二维旋转变换矩阵演示
2020-12-07 12:59
本范例演示了二维平面中的旋转变换及其对应的矩阵表示。通过调整角度参数，可以直观观察向量在旋转前后的位置变化，并展示旋转矩阵如何作用于原向量以得到旋转后的新向量，帮助理解线性代数中的矩阵乘法与几何变换的内在联系。
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="5.0.619.0" app="classic" platform="d" id="2fe35e54-1700-4774-a90a-957a96f2fe5a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1920" height="1040" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.7276041666666667" 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="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="1" visible="true" inframe="false" stylebar="true" location="3" size="1893" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" tab="ALGEBRA" 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="4097" visible="false" inframe="false" stylebar="true" location="1" size="491" window="100,100,700,550" />
<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="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1893" height="868"/>
<coordSystem xZero="606.2423222868523" yZero="559.3835420292852" scale="89.34271570636609" yscale="89.34271570636645"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<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="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" 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="0" 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="">
<element type="numeric" label="θ">
<value val="0.66"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="3"/>
<slider min="0" max="(2 * π)" absoluteScreenLocation="true" width="200.0" x="300.0" y="124.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<ggbscript onUpdate=""/>
<caption val="$\large{θ}$"/>
</element>
<expression label="Mat" exp="{{cos(θ), (-sin(θ))}, {sin(θ), cos(θ)}}" />
<element type="list" label="Mat">
<objColor r="0" g="100" b="0" alpha="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
<symbolic val="true" />
</element>
<command name="Vector">
<input a0="(2.9745869667381406, 1.280278320677176)"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="66" y="-45"/>
<labelMode val="3"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="2.9745869667381406" y="1.280278320677176" z="0.0"/>
<lineStyle thickness="8" type="15" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<caption val="$\huge{\overrightarrow{u}}$"/>
</element>
<expression label="u'" exp="(Mat * u)" type="vector" />
<element type="vector" label="u'">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-71" y="-37"/>
<labelMode val="3"/>
<coords x="1.5649403820130177" y="2.8351793244571555" z="0.0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<caption val="$\large{M \cdot \overrightarrow{u}}$"/>
</element>
<command name="Angle">
<input a0="u" a1="u'"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="0.6599999999999999"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="16" y="-18"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<caption val="$\huge{α}$"/>
</element>
<expression label="text2" exp=""M\cdot\overrightarrow{u}""/>
<element type="text" label="text2">
<show object="true" label="true" ev="40"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="12" style="1"/>
<startPoint exp="Point[(0, 0), u']"/>
</element>
<expression label="text1" exp=""θ=" + (LaTeX[(θ / π * 180)]) + "°\\
\overrightarrow{u}=" + (LaTeX[u]) + "\\
M=
\begin{pmatrix} cosθ & -sinθ \\ sinθ & cosθ \end{pmatrix}=" + (LaTeX[Mat]) + "\\
M\cdot \overrightarrow{u}=" + (LaTeX[Mat]) + "" + (LaTeX[u]) + "=" + (LaTeX[u']) + "\\
α=" + (LaTeX[α]) + """ />
<element type="text" label="text1">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="2.0" size="12" style="1"/>
<absoluteScreenLocation x="939" y="46"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
</construction>
</geogebra>