椭圆轨迹:内切于两定圆的动圆圆心轨迹
2013-06-11 03:20
本GeoGebra范例演示了经典的椭圆定义与轨迹问题。通过构造一个同时内切于给定大圆和小圆的动圆,追踪该动圆圆心的运动轨迹,直观地验证了动圆圆心到两定圆圆心的距离之和为常数,从而形成椭圆。范例包含输入框以动态调整大圆和小圆的半径,帮助学习者深入理解椭圆的几何性质、焦点与轨迹方程之间的关系,是解析几何与几何作图的优秀教学素材。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.2" version="4.2.47.0" id="933436b7-78c5-446c-97f2-088eb5ec4359" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="643" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.205" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="588,229,700,550" />
<view id="4" toolbar="0 59 || 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 | 1011 || 1008 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="773" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="164" window="82,248,250,400" />
<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" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 59 | 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 66 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="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="773" height="477"/>
<coordSystem xZero="426.8388554977139" yZero="230.09386185403707" scale="28.92928392969454" yscale="28.929288915256077"/>
<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="13" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="128" g="128" b="128"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="0" grid="20"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="false" tickDistance="1.0"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="false" tickDistance="1.0"/>
</euclidianView>
<kernel>
<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"/>
<construction title="" author="" date="">
<element type="numeric" label="r">
<value val="1.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="R">
<value val="7.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-7" y="35"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-2.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-8" y="33"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="2.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="B" a1="R"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-45.0" A3="0.0" A4="2.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Ellipse">
<input a0="A" a1="B" a2="(R - r) / 2"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="true" label="false"/>
<condition showObject="f"/>
<objColor r="51" g="153" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="80.0" A1="144.0" A2="-720.0" A3="-0.0" A4="-0.0" A5="-0.0"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="d"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-7" y="-7"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1.6450344448183687" y="1.869917062237004" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="P"/>
<output a0="b"/>
</command>
<element type="segment" label="b">
<show object="true" label="false"/>
<condition showObject="f"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-1.869917062237004" y="3.6450344448183687" z="-3.739834124474008"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="P" a1="Intersect[c, Ray[B, P]]"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="false"/>
<condition showObject="f"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="2"/>
<selectionAllowed val="false"/>
<coords x="-1.3252040277748645" y="2.5832238366084193" z="-2.650408055549728"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="f">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="18" y="22"/>
<labelMode val="0"/>
<caption val="顯示細節"/>
</element>
<command name="Textfield">
<input a0="R"/>
<output a0="inputBox1"/>
</command>
<element type="textfield" label="inputBox1">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="23" y="52"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<linkedGeo exp="R"/>
<length val="4"/>
<caption val="R ="/>
</element>
<command name="Circle">
<input a0="A" a1="r"/>
<output a0="c2"/>
</command>
<element type="conic" label="c2">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="3.0" A3="0.0" A4="-2.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Textfield">
<input a0="r"/>
<output a0="inputBox2"/>
</command>
<element type="textfield" label="inputBox2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="28" y="86"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<linkedGeo exp="r"/>
<length val="4"/>
<caption val="r ="/>
</element>
<command name="Circle">
<input a0="P" a1="Distance[P, A] + r"/>
<output a0="cp"/>
</command>
<element type="conic" label="cp">
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-2.226482960968702" A3="0.0" A4="-1.6450344448183687" A5="-1.869917062237004"/>
<eqnStyle style="specific"/>
</element>
<command name="Segment">
<input a0="P" a1="Intersect[cp, Ray[P, A]]"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="true" label="false"/>
<condition showObject="f"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="2"/>
<coords x="2.8523722054399707" y="0.5414645943052654" z="-5.7047444108799406"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
</construction>
</geogebra>