Introduction to Polar and Cartesian Coordinates
2013-07-22 09:36
This GeoGebra applet introduces the polar coordinate system and compares it with Cartesian coordinates. Users can interactively observe point representations, including radial distance, polar angle, and Cartesian components. Using circle arcs, segments, and vectors, it illustrates quadrant characteristics. It is an excellent tool for learning coordinate geometry.
In collections
Secondary 1 Coordinate Geometry
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.570.0" app="classic" platform="w" id="c14fc88e-bc84-49df-adbb-cbc0ee9b2e97" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1000" height="690" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.10289855072463767" orientation="0" />
</panes>
<views>
<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="true" 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="true" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="2" size="609" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="1" size="191" window="100,100,250,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="0" size="71" 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="4097" visible="false" inframe="false" stylebar="true" location="1" size="604" window="166,78,947,666" />
<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" />
<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="true" location="1" size="480" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="7" help="false" />
<input show="false" cmd="true" top="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="998" height="609"/>
<coordSystem xZero="499" yZero="304.5" scale="28.96281604498158" yscale="28.96281604498155"/>
<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="153" b="255"/>
<gridColor r="128" g="128" b="128"/>
<lineStyle axes="3" grid="0"/>
<axis id="0" show="false" label="x" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1" tickExpression="1"/>
<axis id="1" show="false" label="y" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1" tickExpression="1"/>
<grid distX="1" distY="1" distTheta="0.5235987755982988"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="998" height="71"/>
<coordSystem xZero="199.12534410798563" yZero="70.03527747992996" scale="31.327632970502382" yscale="31.32763297050227"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="250" b="205"/>
<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"/>
<grid distX="1" distY="1" distTheta="0.5235987755982988"/>
</euclidianView>
<algebraView>
<mode val="1"/>
</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" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="R" exp="floor(Min[{abs(x(Corner[1])), abs(y(Corner[1, 1])), abs(x(Corner[1, 3])), abs(y(Corner[1, 3]))}]) - 1" />
<element type="numeric" label="R">
<value val="9"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<labelOffset x="-40" y="3"/>
<selectionAllowed val="false"/>
<caption val="半徑 = %v"/>
</element>
<expression label="listAngle" exp="{°, 5°, 10°, 15°, 30°}"/>
<element type="list" label="listAngle">
<show object="true" label="true" ev="11"/>
<condition showObject="showPolar"/>
<objColor r="0" g="0" b="255" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelOffset x="202" y="5"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="0" playing="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectedIndex val="4"/>
<comboBox val="true"/>
<pointSize val="3"/>
<angleStyle val="0"/>
<caption val="interval ="/>
</element>
<command name="Sequence">
<input a0="Sequence[((r * cos(theta)), (r * sin(theta))), theta, 0°, 359°, SelectedElement[listAngle]]" a1="r" a2="0" a3="R"/>
<output a0="listPoint"/>
</command>
<element type="list" label="listPoint">
<show object="false" label="true" ev="4"/>
<objColor r="160" g="160" b="160" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="1"/>
<angleStyle val="0"/>
</element>
<command name="Point">
<input a0="listPoint"/>
<output a0="P_{pol}"/>
</command>
<element type="point" label="P_{pol}">
<show object="true" label="true" ev="4"/>
<condition showObject="fixPolar"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="9"/>
<labelOffset x="5" y="15"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript val="SetValue[showRectCo,If[showRect,!showRectCo,showRectCo]]
SetValue[showPolarCo,If[showPolar,!showPolarCo,showPolarCo]]"/>
<coords x="1.0000000000000002" y="-1.7320508075688772" z="1"/>
<coordStyle style="polar"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="P"/>
</element>
<element type="boolean" label="showRect">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="34"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="ShowGrid[1,showRect]
ShowAxes[1,showRect]
SetValue[showRectCo,false]"/>
<caption val=" "/>
</element>
<element type="boolean" label="showPolar">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="2"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[showPolarCo,false]
