反射变换（轴对称）探索
2014-03-15 05:57
本GeoGebra范例演示了平面几何中的反射变换（轴对称变换）。通过多边形及其顶点，用户可以直观地观察图形关于特定对称轴的镜像过程。范例使用了镜像、中点、线段和角等构造命令，动态展示了原图形与反射图形之间的对应关系、距离和角度变化，帮助深入理解反射变换的几何性质与作图原理。
所属专题
坐标几何（中一）GeoGebra互动教材
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.309.0" id="9b4c7336-da6c-4ec7-bb95-55887a2ab47a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="960" height="660" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.18333333333333332" orientation="0" />
</panes>
<views>
<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="true" 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="true" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="2" size="529" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="1" size="737" window="100,100,250,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="0" size="121" 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="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="true" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="true" stylebar="true" location="1" size="505" window="37,134,1206,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" 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="false" inframe="true" stylebar="false" location="1" size="480" window="50,50,500,500" />
</views>
<toolbar show="false" 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="7" help="false" />
<input show="false" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="958" height="529"/>
<coordSystem xZero="551.5" yZero="266.5" scale="42.38715725214916" yscale="42.387158641053304"/>
<evSettings axes="true" grid="true" gridIsBold="true" pointCapturing="2" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="153" b="204"/>
<gridColor r="160" g="160" b="160"/>
<lineStyle axes="3" grid="0"/>
<axis id="0" show="true" label="x" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1" tickExpression=""/>
<axis id="1" show="true" label="y" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1" tickExpression=""/>
<grid distX="1" distY="1" distTheta="0.5235987755982988"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="958" height="121"/>
<coordSystem xZero="529.4283874799518" yZero="60.36388040263368" scale="46.292692043955384" yscale="46.292692043955306"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="250" b="205"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="10"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<grid distX="1" distY="1" distTheta="0.5235987755982988"/>
</euclidianView>
<algebraView>
<auxiliary show="true"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="lang">
<value val="2"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="3"/>
<labelMode val="2"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="boolean" label="showAxis">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="571" y="55"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="ShowAxes[1,showAxis] SetCaption[E,If[showAxis,input+" = %v",input]] SetCaption[E',If[showAxis,input+"' = %v",input+"'"]]"/>
<caption val="Show coordinates"/>
</element>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<condition showObject="showPic"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-14" y="41"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-6" y="2" z="1"/>
<pointSize val="6"/>
<pointStyle val="1"/>
</element>
<expression label="listLine" exp="{If[lang ≟ 1, "（請選擇）", "(Please select)"], If[showAxis, If[lang ≟ 1, "x軸", "x-axis"], If[lang ≟ 1, "水平線", "horizontal line"]], If[showAxis, If[lang ≟ 1, "y軸", "y-axis"], If[lang ≟ 1, "鉛垂線", "vertical line"]], If[showAxis, "y=x", If[lang ≟ 1, "對角線(向右上)", "diagonal line (to top right corner)"]], If[showAxis, "y=-x", If[lang ≟ 1, "對角線(向右下)", "diagonal line (to bottom right corner)"]]}" />
<element type="list" label="listLine">
<show object="true" label="true" ev="11"/>
<condition showObject="showPoint ∨ showPic"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelOffset x="251" y="6"/>
<labelMode val="3"/>
<ggbscript onUpdate="SetValue[showImage,false]"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<ggbscript onUpdate="SetValue[showImage,false]"/>
<comboBox val="true"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
<caption val="Line of reflection:"/>
</element>
<element type="point" label="B">
<show object="true" label="true"/>
<condition showObject="showPic"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-30" y="14"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-8" y="4" z="1"/>
<pointSize val="6"/>
<pointStyle val="1"/>
</element>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<condition showObject="showPic"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="10"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-3" y="5" z="1"/>
<pointSize val="6"/>
<pointStyle val="1"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="C"/>
<output a0="poly1" a1="a" a2="b" a3="c"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<condition showObject="showPic"/>
<objColor r="0" g="255" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="a">
<show object="true" label="false" ev="4"/>
<condition showObject="showPic"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-2" y="-2" z="-8"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="true" label="false" ev="4"/>
<condition showObject="showPic"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-1" y="5" z="-28"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c">
<show object="true" label="false" ev="4"/>
<condition showObject="showPic"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="3" y="-3" z="24"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="If">
<input a0="SelectedIndex[listLine] ≟ 2" a1="y = 0" a2="If[SelectedIndex[listLine] ≟ 3, x = 0, If[SelectedIndex[listLine] ≟ 4, y = x, If[SelectedIndex[listLine] ≟ 5, y = (-x)]]]"/>
<output a0="L"/>
</command>
<element type="line" label="L">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="9" type="0" typeHidden="1"/>
<eqnStyle style="explicit"/>
</element>
<command name="Mirror">
<input a0="A" a1="L"/>
