import{_ as l,c as t,j as s,a as e,a4 as i,o as a}from"./chunks/framework.DpC1ZpOZ.js";const D1=JSON.parse('{"title":"mbcp.mp_math.plane","description":"","frontmatter":{"title":"mbcp.mp_math.plane","lastUpdated":false},"headers":[],"relativePath":"ja/api/mp_math/plane.md","filePath":"ja/api/mp_math/plane.md"}'),n={name:"ja/api/mp_math/plane.md"},h=i("",16),o={class:"tip custom-block"},p=s("p",{class:"custom-block-title"},"TIP",-1),r=s("p",null,"平面间夹角计算公式:",-1),d={tabindex:"0",class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.17ex"},xmlns:"http://www.w3.org/2000/svg",width:"22.011ex",height:"5.206ex",role:"img",focusable:"false",viewBox:"0 -1342 9729 2301","aria-hidden":"true"},k=i("",1),T=[k],m=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"θ"),s("mo",null,"="),s("mi",null,"arccos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mfrac",null,[s("mrow",null,[s("mi",null,"n"),s("mn",null,"1"),s("mo",null,"⋅"),s("mi",null,"n"),s("mn",null,"2")]),s("mrow",null,[s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mi",null,"n"),s("mn",null,"1"),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mo",null,"⋅"),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mi",null,"n"),s("mn",null,"2"),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])]),s("mo",{stretchy:"false"},")")])],-1),g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.489ex",height:"1.532ex",role:"img",focusable:"false",viewBox:"0 -666 1100 677","aria-hidden":"true"},y=i("",1),E=[y],u=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"n"),s("mn",null,"1")])],-1),_={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},f={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.489ex",height:"1.532ex",role:"img",focusable:"false",viewBox:"0 -666 1100 677","aria-hidden":"true"},x=i("",1),b=[x],F=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"n"),s("mn",null,"2")])],-1),w={class:"tip custom-block"},C=s("p",{class:"custom-block-title"},"TIP",-1),H=s("p",null,"平面与直线夹角计算公式:",-1),v={tabindex:"0",class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},L={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.17ex"},xmlns:"http://www.w3.org/2000/svg",width:"19.568ex",height:"5.269ex",role:"img",focusable:"false",viewBox:"0 -1370 8649 2329","aria-hidden":"true"},D=i("",1),A=[D],M=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"θ"),s("mo",null,"="),s("mi",null,"arccos"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mfrac",null,[s("mrow",null,[s("mi",null,"n"),s("mo",null,"⋅"),s("mi",null,"d")]),s("mrow",null,[s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mi",null,"n"),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mo",null,"⋅"),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mi",null,"d"),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])]),s("mo",{stretchy:"false"},")")])],-1),V={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},B={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.357ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 600 453","aria-hidden":"true"},Z=s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D45B",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",style:{"stroke-width":"3"}})])])],-1),P=[Z],j=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"n")])],-1),S={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.176ex",height:"1.593ex",role:"img",focusable:"false",viewBox:"0 -694 520 704","aria-hidden":"true"},R=s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D451",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",style:{"stroke-width":"3"}})])])],-1),I=[R],N=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"d")])],-1),z=i("",8),G={class:"tip custom-block"},O=s("p",{class:"custom-block-title"},"TIP",-1),J=s("p",null,"平面和平面之间的距离计算公式: 暂未实现",-1),X=s("li",null,"平行 = 0",-1),$=s("li",null,"相交 = 0",-1),U={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},K={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.389ex"},xmlns:"http://www.w3.org/2000/svg",width:"6.74ex",height:"4.295ex",role:"img",focusable:"false",viewBox:"0 -1284.3 2979.3 