Rolling of a Convex Polygon on a Straight Line
2020-05-01 18:44
This applet demonstrates a convex polygon rolling on a straight line. Combining translation and rotation, it illustrates the geometric features of a polygon tumbling continuously. Users can observe rotation around vertices and subsequent translations, exploring relationships between vertex trajectories, rotation angles, and translation distances to understand rigid body motion.
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.579.0" app="classic" platform="d" id="5c01cb3a-030a-48c2-b43f-be8c486bc812" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1382" height="754" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.5477568740955138" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 || 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="2" visible="false" inframe="false" stylebar="true" location="3,3" size="279" window="100,100,250,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="true" location="1,3" size="300" window="100,100,600,400" />
<view id="32" visible="false" inframe="true" stylebar="true" location="1,1" size="431" window="267,46,941,704" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="354" window="-105,498,1177,514" />
<view id="1" visible="true" inframe="false" stylebar="true" location="3" size="1355" window="383,174,600,400" />
<view id="16" visible="false" inframe="false" stylebar="true" location="1" size="593" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="433,119,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<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 | 100006 100007" visible="false" inframe="false" stylebar="true" location="1" size="534" window="100,100,600,400" />
</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 | 100004 100002 100003 100005 100006 100007 100008 100009" position="1" help="true" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="16"/>
<consProtColumns col0="true" col1="true" col2="false" col3="false" col4="true" col5="true" col6="true" col7="false"/>
<consProtocol useColors="true" addIcons="false" showOnlyBreakpoints="false"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1355" height="572"/>
<coordSystem xZero="724.3632605028383" yZero="226.87177352368195" scale="76.02158800221018" yscale="76.0215880022104"/>
<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="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="593" height="546"/>
<coordSystem xZero="145.19851111172161" yZero="255.7576957854432" scale="68.4108304278266" yscale="183.80596509392095"/>
<evSettings axes="true" 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="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="false" positiveAxis="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="0.25" tickExpression="1 / 4" positiveAxis="true"/>
</euclidianView>
<spreadsheetView>
<prefCellSize height="29"/>
</spreadsheetView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="2" 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="">
<command name="Point">
<input a0="xAxis"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-14" y="41"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-2.0" y="0.0" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<caption val="$\Large{%n}$"/>
</element>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="0" y="-11"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-1.9726509994270893" y="0.5873865388590592" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<caption val="$\Large{%n}$"/>
</element>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-37" y="4"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-3.1058981764986258" y="1.239003665675191" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<caption val="$\Large{%n}$"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-31" y="38"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-4.168317405003191" y="0.0" z="1.0"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<caption val="$\Large %n$"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="C" a3="D"/>
<output a0="q1" a1="a" a2="b" a3="c" a4="d"/>
</command>
<element type="polygon" label="q1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="a">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-0.5873865388590592" y="0.027349000572910676" z="-1.1747730777181185"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-0.6516171268161318" y="-1.1332471770715364" z="-0.6197590394458017"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.239003665675191" y="-1.0624192285045653" z="5.164560544496654"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="2.168317405003191" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="l1" exp="π - {InteriorAngles[q1]}" />
<element type="list" label="l1">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Sum[l1, k]" a1="k" a2="0" a3="4"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<expression label="l3" exp="{a, b, c, d}" />
<element type="list" label="l3">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Sum[l3, k]" a1="k" a2="0" a3="4"/>
<output a0="l4"/>
</command>
<element type="list" label="l4">
<show object="true" label="true" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="numeric" label="n">
<value val="1.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="3" absoluteScreenLocation="true" width="200.0" x="133.0" y="267.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="angle" label="α">
<angleStyle val="3"/>
<value val="4.642575810304916"/>
<slider min="0" max="(n * 360°)" absoluteScreenLocation="true" width="180.0" x="139.0" y="322.0" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="10" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation speed="1" type="0" playing="false"/>
</element>
<expression label="β" exp="α - (floor(α / (360°)) * 360°)" />
<element type="angle" label="β">
<angleStyle val="3"/>
<value val="4.642575810304916"/>
</element>
<expression label="l5" exp="(l2, 0)" />
<element type="list" label="l5">
<show object="true" label="true" ev="11"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="PolyLine">
<input a0="l5"/>
<output a0="f"/>
</command>
<element type="polyline" label="f">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="true" ev="11"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-60" y="53"/>
<labelMode val="3"/>
<caption val="$\Large %n$"/>
</element>
<expression label="E" exp="(β, 0)" type="point" />
<element type="point" label="E">
<show object="true" label="true" ev="11"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-42" y="-9"/>
<labelMode val="3"/>
<coords x="4.642575810304916" y="0.0" z="1.0"/>
<pointSize val="7"/>
<pointStyle val="0"/>
<caption val="$\Large %n=(β,0)$"/>
</element>
<command name="PathParameter">
<input a0="ClosestPoint[f, E]"/>
<output a0="e"/>
</command>
<element type="numeric" label="e">
<value val="0.820078317758619"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<expression label="l6" exp="A + (l4, 0)" />
<element type="list" label="l6">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
<caption val="$\Large %n$"/>
</element>
<command name="PolyLine">
<input a0="l6"/>
<output a0="g"/>
</command>
<element type="polyline" label="g">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-63" y="45"/>
<labelMode val="3"/>
<caption val="$\Large{%n}$"/>
</element>
<command name="Point">
<input a0="g" a1="floor((4 * e)) / 4"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="2"/>
<labelMode val="3"/>
<coords x="1.5273893410073711" y="0.0" z="1.0"/>
<pointSize val="9"/>
<pointStyle val="0"/>
<caption val="$\Large{%n}$"/>
</element>
<command name="Point">
<input a0="q1" a1="floor((4 * e)) / 4"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="6" y="8"/>
<labelMode val="3"/>
<coords x="-4.168317405003191" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\Large{%n}$"/>
</element>
<expression label="I" exp="G + (floor(α / (360°)) * (l6(5) - l6(1)))" type="point" />
<element type="point" label="I">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.5273893410073711" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Translate">
<input a0="Polygon[Rotate[A, (-α), H], Rotate[B, (-α), H], Rotate[C, (-α), H], Rotate[D, (-α), H]]" a1="Vector[H, I]"/>
<output a0="q2"/>
</command>
<element type="polygon" label="q2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="2"/>
<labelMode val="0"/>
</element>
<expression label="text1" exp=""公众号:\ GeoGebra物理数学可视化\\
QQ群:\ 583765627""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<condition showObject="h"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="191" g="255" b="0" alpha="255"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="591" y="433"/>
</element>
<expression label="text2" exp=""文海平原创、肖建伟修改\ \ 2020/5/1\\""/>
<element type="text" label="text2">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<bgColor r="255" g="239" b="213" alpha="255"/>
<layer val="2"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="608" y="375"/>
</element>
<element type="boolean" label="h">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="446" y="352"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<caption val="帮助"/>
</element>
</construction>
</geogebra>