旋转矩阵的几何意义
2013-05-06 13:43
本 GeoGebra 范例直观地展示了二维旋转矩阵的几何意义。用户可以通过拖曳图中的红色向量来改变其方向，或拖曳点 A 来改变其位置，从而实时观察向量与点在旋转变换下的坐标变化。该动态模型结合了多边形、向量与角度等几何元素，是理解线性代数中旋转变换及三角函数关系的绝佳可视化工具。
所属专题
GeoGebra 几何变换互动学习资源
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.2" version="4.2.31.0" id="031c1374-6f94-4a2b-9fe0-ef605472f7cf" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="690" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.2898550724637681" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="865,205,560,550" />
<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="false" 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="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="679" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="299" 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" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" 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 73 , 14 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="679" height="463"/>
<coordSystem xZero="348.99999999999994" yZero="275.00000000000006" scale="49.99999999999998" yscale="50.00000000000003"/>
<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="0" grid="20"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" 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="">
<command name="Point">
<input a0="Circle[(0, 0), 1]"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.8022332212740244" y="0.5970107693202042" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="1"/>
</element>
<expression label="u" exp="(1; Angle[C])" type="vector" />
<element type="vector" label="u">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="0.8022332212740244" y="0.5970107693202042" z="0.0"/>
<lineStyle thickness="9" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<command name="Rotate">
<input a0="u" a1="90*°"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="-0.5970107693202042" y="0.8022332212740244" z="0.0"/>
<lineStyle thickness="9" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<element type="numeric" label="t">
<value val="-3.5"/>
<show object="false" label="true"/>
<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>
<command name="If">
<input a0="t > 0" a1="ceil(t)" a2="floor(t)"/>
<output a0="t2"/>
</command>
<element type="numeric" label="t2">
<value val="-4.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="If" var="x">
<input a0="x > 0" a1="ceil(x)" a2="floor(x)"/>
<output a0="f"/>
</command>
<element type="function" label="f">
<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"/>
</element>
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="3.0" y="2.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Segment[(0, 0) + k * u, (0, 0) + k * u + f(y(A)) * v]" a1="k" a2="0" a3="f(x(A))" a4="sgn(x(A))"/>
<output a0="list1"/>
</command>
<element type="list" label="list1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" 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="Segment[(0, 0) + k * v, (0, 0) + k * v + f(x(A)) * u]" a1="k" a2="0" a3="f(y(A))" a4="sgn(y(A))"/>
<output a0="list2"/>
</command>
<element type="list" label="list2">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" 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>
<expression label="A'" exp="x(A) * u + y(A) * v" type="point" />
<element type="point" label="A'">
<show object="true" label="true"/>
<condition showObject="A' ≠ A"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.212678125181665" y="3.3954987505086613" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="text1" exp=""x\left [ \begin{array}{} " + (LaTeX[x(u)]) + " \\ " + (LaTeX[y(u)]) + " \end{array} \right ]+y\left [ \begin{array}{} " + (LaTeX[x(v)]) + " \\ " + (LaTeX[y(v)]) + " \end{array} \right ] =\left [ \begin{array}{} (" + (LaTeX[x(u)]) + ") x+(" + (LaTeX[x(v)]) + ")y \\ (" + (LaTeX[y(u)]) + ") x+(" + (LaTeX[y(v)]) + ")y\end{array} \right ] =\left [ \begin{array}{} " + (LaTeX[x(u)]) + " & " + (LaTeX[x(v)]) + " \\ " + (LaTeX[y(u)]) + " & " + (LaTeX[y(v)]) + " \\ \end{array} \right ] \left [ \begin{array}{} x \\ y \end{array} \right ] "" />
<element type="text" label="text1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint x="-6.699999999999999" y="5.279999999999999" z="1.0"/>
</element>
<element type="vector" label="u0">
<show object="true" label="false"/>
<objColor r="255" g="192" b="203" 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="9" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<element type="vector" label="v0">
<show object="true" label="false"/>
<objColor r="153" g="204" 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="9" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<command name="Sequence">
<input a0="Segment[(0, 0) + k * u0, (0, 0) + k * u0 + f(y(A)) * v0]" a1="k" a2="0" a3="f(x(A))" a4="sgn(x(A))"/>
<output a0="list3"/>
</command>
<element type="list" label="list3">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" 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="Segment[(0, 0) + k * v0, (0, 0) + k * v0 + f(x(A)) * u0]" a1="k" a2="0" a3="f(y(A))" a4="sgn(y(A))"/>
<output a0="list4"/>
</command>
<element type="list" label="list4">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" 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="Line">
<input a0="(0, 0)" a1="C"/>
<output a0="a"/>
</command>
<element type="line" label="a">
<show object="false" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.5970107693202042" y="0.8022332212740244" z="0.0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<expression label="x1" exp="f(x(A))" />
<element type="numeric" label="x1">
<value val="3.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="y1" exp="f(y(A))" />
<element type="numeric" label="y1">
<value val="2.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<element type="point" label="O">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="O" a1="O + x1 * u0" a2="O + x1 * u0 + y1 * v0" a3="O + y1 * v0"/>
<output a0="poly1" a1="o" a2="b" a3="c" a4="d"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="0" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="segment" label="o">
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="0.0" y="3.0" z="0.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="-2.0" y="0.0" z="6.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c">
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="0.0" y="-3.0" z="6.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d">
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="2.0" y="0.0" z="0.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="O" a1="O + x1 * u" a2="O + x1 * u + y1 * v" a3="O + y1 * v"/>
<output a0="poly2" a1="o_1" a2="e" a3="g" a4="h"/>
</command>
<element type="polygon" label="poly2">
<lineStyle thickness="0" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="segment" label="o_1">
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="-1.7910323079606127" y="2.4066996638220735" z="0.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e">
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="-1.6044664425480486" y="-1.1940215386404085" z="5.999999999999999"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g">
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="1.7910323079606125" y="-2.4066996638220735" z="6.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="h">
<show object="true" label="false"/>
<objColor r="102" g="204" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="1.6044664425480488" y="1.1940215386404085" z="-0.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
</construction>
</geogebra>