三维角锥（金字塔）几何构造与角度分析
2018-02-10 00:54
本 GeoGebra 范例演示三维角锥（金字塔）的几何构造过程。通过 Intersect、Polygon、Pyramid、OrthogonalLine、Angle 等命令，构建以矩形为底面的四棱锥，包含顶点 V 与底面顶点 A、B、C、D 的连线，生成各侧面（faceABF、faceBCF 等）与侧棱（edgeAF、edgeBF 等）。范例还利用正交线与交点构造高线，并标注相关角度 α，帮助学习者直观理解角锥的空间结构、面与棱的关系以及二面角等立体几何核心概念。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.8.0" id="d5d234ae-7690-4655-a838-6c5b67925746" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1032" height="742" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.1937984496124031" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="91,126,700,550" />
<view id="1" visible="false" inframe="false" stylebar="false" location="1,3" size="130" 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="2" visible="true" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
<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="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 22 , 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="true" location="1" size="808" window="100,100,600,400" />
</views>
<toolbar show="true" items="0 39 | 1 501 67 , 5 19 , 72 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="20"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="130" height="543"/>
<coordSystem xZero="215.00000000000003" yZero="314.9999999999999" scale="50.00000000000001" yscale="49.999999999999986"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<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="10"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<uses3D val="true"/>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.32397748976518626" yZero="-0.38945785023061547" zZero="-0.38989972681874835" scale="250.2089956711481" xAngle="18.0" zAngle="367.0"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" gridType="0"/>
<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="true" show="true" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="3" y="28"/>
<labelMode val="0"/>
<coords x="0.0" y="-0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="yAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-14" y="37"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="1.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="4"/>
<output a0="poly1" a1="a" a2="b" a3="c" a4="d" a5="C" a6="D"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
</element>
<element type="segment" label="a">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-1.0" y="0.0" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0.0" y="-0.9999999999999998" z="0.9999999999999998"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0.9999999999999998" y="-2.220446049250313E-16" z="0.9999999999999999"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="1.6653345369377348E-16" y="1.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-15" y="29"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-0.9999999999999998" y="1.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="D">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-29" y="22"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="-1.0" y="1.6653345369377348E-16" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Pyramid">
<input a0="poly1" a1="D"/>
<output a0=""/>
</command>
<command name="Pyramid">
<input a0="poly1" a1="C"/>
<output a0=""/>
</command>
<command name="Pyramid">
<input a0="poly1" a1="A"/>
<output a0=""/>
</command>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="-0.9999999999999998" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="D"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.9999999999999998" y="-1.0" z="1.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="e" a1="f"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.9999999999999998" y="1.0" z="1.9999999999999996"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point3d" label="V">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-0.5" y="0.5" z="1.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Pyramid">
<input a0="poly1" a1="D"/>
<output a0=""/>
</command>
<command name="Pyramid">
<input a0="poly1" a1="A"/>
<output a0=""/>
</command>
<command name="Pyramid">
<input a0="poly1" a1="B"/>
<output a0=""/>
</command>
<command name="Pyramid">
<input a0="poly1" a1="C"/>
<output a0=""/>
</command>
<command name="Pyramid">
<input a0="poly1" a1="D"/>
<output a0=""/>
</command>
<command name="Pyramid">
<input a0="poly1" a1="V"/>
<output a0="g" a1="faceABF" a2="faceBCF" a3="faceCDF" a4="faceADF" a5="edgeAF" a6="edgeBF" a7="edgeCF" a8="edgeDF"/>
</command>
<element type="polyhedron" label="g">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.4"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="polygon3d" label="faceABF">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
</element>
<element type="polygon3d" label="faceBCF">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
</element>
<element type="polygon3d" label="faceCDF">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
</element>
<element type="polygon3d" label="faceADF">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
</element>
<element type="segment3d" label="edgeAF">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeBF">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeCF">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeDF">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="D" a1="edgeAF" a2="space"/>
<output a0="h"/>
</command>
<element type="line3d" label="h">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="-1.0" oy="1.6653345369377348E-16" oz="0.0" ow="1.0" vx="-0.9128709291752768" vy="-0.1825741858350552" vz="-0.36514837167011077" vw="0.0"/>
</element>
<command name="Point">
<input a0="edgeBF"/>
<output a0="Q"/>
</command>
<element type="point3d" label="Q">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-31" y="8"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-0.2144744098877109" y="0.7855255901122891" z="0.4289488197754218" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="A" a1="Q"/>
<output a0="i"/>
</command>
<element type="segment3d" label="i">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="Q" a1="C"/>
<output a0="j"/>
</command>
<element type="segment3d" label="j">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="A" a1="Q" a2="C"/>
<output a0="α"/>
</command>
<element type="angle3d" label="α">
<angleStyle val="1"/>
<value val="1.75236460365736"/>
<show object="true" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.2"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="-0.9999999999999998" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="A" a1="edgeBF" a2="space"/>
<output a0="l"/>
</command>
<element type="line3d" label="l">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.0" oy="-0.0" oz="0.0" ow="1.0" vx="0.18257418583505536" vy="-0.9128709291752768" vz="-0.3651483716701107" vw="0.0"/>
</element>
<command name="Intersect">
<input a0="edgeBF" a1="l"/>
<output a0="K"/>
</command>
<element type="point3d" label="K">
<show object="true" label="true"/>
<objColor r="0" g="102" b="51" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="14" y="8"/>
<labelMode val="0"/>
<coords x="-0.16666666666666666" y="0.8333333333333334" z="0.3333333333333333" w="1.0"/>
<pointSize val="5"/>
</element>
</construction>
</geogebra>