同余运算与多边形几何变换
2019-02-26 12:06
本范例演示了同余(Modulus)运算在几何构造中的应用。通过序列(Sequence)和多边形(Polygon)命令,结合模运算生成特定的点集与多边形结构,直观展示了代数中的同余概念如何与几何变换、多边形构造相结合。适用于探索数论与几何的交叉领域,帮助理解模运算在图形生成和几何变换中的实际效果。
GeoGebra 画板及绘图内容加载中,请稍等……
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.524.0" app="classic" platform="w" id="302184c4-8e6a-4842-89ee-c4ffe050efca" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="923" height="580" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.3001083423618635" 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 | 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="false" stylebar="false" location="1,1,1" size="500" 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="false" inframe="false" stylebar="true" location="1,1" size="393" window="1119,416,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="923" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="277" 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="true" 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" />
</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="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="922" height="580"/>
<coordSystem xZero="588.3789256198343" yZero="284.1960743801652" scale="52.49999999999997" yscale="52.500000000000036"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="96" g="96" b="96"/>
<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>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
<startAnimation val="true"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="a">
<value val="5"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.10000000149011612"/>
<bgColor r="160" g="160" b="160" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="n - 1" absoluteScreenLocation="true" width="200" x="37" y="84" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="30" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="b">
<value val="2"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="102" g="153" b="0" alpha="0.10000000149011612"/>
<bgColor r="160" g="160" b="160" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="n - 1" absoluteScreenLocation="true" width="200" x="37" y="124" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="30" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="n">
<value val="9"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.10000000149011612"/>
<bgColor r="160" g="160" b="160" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="3" max="19" absoluteScreenLocation="true" width="200" x="37" y="164" fixed="true" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="30" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="x0" exp="1"/>
<element type="numeric" label="x0">
<value val="1"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<command name="Mod">
<input a0="(a * x0) + b" a1="n"/>
<output a0="y0"/>
</command>
<element type="numeric" label="y0">
<value val="7"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="r" exp="4"/>
<element type="numeric" label="r">
<value val="4"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="t">
<value val="0.347"/>
<symbolic val="true" />
<show object="true" label="false"/>
<objColor r="255" g="153" b="0" alpha="0.10000000149011612"/>
<bgColor r="160" g="160" b="160" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200" x="43" y="409" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="30" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="true"/>
</element>
<command name="Sequence">
<input a0="(r; π / 2 + ((2 * π / n) * k))" a1="k" a2="0" a3="n - 1"/>
<output a0="pts"/>
</command>
<element type="list" label="pts">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="k" a1="k" a2="0" a3="n - 1"/>
<output a0="s1"/>
</command>
<element type="list" label="s1">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="f[k, a, b, n]" a1="k" a2="0" a3="n - 1"/>
<output a0="s2"/>
</command>
<element type="list" label="s2">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="DivPoint[pts(s1(k) + 1), pts(s2(k) + 1), t]" a1="k" a2="1" a3="n"/>
<output a0="mpts"/>
</command>
<element type="list" label="mpts">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="9"/>
<pointStyle val="-1"/>
<caption val="moving points"/>
</element>
<command name="PolyLine">
<input a0="Join[{mpts, {mpts(1)}}]"/>
<output a0="graph"/>
</command>
<element type="polyline" label="graph">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<expression label="pixel" exp="x(Corner[2] - Corner[1]) / x(Corner[5])" />
<element type="numeric" label="pixel">
<value val="0.0190889370932755"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="pad" exp="(40 * pixel)" />
<element type="numeric" label="pad">
<value val="0.76355748373102"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="off" exp="(pixel * (-5, -15))" type="vector" />
<element type="vector" label="off">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.0954446854663775" y="-0.28633405639913245" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
</element>
<command name="Sequence">
<input a0="Text[k - 1, mpts(k) + (pad; Angle[mpts(k)]) + off]" a1="k" a2="1" a3="n"/>
<output a0="labels"/>
</command>
<element type="list" label="labels">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="204" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
<font serif="false" sizeM="2" size="16" style="0"/>
</element>
<expression label="text1" exp=""" + (LaTeX[a]) + "x+" + (LaTeX[b]) + " \text{ (mod " + (LaTeX[n]) + ")}"" />
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="224" g="224" b="224" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<absoluteScreenLocation x="37" y="204"/>
</element>
<command name="Polygon">
<input a0="pts"/>
<output a0="poly1"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="3" type="20" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="160" g="160" b="160" alpha="0.07000000029802322"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
</construction>
</geogebra>