二面角的夹角
2020-02-06 04:57
本范例演示了如何在三维空间中构造并测量二面角的夹角。通过定义两个相交平面及其交线,利用正交平面和旋转等几何变换,精确构造出二面角的平面角。该构造过程综合运用了交点、直线、平面及正交平面等GeoGebra命令,直观展示了立体几何中二面角夹角的求解与可视化方法。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.553.0" app="classic" platform="d" id="12c0e687-73dd-4d08-a32a-72976bcaa750" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="773" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.129366106080207" orientation="1" />
<pane location="1" divider="0.22897800776196636" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1,1" size="168" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1,3" size="177" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="true" inframe="false" stylebar="false" location="1,1" size="459" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="100" window="733,178,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="70" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="177" height="433"/>
<coordSystem xZero="215.00000000000006" yZero="314.99999999999983" scale="50.000000000000014" yscale="49.99999999999998"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="1"/>
</algebraView>
<kernel>
<uses3D val="true"/>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.0" yZero="0.0" zZero="-1.5" scale="50.0" xAngle="27.0" zAngle="-55.0"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="2" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<plate show="false"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="false" show="false" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<command name="Intersect">
<input a0="xAxis" a1="zAxis"/>
<output a0="A"/>
</command>
<element type="point3d" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="0.0" w="1.0"/>
<pointSize val="4"/>
</element>
<command name="Point">
<input a0="yAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="-2.6615176873657784" z="1.0"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="angle" label="α">
<angleStyle val="3"/>
<value val="1.117010721276371"/>
<slider min="0" max="6.283185307179586" absoluteScreenLocation="true" width="180.0" x="16.0" y="40.0" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="10" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation speed="1" type="0" playing="false"/>
</element>
<command name="Rotate">
<input a0="B" a1="α" a2="zAxis"/>
<output a0="B'"/>
</command>
<element type="point3d" label="B'">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.3921562515242893" y="-1.1667325608099497" z="0.0" w="1.0"/>
<pointSize val="5"/>
</element>
<command name="Line">
<input a0="A" a1="B'"/>
<output a0="a"/>
</command>
<element type="line3d" label="a">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coords ox="0.0" oy="0.0" oz="0.0" ow="1.0" vx="2.3921562515242893" vy="-1.1667325608099497" vz="0.0" vw="0.0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Rotate">
<input a0="B'" a1="90°" a2="A"/>
<output a0="B''"/>
</command>
<element type="point3d" label="B''">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.16673256080995" y="2.3921562515242893" z="0.0" w="1.0"/>
<pointSize val="5"/>
</element>
<element type="angle" label="β">
<angleStyle val="3"/>
<value val="4.293509959906051"/>
<slider min="0" max="6.283185307179586" absoluteScreenLocation="true" width="180.0" x="3.0" y="117.0" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="10" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation speed="1" type="0" playing="false"/>
</element>
<command name="Rotate">
<input a0="B''" a1="β" a2="a"/>
<output a0="B'''"/>
</command>
<element type="point3d" label="B'''">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.47455288515107086" y="-0.9729776034577787" z="-2.4314173937284473" w="1.0"/>
<pointSize val="5"/>
</element>
<command name="Plane">
<input a0="B'''" a1="B'" a2="A"/>
<output a0="p"/>
</command>
<element type="plane3d" label="p">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="255" b="204" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.836813842182645" y="-5.816330318472399" z="2.8811907596367052" w="-0.0"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="Plane">
<input a0="A" a1="B" a2="B'"/>
<output a0="q"/>
</command>
<element type="plane3d" label="q">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="6.366766174374516" w="-0.0"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="IntersectPath">
<input a0="p" a1="q"/>
<output a0="f"/>
</command>
<element type="line3d" label="f">
<show object="false" label="false" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coords ox="-1.0667536983597956E-17" oy="-2.18716920576601E-17" oz="0.0" ow="1.0" vx="-37.031215130639026" vy="18.061330413405873" vz="0.0" vw="0.0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="OrthogonalPlane">
<input a0="A" a1="f"/>
<output a0="r"/>
</command>
<element type="plane3d" label="r">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="204" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-37.031215130639026" y="18.061330413405873" z="0.0" w="-0.0"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="IntersectPath">
<input a0="p" a1="r"/>
<output a0="g"/>
</command>
<element type="line3d" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coords ox="7.301342125259285E-17" oy="1.496996980810385E-16" oz="-7.415558663538439E-17" ow="1.0" vx="-52.038138293850395" vy="-106.69399485251611" vz="-266.6224114191938" vw="0.0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="IntersectPath">
<input a0="q" a1="r"/>
<output a0="h"/>
</command>
<element type="line3d" label="h">
<show object="true" label="false" ev="4"/>
<objColor r="21" g="101" b="192" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coords ox="0.0" oy="0.0" oz="0.0" ow="1.0" vx="-114.9922675402742" vy="-235.76908788973833" vz="0.0" vw="0.0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
</construction>
</geogebra>