1898.3","aria-hidden":"true"},W=i("",1),Y=[W],s3=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mfrac",null,[s("mrow",null,[s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mn",null,"1")]),s("msub",null,[s("mi",null,"P"),s("mn",null,"2")])]),s("mo",{stretchy:"false"},"→")])]),s("mo",null,"⋅"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"n"),s("mo",{stretchy:"false"},"→")])]),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")]),s("mrow",null,[s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"n"),s("mo",{stretchy:"false"},"→")])]),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])])])],-1),t3={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},a3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.339ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.44ex",height:"1.885ex",role:"img",focusable:"false",viewBox:"0 -683 1078.6 833","aria-hidden":"true"},i3=i("",1),e3=[i3],l3=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"P"),s("mn",null,"1")])])],-1),n3={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},h3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.339ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.44ex",height:"1.885ex",role:"img",focusable:"false",viewBox:"0 -683 1078.6 833","aria-hidden":"true"},o3=i("",1),p3=[o3],r3=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"P"),s("mn",null,"2")])])],-1),d3={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},Q3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.357ex",height:"1.937ex",role:"img",focusable:"false",viewBox:"0 -845 600 856","aria-hidden":"true"},k3=i("",1),T3=[k3],m3=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"n"),s("mo",{stretchy:"false"},"→")])])])],-1),g3={class:"tip custom-block"},c3=s("p",{class:"custom-block-title"},"TIP",-1),y3=s("p",null,"平面和点之间的距离计算公式:",-1),E3={tabindex:"0",class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},u3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.386ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.385ex",height:"6.448ex",role:"img",focusable:"false",viewBox:"0 -1795.5 4148 2850","aria-hidden":"true"},_3=i("",1),f3=[_3],x3=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mfrac",null,[s("mrow",null,[s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mrow",null,[s("msub",null,[s("mi",null,"P"),s("mn",null,"1")]),s("mi",null,"P")]),s("mo",{stretchy:"false"},"→")])]),s("mo",null,"⋅"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"n"),s("mo",{stretchy:"false"},"→")])]),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")]),s("mrow",null,[s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"n"),s("mo",{stretchy:"false"},"→")])]),s("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])])])],-1),b3={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},F3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.339ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.44ex",height:"1.885ex",role:"img",focusable:"false",viewBox:"0 -683 1078.6 833","aria-hidden":"true"},w3=i("",1),C3=[w3],H3=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"P"),s("mn",null,"1")])])],-1),v3={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},L3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"1.699ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 751 683","aria-hidden":"true"},D3=s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D443",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",style:{"stroke-width":"3"}})])])],-1),A3=[D3],M3=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"P")])],-1),V3={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},B3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.357ex",height:"1.937ex",role:"img",focusable:"false",viewBox:"0 -845 600 856","aria-hidden":"true"},Z3=i("",1),P3=[Z3],j3=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mover",null,[s("mi",null,"n"),s("mo",{stretchy:"false"},"→")])])])],-1),S3=i("",8),q3={class:"tip custom-block"},R3=s("p",{class:"custom-block-title"},"TIP",-1),I3=s("p",null,"计算两平面交线的一般步骤:",-1),N3=s("ol",null,[s("li",null,"求两平面的法向量的叉乘得到方向向量")],-1),z3={tabindex:"0",class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},G3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"11.937ex",height:"1.756ex",role:"img",focusable:"false",viewBox:"0 -694 5276 776","aria-hidden":"true"},O3=i("",1),J3=[O3],X3=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"d"),s("mo",null,"="),s("mi",null,"n"),s("mn",null,"1"),s("mo",null,"×"),s("mi",null,"n"),s("mn",null,"2")])],-1),$3={start:"2"},U3={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},K3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.442ex",height:"1.692ex",role:"img",focusable:"false",viewBox:"0 -666 2405.6 748","aria-hidden":"true"},W3=i("",1),Y3=[W3],s1=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"x"),s("mo",null,"="),s("mn",null,"0")])],-1),t1={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},a1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.464ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.257ex",height:"1.971ex",role:"img",focusable:"false",viewBox:"0 -666 2323.6 871","aria-hidden":"true"},i1=i("",1),e1=[i1],l1=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"y"),s("mo",null,"="),s("mn",null,"0")])],-1),n1={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},h1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.2ex",height:"1.692ex",role:"img",focusable:"false",viewBox:"0 -666 2298.6 748","aria-hidden":"true"},o1=i("",1),p1=[o1],r1=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"z"),s("mo",null,"="),s("mn",null,"0")])],-1),d1={tabindex:"0",class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},Q1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-3.281ex"},xmlns:"http://www.w3.org/2000/svg",width:"13.363ex",height:"7.692ex",role:"img",focusable:"false",viewBox:"0 -1950 5906.6 3400","aria-hidden":"true"},k1=i("",1),T1=[k1],m1=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"{"),s("mtable",{columnalign:"left left",columnspacing:"1em",rowspacing:".2em"},[s("mtr",null,[s("mtd",null,[s("mi",null,"x"),s("mo",null,"="),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")]),s("mo",null,"+"),s("mi",null,"d"),s("mi",null,"t")])]),s("mtr",null,[s("mtd",null,[s("mi",null,"y"),s("mo",null,"="),s("msub",null,[s("mi",null,"y"),s("mn",null,"0")]),s("mo",null,"+"),s("mi",null,"d"),s("mi",null,"t")])]),s("mtr",null,[s("mtd",null,[s("mi",null,"z"),s("mo",null,"="),s("msub",null,[s("mi",null,"z"),s("mn",null,"0")]),s("mo",null,"+"),s("mi",null,"d"),s("mi",null,"t")])])]),s("mo",{"data-mjx-texclass":"CLOSE",fence:"true",stretchy:"true",symmetric:"true"})])])],-1),g1=s("p",null,"或",-1),c1={tabindex:"0",class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},y1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.991ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.19ex",height:"4.839ex",role:"img",focusable:"false",viewBox:"0 -1259 12018.1 2139","aria-hidden":"true"},E1=i("",1),u1=[E1],_1=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mfrac",null,[s("mrow",null,[s("mi",null,"x"),s("mo",null,"−"),s("msub",null,[s("mi",null,"x"),s("mn",null,"0")])]),s("mi",null,"m")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"y"),s("mo",null,"−"),s("msub",null,[s("mi",null,"y"),s("mn",null,"0")])]),s("mi",null,"n")]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mi",null,"z"),s("mo",null,"−"),s("msub",null,[s("mi",null,"z"),s("mn",null,"0")])]),s("mi",null,"p")])])],-1),f1=i("",82);function x1(b1,F1,w1,C1,H1,v1){return a(),t("div",null,[h,s("div",o,[p,r,s("mjx-container",d,[(a(),t("svg",Q,T)),m]),s("p",null,[e("其中 "),s("mjx-container",g,[(a(),t("svg",c,E)),u]),e(" 和 "),s("mjx-container",_,[(a(),t("svg",f,b)),F]),e(" 分别为两个平面的法向量")])]),s("div",w,[C,H,s("mjx-container",v,[(a(),t("svg",L,A)),M]),s("p",null,[e("其中 "),s("mjx-container",V,[(a(),t("svg",B,P)),j]),e(" 为平面的法向量，"),s("mjx-container",S,[(a(),t("svg",q,I)),N]),e(" 为直线的方向向量")])]),z,s("div",G,[O,J,s("ul",null,[X,$,s("li",null,[e("不平行 = "),s("mjx-container",U,[(a(),t("svg",K,Y)),s3]),e(" 其中，"),s("mjx-container",t3,[(a(),t("svg",a3,e3)),l3]),e("和"),s("mjx-container",n3,[(a(),t("svg",h3,p3)),r3]),e("分别为两个平面上的点，"),s("mjx-container",d3,[(a(),t("svg",Q3,T3)),m3]),e("为平面的法向量。")])])]),s("div",g3,[c3,y3,s("mjx-container",E3,[(a(),t("svg",u3,f3)),x3]),s("p",null,[e("其中，"),s("mjx-container",b3,[(a(),t("svg",F3,C3)),H3]),e("为平面上的点，"),s("mjx-container",v3,[(a(),t("svg",L3,A3)),M3]),e("为点，"),s("mjx-container",V3,[(a(),t("svg",B3,P3)),j3]),e("为平面的法向量。")])]),S3,s("div",q3,[R3,I3,N3,s("mjx-container",z3,[(a(),t("svg",G3,J3)),X3]),s("ol",$3,[s("li",null,[e("寻找直线上的一点，依次假设"),s("mjx-container",U3,[(a(),t("svg",K3,Y3)),s1]),e(", "),s("mjx-container",t1,[(a(),t("svg",a1,e1)),l1]),e(", "),s("mjx-container",n1,[(a(),t("svg",h1,p1)),r1]),e("，并代入两平面方程求出合适的点 直线最终可用参数方程或点向式表示")])]),s("mjx-container",d1,[(a(),t("svg",Q1,T1)),m1]),g1,s("mjx-container",c1,[(a(),t("svg",y1,u1)),_1])]),f1])}const A1=l(n,[["render",x1]]);export{D1 as __pageData,A1 as default};