SetVisibleInView(g,1,false)
SetVisibleInView(α,1,false)"/>
<caption val=" "/>
</element>
<command name="Sequence">
<input a0="Circle[(0, 0), r]" a1="r" a2="1" a3="R"/>
<output a0="listCircle"/>
</command>
<element type="list" label="listCircle">
<show object="true" label="true" ev="4"/>
<condition showObject="showPolar"/>
<objColor r="0" g="102" b="153" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="Vector">
<input a0="(R + 1, 0)"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false"/>
<condition showObject="showPolar ∧ (¬showRect)"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="10" y="0" z="0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
</element>
<command name="Text">
<input a0=""180°"" a1="((-R) - 1.8, -0.2)"/>
<output a0="text3"/>
</command>
<element type="text" label="text3">
<show object="true" label="false"/>
<condition showObject="showPolar ∧ (¬showRect)"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint x="-10.8" y="-0.2" z="1"/>
</element>
<command name="Text">
<input a0=""0°"" a1="(R + 1.2, -0.2)"/>
<output a0="text1"/>
</command>
<element type="text" label="text1">
<show object="false" label="false"/>
<condition showObject="showPolar ∧ (¬showRect)"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint x="10.2" y="-0.2" z="1"/>
</element>
<command name="Sequence">
<input a0="Text[r, (r - 0.2, -0.7)]" a1="r" a2="1" a3="R"/>
<output a0="listTextAxis"/>
</command>
<element type="list" label="listTextAxis">
<show object="false" label="false" ev="4"/>
<condition showObject="showPolar ∧ (¬showRect)"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<element type="boolean" label="showPolarCo">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="showPolar"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="382" y="2"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val=" "/>
</element>
<element type="boolean" label="showRectCo">
<value val="false"/>
<show object="false" label="true" ev="3"/>
<condition showObject="showRect"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="382" y="34"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val=" "/>
</element>
<command name="SelectedElement">
<input a0="listAngle"/>
<output a0="angle"/>
</command>
<element type="angle" label="angle">
<angleStyle val="0"/>
<value val="0.5235987755982988"/>
<show object="false" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<selectionAllowed val="false"/>
</element>
<command name="Sequence">
<input a0="Segment[(0, 0), ((R * cos(theta)), (R * sin(theta)))]" a1="theta" a2="0°" a3="360°" a4="angle"/>
<output a0="listRadiusThin"/>
</command>
<element type="list" label="listRadiusThin">
<show object="true" label="true" ev="4"/>
<condition showObject="showPolar ∧ angleIndex ≤ 2"/>
<objColor r="0" g="102" b="153" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="SelectedIndex">
<input a0="listAngle"/>
<output a0="angleIndex"/>
</command>
<element type="numeric" label="angleIndex">
<value val="5"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<command name="Sequence">
<input a0="Text[((theta / π * 180))°, (((R + 0.5) * cos(theta)), ((R + 0.5) * sin(theta)))]" a1="theta" a2="If[angleIndex ≥ 3, angle, 10°]" a3="89°" a4="If[angleIndex ≥ 3, angle, 10°]"/>
<output a0="listTextQ1"/>
</command>
<element type="list" label="listTextQ1">
<show object="true" label="false" ev="4"/>
<condition showObject="showPolar"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text[((theta / π * 180))°, (((R + 0.4) * cos(theta)) - 1, ((R + 0.4) * sin(theta)))]" a1="theta" a2="90° + If[angleIndex ≥ 3, angle, 10°]" a3="179°" a4="If[angleIndex ≥ 3, angle, 10°]"/>
<output a0="listTextQ2"/>
</command>
<element type="list" label="listTextQ2">
<show object="true" label="false" ev="4"/>
<condition showObject="showPolar"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text[((theta / π * 180))°, (((R + 0.4) * cos(theta)) - 1, ((R + 0.4) * sin(theta)) - 0.6)]" a1="theta" a2="180° + If[angleIndex ≥ 3, angle, 10°]" a3="269°" a4="If[angleIndex ≥ 3, angle, 10°]"/>
<output a0="listTextQ3"/>
</command>
<element type="list" label="listTextQ3">
<show object="true" label="false" ev="4"/>
<condition showObject="showPolar"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text[((theta / π * 180))°, (((R + 0.4) * cos(theta)), ((R + 0.4) * sin(theta)) - 0.6)]" a1="theta" a2="270° + If[angleIndex ≥ 3, angle, 10°]" a3="359°" a4="If[angleIndex ≥ 3, angle, 10°]"/>
<output a0="listTextQ4"/>
</command>
<element type="list" label="listTextQ4">
<show object="true" label="false" ev="4"/>
<condition showObject="showPolar"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Segment[(0, 0), ((R * cos(theta)), (R * sin(theta)))]" a1="theta" a2="0°" a3="360°" a4="If[angleIndex ≥ 3, angle, 10°]"/>
<output a0="listRadiusThick"/>
</command>
<element type="list" label="listRadiusThick">
<show object="false" label="true" ev="4"/>
