圆内迭代反射几何作图
2020-08-18 08:48
本范例展示了圆内迭代反射的几何构造过程。通过运用圆、向量、线段、角以及镜像反射等GeoGebra命令，结合迭代列表与数列功能，动态演示了点或图形在圆内的连续反射轨迹。该构造深入探讨了几何变换中的反射原理与迭代规律，适用于研究圆内反射路径、多边形构图及相关几何性质，是理解几何变换与迭代序列的直观教学工具。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.599.0" app="classic" platform="d" id="60b70901-5d15-4365-adfe-332fbf260f84" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/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="0.3440702781844802" orientation="1" />
<pane location="1" divider="0.4707174231332357" orientation="1" />
</panes>
<views>
<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 , 538 | 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="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1,3" size="643" window="100,100,600,400" />
<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="true" inframe="false" stylebar="true" location="1,1" size="216" window="100,100,700,550" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="470" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" 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="true" cmd="true" top="true" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="643" height="497"/>
<coordSystem xZero="367.1459017948816" yZero="258.0034070368534" scale="179.1562347559648" yscale="179.15623475596652"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="0" g="51" b="0"/>
<axesColor r="255" g="255" b="255"/>
<gridColor r="255" g="255" b="255"/>
<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>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="2" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="O" exp="(0, 0)" type="point"/>
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-9" y="31"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="10"/>
</element>
<expression label="P0" exp="(1, 0)" type="point"/>
<element type="point" label="P0">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="4" y="18"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="1.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="10"/>
</element>
<command name="Circle">
<input a0="O" a1="P0"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="3" type="0" typeHidden="1" opacity="178"/>
<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="-1.0" A3="0.0" A4="-0.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<element type="numeric" label="inDeg">
<value val="9.0"/>
<show object="true" label="true"/>
<objColor r="255" g="0" b="255" alpha="0.10000000149011612"/>
<bgColor r="255" g="0" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="30" absoluteScreenLocation="true" width="100.0" x="60.0" y="38.0" fixed="false" horizontal="true" showAlgebra="true"/>
<pointSize val="3.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate=""/>
</element>
<expression label="inAngle" exp="(inDeg / 180 * π)" />
<element type="numeric" label="inAngle">
<value val="0.15707963267948966"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<ggbscript onUpdate=""/>
</element>
<expression label="P1" exp="(1; π - (2 * inAngle))" type="point" />
<element type="point" label="P1">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-27" y="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="-0.9510565162951535" y="0.3090169943749475" z="1.0"/>
<coordStyle style="polar"/>
<pointSize val="3"/>
<pointStyle val="10"/>
</element>
<command name="Vector">
<input a0="P0" a1="P1"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="-1.9510565162951536" y="0.3090169943749475" z="0.0"/>
<lineStyle thickness="3" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="P0"/>
</element>
<command name="Segment">
<input a0="P1" a1="O"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="0.3090169943749475" y="0.9510565162951535" z="-0.0"/>
<lineStyle thickness="3" type="10" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="O" a1="P1" a2="P0"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="1"/>
<value val="0.1570796326794897"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="22" y="18"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2"/>
<arcSize val="45"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<command name="Mirror">
<input a0="P0" a1="Line[O, P1]"/>
<output a0="P2"/>
</command>
<element type="point" label="P2">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-11" y="31"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="0.8090169943749473" y="-0.5877852522924734" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="10"/>
</element>
<command name="Vector">
<input a0="P1" a1="P2"/>
<output a0="v2"/>
</command>
<element type="vector" label="v2">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="1.7600735106701009" y="-0.8968022466674208" z="0.0"/>
<lineStyle thickness="3" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="P1"/>
</element>
<command name="Mirror">
<input a0="P1" a1="Line[O, P2]"/>
<output a0="P3"/>
</command>
<element type="point" label="P3">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="-0.5877852522924729" y="0.8090169943749476" z="1.0"/>
<coordStyle style="polar"/>
<pointSize val="3"/>
<pointStyle val="10"/>
</element>
<command name="Mirror">
<input a0="P2" a1="Line[O, P3]"/>
<output a0="P4"/>
</command>
<element type="point" label="P4">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-33" y="10"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="0.30901699437494756" y="-0.9510565162951536" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="10"/>
</element>
<element type="numeric" label="n">
<value val="20.0"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="0" alpha="1.0"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="30" absoluteScreenLocation="true" width="100.0" x="58.0" y="75.0" fixed="false" horizontal="true" showAlgebra="true"/>
<pointSize val="3.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate=""/>
</element>
<command name="IterationList">
<input a0="Mirror[a, Line[O, b]]" a1="a" a2="b" a3="{P0, P1}" a4="n"/>
<output a0="Ps"/>
</command>
<element type="list" label="Ps">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="5" y="55"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Vector[Ps(k), Ps(k + 1)]" a1="k" a2="1" a3="n"/>
<output a0="Vs"/>
</command>
<element type="list" label="Vs">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text["$P_{" + (k) + "}$", (1.15 * Ps(k + 1)) + (-0.1, 0.1), true, true]" a1="k" a2="0" a3="n"/>
<output a0="l1"/>
</command>
<element type="list" label="l1">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<expression label="a" exp="180 - (inDeg * 2)" />
<element type="numeric" label="a">
<value val="162.0"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<ggbscript onUpdate=""/>
</element>
<element type="numeric" label="shiftDeg">
<value val="1.0"/>
<show object="false" label="true"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="100.0" x="30.0" y="130.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
<ggbscript onUpdate=""/>
</element>
<command name="GCD">
<input a0="shiftDeg" a1="360"/>
<output a0="b"/>
</command>
<element type="numeric" label="b">
<value val="1.0"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<ggbscript onUpdate=""/>
</element>
</construction>
</geogebra>