椭圆的轨迹:动圆与定圆相切的几何构造
2013-06-19 07:00
本范例通过几何作图直观演示椭圆的轨迹生成原理。通过构造一个动圆,使其与给定的大圆和小圆分别相切,利用动圆圆心到两定圆圆心距离之和为定值的几何性质,生动展示椭圆的定义与轨迹形成过程。该构造涉及圆、点、线段、角平分线及交点等基础几何对象,完美契合解析几何中圆锥曲线的教学与探索。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.0" id="a5bd0082-5bff-4a0a-b56b-5031de1efa1a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1366" height="728" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.23938506588579794" 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="1" visible="true" inframe="false" stylebar="false" location="1" size="1018" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="327" window="100,100,250,400" />
<view id="16" visible="false" inframe="true" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="true" stylebar="false" 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 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6" />
<input show="true" cmd="true" top="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
<consProtColumns col0="true" col1="true" col2="false" col3="true" col4="false" col5="true" col6="true" col7="false"/>
<consProtocol useColors="true" addIcons="false" showOnlyBreakpoints="false"/>
</gui>
<euclidianView>
<size width="1018" height="570"/>
<coordSystem xZero="556.1111111111112" yZero="279.55555555555554" scale="22.2222222222222" yscale="22.2222222222222"/>
<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="13" 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="3" grid="10"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
<startAnimation val="true"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="F_1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-3.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="F_2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="3.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="F_1" a1="10"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="false" label="false"/>
<objColor r="0" g="0" 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="-91.0" A3="0.0" A4="3.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="true"/>
<coords x="-5.698202459159328" y="9.629107097202033" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="F_1"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.629107097202033" y="2.6982024591593277" z="28.8873212916061"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="F_2"/>
<output a0="b"/>
</command>
<element type="segment" label="b">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.629107097202033" y="8.698202459159328" z="-28.8873212916061"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="LineBisector">
<input a0="b"/>
<output a0="d"/>
</command>
<element type="line" label="d">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-8.698202459159328" y="9.629107097202033" z="-58.09460737747798"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="a" a1="d"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<trace val="true"/>
<coords x="-434.91012295796634" y="308.131427110465" z="116.18921475495596"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Ellipse">
<input a0="F_1" a1="F_2" a2="B"/>
<output a0="e"/>
</command>
<element type="conic" label="e">
<show object="false" label="false"/>
<objColor r="0" g="0" 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="256.0" A1="400.0" A2="-6400.0" A3="0.0" A4="-0.0" A5="-0.0"/>
<eqnStyle style="implicit"/>
</element>
<command name="Circle">
<input a0="F_2" a1="1"/>
<output a0="f"/>
</command>
<element type="conic" label="f">
<show object="true" label="false"/>
<objColor r="0" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" 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="8.0" A3="0.0" A4="-3.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Circle">
<input a0="F_1" a1="11"/>
<output a0="g"/>
</command>
<element type="conic" label="g">
<show object="true" label="false"/>
<objColor r="0" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" 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="-112.0" A3="0.0" A4="3.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Line">
<input a0="B" a1="A"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-6.977127498576491" y="-1.9550828944458947" z="-20.931382495729473"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="g" a1="h"/>
<output a0="C" a1="D"/>
</command>
<element type="point" label="C">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.03197729492473966" y="-10.592017806922236" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.96802270507526" y="10.592017806922236" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="B" a1="D"/>
<output a0="k"/>
</command>
<element type="conic" label="k">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" 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="1.0" A1="1.0" A2="-46.95046086593673" A3="0.0" A4="3.743119564713433" A5="-2.651979598625542"/>
<eqnStyle style="specific"/>
</element>
</construction>
</geogebra>