射线上滚动椭圆的几何构造与运动轨迹
2019-04-02 06:47
本GeoGebra范例演示了椭圆在射线上无滑动滚动的几何构造过程。通过运用圆、射线、交点、缩放、旋转及椭圆等命令,精确模拟了椭圆的焦点、中心及轮廓在滚动过程中的位置变化。该动态模型直观展示了圆锥曲线的运动学特征、角度变化以及相关的几何变换,适用于解析几何与运动轨迹的深入探究。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.531.0" app="classic" platform="d" id="230ce281-0689-4668-9ff3-011cce93df21" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1513" height="1005" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.13218770654329148" 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 | 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="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="513" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1281" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="200" 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" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="70" 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="1281" height="812"/>
<coordSystem xZero="-103.95612560503307" yZero="634.8037752541966" scale="37.194059908517104" yscale="37.19405990851716"/>
<evSettings axes="false" 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="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" 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="point" label="O">
<show object="false" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="33.43261244565966" y="7.073140251109304" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="a">
<value val="4.42"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="3" max="5" absoluteScreenLocation="true" width="200.0" x="43.0" y="40.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.001" speed="1" type="0" playing="false"/>
</element>
<command name="Circle">
<input a0="O" a1="a"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-19" y="-36"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="1148.2324879535397" A3="0.0" A4="-33.43261244565966" A5="-7.073140251109304"/>
<eqnStyle style="specific"/>
</element>
<command name="Intersect">
<input a0="c" a1="Ray[O, O - (0, 1)]"/>
<output a0="B" a1=""/>
</command>
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="33.43261244565966" y="2.653140251109303" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="X">
<show object="true" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="79" y="-14"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="5.374255263850255" y="5.341018101841691" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="X" a1="X + (1, 0)"/>
<output a0="f"/>
</command>
<element type="ray" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="1.0" z="-5.341018101841694"/>
<lineStyle thickness="6" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="Q"/>
</command>
<element type="point" label="Q">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="255" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-5" y="30"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="9.43581116092861" y="5.341018101841694" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="e'">
<value val="0.615"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200.0" x="44.0" y="86.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.001" speed="1" type="0" playing="false"/>
</element>
<expression label="T" exp="X + ((a * Integral[sqrt(1 - (e'^(2) * cos(x)^(2))), 0, (2 * π)]), 0)" type="point" />
<element type="point" label="T">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="30.296333061135407" y="5.341018101841691" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Dilate">
<input a0="T" a1="floor(Distance[X, Q] / Distance[X, T])" a2="X"/>
<output a0="Q_1"/>
</command>
<element type="point" label="Q_1">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.374255263850255" y="5.341018101841691" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="Q_2" exp="Q_1 + T - X" type="point" />
<element type="point" label="Q_2">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="30.296333061135403" y="5.341018101841691" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="β" exp="((Distance[Q_1, Q] / Distance[Q_1, Q_2] * 2) * π)" />
<element type="numeric" label="β">
<value val="1.023971939434009"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="W" exp="Q_1 + ((a * Integral[sqrt(1 - (e'^(2) * cos(x)^(2))), 0, β]), 0)" type="point" />
<element type="point" label="W">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.231416940226273" y="5.341018101841691" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="2"/>
</element>
<command name="Rotate">
<input a0="B" a1="(-β)" a2="O"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="29.657136523904086" y="4.774840539530347" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Dilate">
<input a0="P" a1="e'" a2="O"/>
<output a0="F_1"/>
</command>
<element type="point" label="F_1">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="31.11069475377998" y="5.659685928488246" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="F_2" exp="(2 * O) - F_1" type="point" />
<element type="point" label="F_2">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="35.754530137539334" y="8.486594573730363" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Ellipse">
<input a0="F_1" a1="F_2" a2="a"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="0.8541800727953782" y0="0.5199773103119812" z0="1.0" x1="-0.5199773103119812" y1="0.8541800727953782" z1="1.0"/>
<matrix A0="226.3215717141785" A1="280.6167500458215" A2="238375.797310084" A3="-52.51079357372223" A4="-7195.1051875687135" A5="-229.2686191202879"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="d" a1="Segment[B, Intersect[OrthogonalLine[B, Line[P, O]], Line[P, O]]]"/>
<output a0="A" a1=""/>
</command>
<element type="point" label="A">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="33.0174569391266" y="3.335126856599912" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="W + F_1 - A" a1="π + Angle[OrthogonalVector[Tangent[A, d]], Vector[(0, 1)]]" a2="W"/>
<output a0="F_1'"/>
</command>
<element type="point" label="F_1'">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="7.564485173484632" y="7.843148273354309" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="W + F_2 - A" a1="π + Angle[OrthogonalVector[Tangent[A, d]], Vector[(0, 1)]]" a2="W"/>
<output a0="F_2'"/>
</command>
<element type="point" label="F_2'">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="12.465683558350818" y="10.195779556021673" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Ellipse">
<input a0="F_1'" a1="F_2'" a2="W"/>
<output a0="e"/>
</command>
<element type="conic" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="0.9015190348501244" y0="0.4327394479394041" z0="1.0" x1="-0.4327394479394041" y1="0.9015190348501244" z1="1.0"/>
<matrix A0="216.49541756874362" A1="290.4429041912629" A2="33213.013966324426" A3="-46.12285057117979" A4="-1752.2164854162515" A5="-2157.7150539632953"/>
<eqnStyle style="implicit"/>
</element>
<command name="Rotate">
<input a0="W + P - A" a1="π + Angle[OrthogonalVector[Tangent[A, d]], Vector[(0, 1)]]" a2="W"/>
<output a0="P'"/>
</command>
<element type="point" label="P'">
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.030370231880176" y="7.106755554795825" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Center">
<input a0="e"/>
<output a0="O'"/>
</command>
<element type="point" label="O'">
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="10.015084365917726" y="9.019463914687993" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="O'" a1="P'"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.912708359892168" y="-3.9847141340375503" z="16.78404975058271"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""拖动Q点滚动椭圆""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<startPoint x="4.193038511757639" y="3.463020051347002" z="1.0"/>
</element>
</construction>
</geogebra>