<condition showObject="showPolar"/>
<objColor r="0" g="102" b="153" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<pointSize val="3"/>
<angleStyle val="0"/>
</element>
<expression label="P_{rec}" exp="(0, 0)" type="point"/>
<element type="point" label="P_{rec}">
<show object="true" label="true" ev="4"/>
<condition showObject="¬fixPolar"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="0" y="1"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript val="SetValue[showRectCo,If[showRect,!showRectCo,showRectCo]]
SetValue[showPolarCo,If[showPolar,!showPolarCo,showPolarCo]]"/>
<coords x="0" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="P"/>
</element>
<element type="boolean" label="fixPolar">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<condition showObject="showPolar"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="664" y="2"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[P_{pol},(4;120deg)]
SetValue[P_{rec},(0,0)]
SetValue[Q_{pol},(3;30deg)]
SetValue[Q_{rec},(4,0)]"/>
<caption val=" "/>
</element>
<expression label="Export_1" exp="(R + 2.5, R + 2.5)" type="point" />
<element type="point" label="Export_1">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="11.5" y="11.5" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="Export_2" exp="((-R) - 2.5, (-R) - 2.5)" type="point" />
<element type="point" label="Export_2">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-11.5" y="-11.5" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="listPoint"/>
<output a0="Q_{pol}"/>
</command>
<element type="point" label="Q_{pol}">
<show object="true" label="true" ev="4"/>
<condition showObject="fixPolar ∧ showQ"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="7" y="17"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1.0000000000000002" y="-1.7320508075688772" z="1"/>
<coordStyle style="polar"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="Q"/>
</element>
<expression label="Q_{rec}" exp="(4, 0)" type="point"/>
<element type="point" label="Q_{rec}">
<show object="true" label="true" ev="4"/>
<condition showObject="(¬fixPolar) ∧ showQ"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="1" y="3"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="4" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="Q"/>
</element>
<command name="If">
<input a0="fixPolar" a1="Segment[P_{pol}, Q_{pol}]" a2="Segment[P_{rec}, Q_{rec}]"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="true" label="false" ev="4"/>
<condition showObject="showQ"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="0"/>
<lineStyle thickness="5" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="showQ">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="664" y="34"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val=" "/>
</element>
<expression label="textRect" exp=""" + (If[fixPolar, "P(" + (x(P_{pol})) + ", " + (y(P_{pol})) + ")", "P(" + (x(P_{rec})) + ", " + (y(P_{rec})) + ")"]) + "
" + (If[showQ, If[fixPolar, "Q(" + (x(Q_{pol})) + ", " + (y(Q_{pol})) + ")", "Q(" + (x(Q_{rec})) + ", " + (y(Q_{rec})) + ")"], ""]) + """ />
<element type="text" label="textRect">
<show object="true" label="false" ev="8"/>
<condition showObject="showRectCo"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.6" size="12" style="0"/>
<absoluteScreenLocation x="24" y="173"/>
</element>
<command name="Text">
<input a0=""90°"" a1="(-0.3, R + 0.4)"/>
<output a0="text2"/>
</command>
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<condition showObject="showPolar ∧ (¬showRect)"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint x="-0.3" y="9.4" z="1"/>
</element>
<command name="Text">
<input a0=""270°"" a1="(-0.5, (-R) - 1)"/>
<output a0="text4"/>
</command>
<element type="text" label="text4">
<show object="true" label="false" ev="8"/>
<condition showObject="showPolar ∧ (¬showRect)"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint x="-0.5" y="-10" z="1"/>
</element>
<element type="numeric" label="lang">
<value val="2"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="5"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="button" label="button0">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="5"/>
<labelOffset x="920" y="39"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[lang,If[lang==1,2,1]]
RunClickScript[buttonLang]
"/>
<caption val="中文"/>
</element>
<element type="button" label="buttonLang">
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="5"/>
<labelOffset x="879" y="39"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetCaption[button0,If[lang==1,"ENG","中文"]]
SetCaption[showPolar,If[lang==1,"顯示極坐標平面"," "]]
SetCaption[showRect,If[lang==1,"顯示直角坐標平面"," "]]
SetCaption[showPolarCo,If[lang==1,"顯示極坐標"," "]]
SetCaption[showRectCo,If[lang==1,"顯示直角坐標"," "]]
SetCaption[fixPolar,If[lang==1,"局限點在極坐標平面格線"," "]]
SetCaption[showQ,If[lang==1,"顯示兩點"," "]]
SetCaption[listAngle,If[lang==1,"間隔 =","interval ="]]
