一般线性变换的矩阵演示
2015-05-12 12:39
本GeoGebra范例演示一般线性变换及其矩阵表示。通过可交互的矩阵M及其逆矩阵invM，展示基向量i、j与任意向量u、v在变换前后的变化。利用多边形（正方形、平行四边形）顶点的变换，直观呈现线性变换对平面图形的作用效果，包括旋转、缩放、剪切等。结合网格线的变换，帮助学习者理解线性变换的几何意义与代数计算之间的联系。
所属专题
GeoGebra 几何变换互动学习资源
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.109.0" id="704a680b-a4f1-411e-a1a4-51d1b9ead004" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="937" height="673" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.47491995731056563" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="338,157,700,550" />
<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="2" visible="false" inframe="true" stylebar="true" location="3,3" size="200" window="1138,0,250,721" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="true" stylebar="true" location="1,3" size="305" window="342,254,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3" size="445" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="1" size="460" window="320,114,500,500" />
<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" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 | 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 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="445" height="506"/>
<coordSystem xZero="220.99999999999986" yZero="251.99999999999966" scale="49.99999999999998" yscale="49.99999999999998"/>
<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="64" g="64" b="64"/>
<lineStyle axes="0" grid="20"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="460" height="506"/>
<coordSystem xZero="228.00000000000003" yZero="252.0000000000001" scale="50.00000000000002" yscale="49.99999999999998"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" 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="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"/>
</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="point" label="O">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="vector" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="1.0" y="0.0" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<element type="vector" label="j">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0.0" y="1.0" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<element type="point" label="A">
<show object="true" label="false" ev="11"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.84" y="0.22" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="false" ev="11"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="52" y="3"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-0.24" y="0.84" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="A"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="11"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.84" y="0.22" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<command name="Vector">
<input a0="B"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.24" y="0.84" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<expression label="M" exp="{{x(u), x(v)}, {y(u), y(v)}}" />
<element type="list" label="M">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Invert">
<input a0="M"/>
<output a0="invM"/>
</command>
<element type="list" label="invM">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Corner">
<input a0="2" a1="1"/>
<output a0="C1"/>
</command>
<element type="point" label="C1">
<show object="false" label="false" ev="11"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.579999999999998" y="-5.1" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Corner">
<input a0="2" a1="2"/>
<output a0="C2"/>
</command>
<element type="point" label="C2">
<show object="false" label="false" ev="11"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.6599999999999975" y="-5.1" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Corner">
<input a0="2" a1="3"/>
<output a0="C3"/>
</command>
<element type="point" label="C3">
<show object="false" label="false" ev="11"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.6599999999999975" y="5.060000000000004" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Corner">
<input a0="2" a1="4"/>
<output a0="C4"/>
</command>
<element type="point" label="C4">
<show object="false" label="false" ev="11"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.579999999999998" y="5.060000000000004" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="corners" exp="{C1, C2, C3, C4}" />
<element type="list" label="corners">
<show object="false" label="true" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Zip">
<input a0="(invM * p)" a1="p" a2="corners"/>
<output a0="cornersS2"/>
</command>
<element type="list" label="cornersS2">
<show object="false" label="true" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Min">
<input a0="Zip[x(p), p, cornersS2]"/>
<output a0="a1"/>
</command>
<element type="numeric" label="a1">
<value val="-6.686708860759491"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<command name="Max">
<input a0="Zip[x(p), p, cornersS2]"/>
<output a0="a2"/>
</command>
<element type="numeric" label="a2">
<value val="6.7626582278481"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<command name="Min">
<input a0="Zip[y(p), p, cornersS2]"/>
<output a0="b1"/>
</command>
<element type="numeric" label="b1">
<value val="-7.000527426160337"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<command name="Max">
<input a0="Zip[y(p), p, cornersS2]"/>
<output a0="b2"/>
</command>
<element type="numeric" label="b2">
<value val="6.933016877637136"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<command name="Sequence">
<input a0="Line[O + (k * u), O + (k * u) + v]" a1="k" a2="floor(a1)" a3="ceil(a2)"/>
<output a0="linesX"/>
</command>
<element type="list" label="linesX">
<show object="true" label="true" ev="11"/>
<condition showObject="Distance[O, A] ≥ hiddenUnder"/>
<objColor r="32" g="32" b="32" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Line[O + (k * v), O + (k * v) + u]" a1="k" a2="floor(b1)" a3="ceil(b2)"/>
<output a0="linesY"/>
</command>
<element type="list" label="linesY">
<show object="true" label="true" ev="11"/>
<condition showObject="Distance[O, B] ≥ hiddenUnder"/>
<objColor r="32" g="32" b="32" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<element type="point" label="C">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1.0" y="1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="Line[C, C + (1, 0)]"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="3.0" y="1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="Line[C, C + (0, 1)]"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1.0" y="3.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="F" exp="D + E - C" type="point" />
