4B-144：圆锥截痕
2020-08-03 07:30
本范例演示了圆锥截痕（圆锥曲线）的几何构造。通过定义顶点、轴线和底面圆，构建无限圆锥，并利用平面与圆锥相交，展示不同截面产生的圆锥曲线（如椭圆、抛物线、双曲线）。涉及点、角、向量、平面、圆弧及交线等几何对象的动态交互，直观解析立体几何中圆锥截面的形成原理与几何性质。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.593.0" app="classic" platform="w" id="1fcb4543-1425-4e9e-b2c8-d27c744b01c3" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="912" height="706" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.9978070175438597" orientation="1" />
</panes>
<views>
<view id="1" visible="false" inframe="false" stylebar="true" location="3,3,2" size="305" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3,0" size="456" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 | 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="300" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="383" window="100,100,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" 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 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="true" location="3" size="910" window="100,100,600,400" />
</views>
<toolbar show="false" items="0 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="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="694" height="257"/>
<coordSystem xZero="280.41140045463663" yZero="384.0011370618615" scale="32.08116795470612" yscale="32.0811679547061"/>
<evSettings axes="false" 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="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" 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" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="-0.38552908816283127" yZero="-0.5956733720700218" zZero="-2.2435934946273863" scale="51.068105393561616" xAngle="17" zAngle="-365"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="2" showNumbers="true" selectionAllowed="false"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="2" showNumbers="true" selectionAllowed="false"/>
<axis id="2" show="false" label="" unitLabel="" tickStyle="2" showNumbers="false" selectionAllowed="false"/>
<plate show="false"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="false" show="false" size="1"/>
<projection type="1"/>
</euclidianView3D>
<construction title="" author="" date="">
<expression label="O" exp="(0, 0, 0)" type="point"/>
<element type="point3d" label="O">
<show object="false" label="true" ev="20"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="0" w="1"/>
<pointSize val="5"/>
</element>
<expression label="i" exp="(1, 0, 0)" type="vector"/>
<element type="vector3d" label="i">
<show object="false" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="1" y="0" z="0" w="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<expression label="j" exp="(0, 1, 0)" type="vector"/>
<element type="vector3d" label="j">
<show object="false" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0" y="1" z="0" w="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<expression label="k" exp="(0, 0, 1)" type="vector"/>
<element type="vector3d" label="k">
<show object="false" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="1" w="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<command name="Point">
<input a0="Ray[O, O + k]"/>
<output a0="V"/>
</command>
<element type="point3d" label="V">
<show object="true" label="true" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="2.853073856645165" w="1"/>
<pointSize val="5"/>
</element>
<command name="Point">
<input a0="Ray[O, O + i]"/>
<output a0="A"/>
</command>
<element type="point3d" label="A">
<show object="false" label="true" ev="37"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="2.2715314512684097" y="0" z="0" w="1"/>
<pointSize val="5"/>
</element>
<command name="Angle">
<input a0="O" a1="V" a2="A"/>
<output a0="a"/>
</command>
<element type="angle3d" label="a">
<angleStyle val="1"/>
<value val="0.6724011318464207"/>
<show object="false" label="true" ev="20"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<selectionAllowed val="false"/>
</element>
<command name="InfiniteCone">
<input a0="V" a1="k" a2="a"/>
<output a0="b"/>
</command>
<element type="quadric" label="b">
<show object="true" label="false" ev="20"/>
<objColor r="102" g="153" b="255" alpha="0.8500000238418579"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="0" x1="0" y1="1" z1="0" x2="0" y2="0" z2="1"/>
<matrix A0="1" A1="1" A2="-0.633886467322262" A3="-5.159855134101568" A4="0" A5="0" A6="0" A7="0" A8="0" A9="1.8085249079983055"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="yAxis"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="false" label="true" ev="21"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0" y="3.766934133548802" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="r" exp="abs(y(B))" />
<element type="numeric" label="r">
<value val="3.766934133548802"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<command name="Circle">
<input a0="O" a1="B" a2="i"/>
<output a0="circ"/>
</command>
<element type="conic3d" label="circ">
<show object="false" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="10" typeHidden="2"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-14.189792766495062" A3="0" A4="0" A5="0"/>
<eqnStyle style="parametric"/>
</element>
<command name="Point">
<input a0="circ"/>
<output a0="C"/>
</command>
<element type="point3d" label="C">
<show object="true" label="false" ev="37"/>
<objColor r="153" g="204" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="28" y="75"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0" y="3.551451939181599" z="1.2557793955063616" w="1"/>
<pointSize val="7"/>
</element>
<command name="UnitVector">
<input a0="Vector[C]"/>
<output a0="v"/>
</command>
<element type="vector3d" label="v">
<show object="false" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="0.9427963997437357" z="0.3333690876944803" w="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<expression label="n" exp="i ⊗ v" type="vector" />
<element type="vector3d" label="n">
<show object="false" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="-0.3333690876944803" z="0.9427963997437357" w="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<command name="Plane">
<input a0="O" a1="C" a2="A"/>
<output a0="d"/>
</command>
<element type="plane3d" label="d">
<show object="true" label="false" ev="20"/>
<objColor r="255" g="102" b="153" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="2.852542392747532" z="-8.067234777519186" w="0"/>
<fading val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="CircleArc">
<input a0="O" a1="B" a2="C"/>
<output a0="arc"/>
</command>
<element type="conicpart" label="arc">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="10" typeHidden="2"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-14.189792766495062" A3="0" A4="0" A5="0"/>
<eqnStyle style="parametric"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="IntersectPath">
<input a0="d" a1="b"/>
<output a0="conic"/>
</command>
<element type="conic3d" label="conic">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="2"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="0.8184181113864796" A1="1" A2="-5.159855134101568" A3="0" A4="0.6029062986521391" A5="0"/>
<eqnStyle style="parametric"/>
</element>
<command name="Line">
<input a0="O" a1="O + k"/>
<output a0="ax"/>
</command>
<element type="line3d" label="ax">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coords ox="0" oy="0" oz="0" ow="1" vx="0" vy="0" vz="1" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<expression label="text1" exp=""V""/>
<element type="text" label="text1">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<startPoint exp="V"/>
<labelOffset x="-49" y="-13"/>
</element>
<expression label="text2" exp=""E""/>
<element type="text" label="text2">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<startPoint exp="C"/>
<labelOffset x="-13" y="-51"/>
</element>
</construction>
</geogebra>