SetCaption[button1,If[lang==1,"動畫","Animate"]]"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
</element>
<expression label="text5" exp=""Cartesian plane""/>
<element type="text" label="text5">
<show object="true" label="false" ev="43"/>
<condition showObject="lang ≠ 1"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="0.85" size="-3" style="0"/>
<absoluteScreenLocation x="43" y="59"/>
</element>
<expression label="text6" exp=""Polar plane""/>
<element type="text" label="text6">
<show object="true" label="false" ev="43"/>
<condition showObject="lang ≠ 1"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="0.85" size="-3" style="0"/>
<absoluteScreenLocation x="43" y="27"/>
</element>
<expression label="text7" exp=""Cartesian coordinates""/>
<element type="text" label="text7">
<show object="true" label="false" ev="43"/>
<condition showObject="lang ≠ 1 ∧ showRect"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="0.85" size="-3" style="0"/>
<absoluteScreenLocation x="419" y="59"/>
</element>
<expression label="text8" exp=""Polar coordinates""/>
<element type="text" label="text8">
<show object="true" label="false" ev="43"/>
<condition showObject="lang ≠ 1 ∧ showPolar"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="0.85" size="-3" style="0"/>
<absoluteScreenLocation x="419" y="28"/>
</element>
<expression label="text9" exp=""restrict points on polar grid lines""/>
<element type="text" label="text9">
<show object="true" label="false" ev="43"/>
<condition showObject="lang ≠ 1 ∧ showPolar"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="0.85" size="-3" style="0"/>
<absoluteScreenLocation x="701" y="29"/>
</element>
<expression label="text10" exp=""show two points""/>
<element type="text" label="text10">
<show object="true" label="false" ev="43"/>
<condition showObject="lang ≠ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="0.85" size="-3" style="0"/>
<absoluteScreenLocation x="700" y="59"/>
</element>
<expression label="textPolar" exp=""" + (LaTeX[If[fixPolar, "P(" + (Length[P_{pol}]) + ", " + (Angle[P_{pol}]) + ")", "P(" + (Length[P_{rec}]) + ", " + (Angle[P_{rec}]) + ")"]]) + "\\n" + (LaTeX[If[showQ, If[fixPolar, "Q(" + (Length[Q_{pol}]) + ", " + (Angle[Q_{pol}]) + ")", "Q(" + (Length[Q_{rec}]) + ", " + (Angle[Q_{rec}]) + ")"], ""]]) + """ />
<element type="text" label="textPolar">
<show object="true" label="false" ev="8"/>
<condition showObject="showPolarCo"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.6" size="12" style="0"/>
<absoluteScreenLocation x="23" y="59"/>
</element>
<command name="Segment">
<input a0="(0, 0)" a1="(Distance[P_{pol}, (0, 0)], 0)"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="2" z="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CircleArc">
<input a0="(0, 0)" a1="(Distance[P_{pol}, (0, 0)], 0)" a2="P_{pol}"/>
<output a0="c"/>
</command>
<element type="conicpart" label="c">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-4" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="numeric" label="t">
<value val="1"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="5"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200" x="32" y="355" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.001" speed="5" type="3" playing="false"/>
</element>
<command name="Point">
<input a0="{f, c}" a1="t"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<condition showObject="fixPolar"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="1.0000000000000002" y="-1.7320508075688772" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="(Distance[P_{pol}, (0, 0)], 0)" a1="(0, 0)" a2="A"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<emphasizeRightAngle val="false"/>
<value val="5.235987755982989"/>
<show object="true" label="false" ev="5"/>
<condition showObject="showPolar ∧ fixPolar"/>
<objColor r="127" g="0" b="255" alpha="0.5"/>
<layer val="8"/>
<labelMode val="1"/>
<decoration type="6"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="70"/>
<selectionAllowed val="false"/>
</element>
<element type="button" label="button1">
<show object="true" label="true" ev="3"/>
<condition showObject="showPolar ∧ fixPolar"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="153" b="153" alpha="255"/>
<layer val="8"/>
<labelOffset x="568" y="8"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetVisibleInView(g,1,true)
SetVisibleInView(α,1,true)
SetValue(t,0)
StartAnimation(t)"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<caption val="Animate"/>
</element>
<command name="Segment">
<input a0="(0, 0)" a1="A"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="5"/>
<condition showObject="showPolar ∧ fixPolar"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="8"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="1.7320508075688772" y="1.0000000000000002" z="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
</construction>
</geogebra>