Rotation Transformation in Cartesian Coordinates
2019-04-30 02:45
This GeoGebra applet demonstrates rotation transformations of points and polygons in the Cartesian coordinate system. Using the Rotate command, point P is rotated about the origin O by a specified angle α to produce its image P', while DynamicCoordinates displays the real-time numerical relationship between original and rotated coordinates. The construction includes concentric circle arcs, polygon
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<expression label="text1" exp=""P'" + (If[b, text3, ""]) + """ />
<element type="text" label="text1">
<show object="true" label="false" ev="4"/>
<condition showObject="(a ∨ f) ∧ rotated"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="P'"/>
<labelOffset x="-49" y="-14"/>
</element>
<element type="boolean" label="c">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="760" y="7"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="ShowGrid(1, c)"/>
<caption val="格線"/>
</element>
<command name="CircleArc">
<input a0="O" a1="P" a2="P'"/>
<output a0="d"/>
</command>
<element type="conicpart" label="d">
<show object="true" label="false" ev="4"/>
<condition showObject="a ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-74" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CircleArc">
<input a0="O" a1="P'" a2="P"/>
<output a0="e"/>
</command>
<element type="conicpart" label="e">
<show object="true" label="false" ev="4"/>
<condition showObject="f ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-74" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="P'" a1="O" a2="P"/>
<output a0="ζ"/>
</command>
<element type="angle" label="ζ">
<angleStyle val="0"/>
<value val="0"/>
<show object="true" label="true" ev="4"/>
<condition showObject="f ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="49" y="31"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="60"/>
</element>
<element type="boolean" label="f">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="350" y="7"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue(P', P)
SetValue(b, false)
SetValue[a, false]
SetValue[hint, false]"/>
<caption val="順時針旋轉"/>
</element>
<command name="Rotate">
<input a0="Q" a1="ε" a2="O"/>
<output a0="Q'"/>
</command>
<element type="point" label="Q'">
<show object="true" label="true" ev="4"/>
<condition showObject="(a ∨ f) ∧ rotated ∧ x(P) ≠ 0 ∧ y(P) ≠ 0 ∧ hint"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="4" y="12"/>
<labelMode val="0"/>
<coords x="7" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="polyhedron" label="Archimedean">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="true" label="true"/>
<objColor r="211" g="47" b="47" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="polyhedron" label="Prism">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="true" label="true"/>
<objColor r="216" g="27" b="96" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="polyhedron" label="Pyramid">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="true" label="true"/>
<objColor r="219" g="97" b="20" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="boolean" label="hint">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="a ∨ f"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="520" y="57"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="提示"/>
</element>
<expression label="text2" exp=""P(" + (x(P)) + ", " + (y(P)) + ")"" />
<element type="text" label="text2">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="P"/>
<labelOffset x="8" y="-5"/>
</element>
<expression label="text4" exp=""拖拉紅點，把 P 繞 O " + (If[a, "逆時針", "順時針"]) + " 90°、180° 或 270°。"" />
<element type="text" label="text4">
<show object="true" label="false" ev="11"/>
<condition showObject="a ∨ f"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="0.9" size="-3" style="0"/>
<absoluteScreenLocation x="13" y="81"/>
</element>
</construction>
</geogebra>