<element type="point" label="F">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.0" y="3.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="C" a1="D" a2="F" a3="E"/>
<output a0="poly1" a1="c" a2="d" a3="f" a4="e"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0.0" y="2.0" z="-2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-2.0" y="0.0" z="6.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0.0" y="-2.0" z="6.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="2.0" y="0.0" z="-2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="Zip[(M * p), p, {C, D, F, E}]"/>
<output a0="poly2"/>
</command>
<element type="polygon" label="poly2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="11"/>
<objColor r="153" g="51" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="point" label="G">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-2.0" y="2.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="3"/>
</element>
<command name="Point">
<input a0="Line[G, G + (1, 0)]"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-1.0" y="2.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="G" a1="H"/>
<output a0="g"/>
</command>
<element type="conic" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<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="7.0" A3="0.0" A4="2.0" A5="-2.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Radius">
<input a0="g"/>
<output a0="r"/>
</command>
<element type="numeric" label="r">
<value val="1.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<cascell caslabel="h" >
<cellPair>
<inputCell>
<expression value="h:=(((invM (x, y)) - G) ((invM (x, y)) - G)) = r²" eval="Evaluate[(((invM * (x, y)) - G) * ((invM * (x, y)) - G)) = r^(2)]" evalCmd=""/>
</inputCell>
<outputCell>
<expression value="h: (1.3109154960921503 * x^(2)) + (1.326910751482135 * y^(2)) + ((0.0584174544677669 * x) * y) + (5.5907172995780588 * x) - (3.164556962025317 * y) + 8 = 1" native="true"/>
</outputCell>
</cellPair>
</cascell>
<element type="conic" label="h">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="153" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="0.7950122126904888" y0="-0.6065934236974325" z0="1.0" x1="0.6065934236974325" y1="0.7950122126904888" z1="1.0"/>
<matrix A0="1.3109154960921503" A1="1.326910751482135" A2="7.0" A3="0.02920872723388345" A4="2.7953586497890295" A5="-1.5822784810126584"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="O" a1="O + u"/>
<output a0="a"/>
</command>
<element type="line" label="a">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.22" y="0.84" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="O" a1="O + v"/>
<output a0="b"/>
</command>
<element type="line" label="b">
<show object="true" label="false" ev="7"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.84" y="-0.24" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<element type="point" label="I">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-1.0" y="-2.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="Circle[I, 1]"/>
<output a0="J"/>
</command>
<element type="point" label="J">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-0.9999999999999999" y="-1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="K" exp="I + Vector[I, J] + ((-2) * OrthogonalVector[Segment[I, J]])" type="point" />
<element type="point" label="K">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.0" y="-1.0000000000000002" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="L" exp="I + (0 * Vector[I, J]) + ((-2) * OrthogonalVector[Segment[I, J]])" type="point" />
<element type="point" label="L">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.0" y="-2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="N" exp="I + (0 * Vector[I, J]) + ((-4) * OrthogonalVector[Segment[I, J]])" type="point" />
<element type="point" label="N">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.0" y="-2.0000000000000004" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="P" exp="I + (-1 * Vector[I, J]) + ((-3) * OrthogonalVector[Segment[I, J]])" type="point" />
<element type="point" label="P">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.0" y="-3.0000000000000004" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="Q" exp="I + (-1 * Vector[I, J]) + (0 * OrthogonalVector[Segment[I, J]])" type="point" />
<element type="point" label="Q">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="-3.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="R" exp="I + (0 * Vector[I, J]) + OrthogonalVector[Segment[I, J]]" type="point" />
<element type="point" label="R">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.0" y="-2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="I" a1="J" a2="K" a3="L" a4="N" a5="P" a6="Q" a7="R"/>
<output a0="poly3" a1="i_1" a2="j_1" a3="k" a4="l" a5="n" a6="p" a7="q" a8="r_1"/>
</command>
<element type="polygon" label="poly3">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="i_1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-1.0" y="1.1102230246251565E-16" z="-0.9999999999999998"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="j_1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="2.220446049250313E-16" y="2.0" z="2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="k">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0.9999999999999998" y="0.0" z="-0.9999999999999998"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="l">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="4.440892098500626E-16" y="2.0" z="3.9999999999999996"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="n">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="1.0" y="-1.0" z="-5.000000000000001"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="p">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-4.440892098500626E-16" y="-3.0" z="-9.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="q">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-1.0" y="-1.0" z="-4.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="r_1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0.0" y="1.0" z="2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="Zip[(M * p), p, {I, J, K, L, N, P, Q, R}]"/>
<output a0="poly4"/>
</command>
<element type="polygon" label="poly4">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="11"/>
<objColor r="0" g="153" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="numeric" label="hiddenUnder">
<value val="0.15"/>
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
</construction>
</geogebra>