<output a0="A'"/>
</command>
<element type="point" label="A'">
<show object="true" label="true"/>
<condition showObject="showPic ∧ showImage"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-12" y="-2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="6"/>
<pointStyle val="1"/>
</element>
<command name="Mirror">
<input a0="B" a1="L"/>
<output a0="B'"/>
</command>
<element type="point" label="B'">
<show object="true" label="true"/>
<condition showObject="showPic ∧ showImage"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-34" y="33"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="6"/>
<pointStyle val="1"/>
</element>
<command name="Mirror">
<input a0="C" a1="L"/>
<output a0="C'"/>
</command>
<element type="point" label="C'">
<show object="true" label="true"/>
<condition showObject="showPic ∧ showImage"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="7" y="22"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="6"/>
<pointStyle val="1"/>
</element>
<command name="Polygon">
<input a0="A'" a1="B'" a2="C'"/>
<output a0="poly1'" a1="g" a2="h" a3="i"/>
</command>
<element type="polygon" label="poly1'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<condition showObject="showPic ∧ showImage"/>
<objColor r="0" g="255" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<condition showObject="showPic ∧ showImage"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<condition showObject="showPic ∧ showImage"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<condition showObject="showPic ∧ showImage"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<condition showObject="showPoint"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="10" y="6"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1" y="2" z="1"/>
<pointSize val="7"/>
<pointStyle val="0"/>
<caption val="P = %v"/>
</element>
<command name="Mirror">
<input a0="E" a1="L"/>
<output a0="E'"/>
</command>
<element type="point" label="E'">
<show object="true" label="true"/>
<condition showObject="showPoint ∧ showImage"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-43" y="4"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="7"/>
<pointStyle val="0"/>
<caption val="P' = %v"/>
</element>
<command name="Midpoint">
<input a0="E" a1="E'"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="NaN" y="NaN" z="1"/>
<pointSize val="7"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="E" a1="F"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="false" ev="4"/>
<condition showObject="showPoint ∧ showImage"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="2"/>
<selectionAllowed val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="3" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E'" a1="F"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<condition showObject="showPoint ∧ showImage"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="2"/>
<selectionAllowed val="false"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="3" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="e" a1="L"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="1"/>
<value val="NaN"/>
<show object="true" label="false"/>
<condition showObject="showPoint ∧ showImage"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="9" y="35"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="25"/>
<selectionAllowed val="false"/>
</element>
<command name="Angle">
<input a0="f" a1="L"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="1"/>
<value val="NaN"/>
<show object="true" label="false"/>
<condition showObject="showPoint ∧ showImage"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-66" y="32"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="25"/>
<selectionAllowed val="false"/>
</element>
<element type="boolean" label="showPoint">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="15" y="9"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[showPic,false]"/>
<caption val="Point"/>
</element>
<element type="boolean" label="showPic">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="15" y="55"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[showPoint,false]"/>
<caption val="Triangle ABC"/>
</element>
<element type="boolean" label="showImage">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="(SelectedIndex[listLine] > 1) ∧ (showPoint ∨ showPic)"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="247" y="55"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="Show image"/>
</element>
<expression label="input" exp=""P""/>
<element type="text" label="input">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<startPoint x="8.285736943018374" y="-1.4336403940943168" z="1"/>
</element>
<command name="Textfield">
<input a0="input"/>
<output a0="inputBox1"/>
</command>
<element type="textfield" label="inputBox1">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="116" y="8"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript onUpdate="SetCaption[E,If[showAxis,input+" = %v",input]] SetCaption[E',If[showAxis,input+"' = %v",input+"'"]]"/>
<linkedGeo exp="input"/>
<length val="2"/>
<caption val="_"/>
</element>
<element type="button" label="button0">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="3"/>
<labelOffset x="853" y="60"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[lang,If[lang==1,2,1]] RunClickScript[buttonLang] "/>
<font serif="false" sizeM="1.4" size="10" style="0"/>
<caption val="中文"/>
</element>
<element type="button" label="buttonLang">
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="3"/>
<labelOffset x="946" y="60"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetCaption[button0,If[lang==1,"ENG","中文"]] SetCaption[showPoint,If[lang==1,"點","Point"]] SetCaption[showPic,If[lang==1,"三角形ABC","Triangle ABC"]] SetCaption[showAxis,If[lang==1,"顯示坐標","Show coordinates"]] SetCaption[showImage,If[lang==1,"顯示反射影像","Show image"]] SetCaption[listLine,If[lang==1,"反射軸：","Line of reflection:"]]"/>
<font serif="false" sizeM="2" size="24" style="0"/>
</element>
</construction>
</geogebra>