圆锥曲线:圆锥截痕的三维演示
2020-03-30 08:41
本GeoGebra范例通过三维模型,动态演示平面截取圆锥所得的圆锥截痕。用户可操作截面与圆锥的相对位置及角度,直观观察截面图形如何从圆、椭圆过渡到抛物线和双曲线。该模型结合正交线与交线等几何构造,帮助学习者深入理解圆锥曲线的几何定义与空间形成过程,是立体几何与解析几何结合的经典教学工具。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="w" id="e76c6b52-f234-4a25-972c-a462a59fec6a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1028" height="648" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.2519455252918288" orientation="1" />
</panes>
<views>
<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="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="395" window="859,209,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3" size="259" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" window="100,100,600,400" />
<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" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="150" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" 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 , 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" size="760" window="100,100,600,400" />
</views>
<toolbar show="false" 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="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="2"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="257" height="645"/>
<coordSystem xZero="389.99999999999966" yZero="438.99999999999983" scale="49.99999999999994" yscale="49.999999999999964"/>
<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="-0.35370335292456745" yZero="-0.5838596535122604" zZero="0.4193651383103387" scale="80.0015507907888" xAngle="13" zAngle="-18"/>
<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="0" g="0" b="0"/>
<clipping use="false" show="false" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="numeric" label="a">
<value val="0.77"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="14" y="-37"/>
<labelMode val="3"/>
<slider min="0" max="3" absoluteScreenLocation="true" width="200" x="9" y="49" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<caption val="圓錐開口大小"/>
</element>
<element type="numeric" label="m">
<value val="-0.35"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="28" y="28"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200" x="9" y="118" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="b">
<value val="0.3"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="80" y="1"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="200" x="13" y="180" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="h">
<value val="0"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200" x="10" y="239" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="angle" label="α">
<angleStyle val="3"/>
<value val="0.767944870877505"/>
<slider min="0" max="6.283185307179586" absoluteScreenLocation="true" width="180" x="23" y="306" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="true" label="false" 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>
<element type="numeric" label="dac">
<value val="0.42"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200" x="16" y="362" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="dap">
<value val="0.56"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="80" y="-5"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200" x="12" y="418" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<expression label="p" exp="(cos(α) * (x - (b * cos(α)))) + (sin(α) * (y - (b * sin(α)))) = (m * (z - h))" type="plane" />
<element type="plane3d" label="p">
<show object="true" label="false" ev="4"/>
<condition showObject="c"/>
<objColor r="102" g="204" b="255" alpha="0.56" dynamicr="0.4" dynamicg="0.8" dynamicb="1" dynamica="dap" colorSpace="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0.7193398003386512" y="0.6946583704589973" z="0.35" w="-0.3"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="con1" exp="x^(2) + y^(2) = (a^(2) * z^(2))" type="quadric" />
<element type="quadric" label="con1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.42" dynamicr="1" dynamicg="1" dynamicb="0" dynamica="dac" colorSpace="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<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="-0.5929" A3="0" A4="0" A5="0" A6="0" A7="0" A8="0" A9="0"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="((b * cos(α)), (b * sin(α)), h)" a1="p"/>
<output a0="Np"/>
</command>
<element type="line3d" label="Np">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coords ox="0.21580194010159534" oy="0.20839751113769917" oz="0" ow="1" vx="-0.6789548816162985" vy="-0.6556591077773252" vz="-0.3303504247281061" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="IntersectPath">
<input a0="p" a1="con1"/>
<output a0="curv1"/>
</command>
<element type="conic3d" label="curv1">
<show object="true" label="false" ev="4"/>
<condition showObject="d"/>
<objColor r="255" g="127" b="0" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="9" type="0" typeHidden="1"/>
<eigenvectors x0="0" y0="1" z0="1.0" x1="-1" y1="0" z1="1.0"/>
<matrix A0="1" A1="-0.4190645879732739" A2="0.06624036785531816" A3="0" A4="0" A5="0.14063657680607577"/>
<eqnStyle style="parametric"/>
</element>
<command name="Line">
<input a0="(0, 0, 0)" a1="(cos(α), sin(α), 0)"/>
<output a0="H1"/>
</command>
<element type="line3d" label="H1">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coords ox="0" oy="0" oz="0" ow="1" vx="0.7193398003386512" vy="0.6946583704589973" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Line">
<input a0="(0, 0, 0)" a1="(cos(α + 90°), sin(α + 90°), 0)"/>
<output a0="H2"/>
</command>
<element type="line3d" label="H2">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coords ox="0" oy="0" oz="0" ow="1" vx="-0.6946583704589974" vy="0.7193398003386511" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="16" y="494"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetViewDirection(Np)"/>
<font serif="false" sizeM="2" size="24" style="0"/>
<caption val="俯瞰截面"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="44" y="546"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetViewDirection(H1)"/>
<font serif="false" sizeM="2" size="24" style="0"/>
<caption val="水平視角1"/>
</element>
<element type="button" label="button3">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="109" y="602"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetViewDirection(H2)"/>
<font serif="false" sizeM="2" size="24" style="0"/>
<caption val="水平視角2"/>
</element>
<expression label="text1" exp=""圓錐開口大小:" + (LaTeX[a]) + """ />
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<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="0.6" size="-10" style="0"/>
<absoluteScreenLocation x="43" y="16"/>
</element>
<expression label="text2" exp=""截面斜度(相對xy平面):" + (LaTeX[m]) + """ />
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<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="0.6" size="-10" style="0"/>
<absoluteScreenLocation x="9" y="82"/>
</element>
<expression label="text3" exp=""平移截面:" + (LaTeX[b]) + """ />
<element type="text" label="text3">
<show object="true" label="false" ev="40"/>
<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="0.6" size="-10" style="0"/>
<absoluteScreenLocation x="57" y="146"/>
</element>
<expression label="text4" exp=""平移截面(z方向):" + (LaTeX[h]) + """ />
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<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="0.6" size="-10" style="0"/>
<absoluteScreenLocation x="37" y="203"/>
</element>
<expression label="text6" exp=""圓錐透明度" + (LaTeX[(dac * 100)]) + "\%"" />
<element type="text" label="text6">
<show object="true" label="false" ev="40"/>
<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="0.6" size="-10" style="0"/>
<absoluteScreenLocation x="65" y="328"/>
</element>
<expression label="text7" exp=""截面透平度:" + (LaTeX[(dap * 100)]) + "\%"" />
<element type="text" label="text7">
<show object="true" label="false" ev="40"/>
<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="0.6" size="-10" style="0"/>
<absoluteScreenLocation x="61" y="386"/>
</element>
<expression label="text5" exp=""旋轉截面:" + (LaTeX[α]) + """ />
<element type="text" label="text5">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.6" size="-10" style="0"/>
<absoluteScreenLocation x="54" y="271"/>
</element>
<element type="boolean" label="c">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="13" y="442"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="截平面"/>
</element>
<element type="boolean" label="d">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="146" y="442"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="截痕"/>
</element>
</construction>
</geogebra>