta-item meta-tags"><a class="tag" href="/tags/Math/" rel="nofollow"><i class="fa-duotone fa-hashtag fa-fw" aria-hidden="true"></i><p>Math</p></a></div><div class="new-meta-item meta-tags"><a class="tag" href="/tags/%E8%B7%9D%E7%A6%BB/" rel="nofollow"><i class="fa-duotone fa-hashtag fa-fw" aria-hidden="true"></i><p>距离</p></a></div> <span hidden itemprop="keywords">Math 距离</span><div class="new-meta-item date" itemprop="dateCreated datePublished" datetime="2020-02-23T09:40:00+08:00"><a class="notlink"><i class="fa-duotone fa-calendar-alt fa-fw" 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