台球轨迹模拟:椭圆与圆形台面上的反射
2020-12-23 05:00
本 GeoGebra 范例模拟台球在椭圆或圆形台面上的运动轨迹与反射规律。通过 Ellipse、Circle 构建台面边界,利用 Ray、Intersect、Tangent、OrthogonalLine 和 Mirror 等命令计算球体碰撞点及反射路径,并借助 Angle 命令展示入射角与反射角的关系。该演示直观呈现了椭圆的光学反射性质(从一个焦点发出的射线经椭圆反射后必过另一焦点)以及圆形台面上的周期轨道特征,用户可通过交互按钮与滑块调整初始发射角度与位置,观察不同条件下轨迹的变化与规律。
所属专题
点、线、角:基础几何互动教程
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.620.0" app="classic" platform="w" id="FF796BEE-CAC2-48E6-996A-378C6178E863" xsi:noNamespaceSchemaLocation="https://ggb123.cn/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.475" 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="true" location="1,1,1" size="584" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 || 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="2" visible="false" inframe="false" stylebar="true" location="3" size="380" tab="ALGEBRA" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 | 1009 || 6" visible="false" inframe="false" stylebar="true" location="3" size="300" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="100" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="22" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" 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="300" window="100,100,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" window="50,50,500,500" />
<view id="1" visible="true" inframe="false" stylebar="true" location="" size="598" window="100,100,600,400" />
</views>
<toolbar show="false" items="0 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 | 100001 100001" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="798" height="598"/>
<coordSystem xZero="399.44043163643266" yZero="334.7979539313182" scale="4.431899869150135" yscale="4.431899869150245"/>
<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="11" 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>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="3"/>
<angleUnit val="degree"/>
<algebraStyle val="3" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="人人都是神球手" author="" date="">
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<condition showObject="¬t"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="5"/>
<labelOffset x="23" y="50"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-40.714013620310894" y="1.533869025028689" z="1"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="5"/>
<labelOffset x="-15" y="7"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="24.946314589180542" y="-0.9481365086633652" z="1"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="20.12058629111658" y="15.634421720177208" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="改变球桌形状"/>
</element>
<command name="Ellipse">
<input a0="A" a1="B" a2="C"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="0.9992863187594154" y0="-0.03777370964382575" z0="1.0" x1="0.03777370964382575" y1="0.9992863187594154" z1="1.0"/>
<matrix A0="8174.617343980112" A1="25395.09074041555" A2="-12440875.915858645" A3="651.877191839977" A4="64256.54015329952" A5="-2298.0635182590786"/>
<eqnStyle style="implicit"/>
</element>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-46.112711287055376" y="21.772421280223178" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="A" a1="D"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" 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="1221.2387253776533" A3="0" A4="40.714013620310894" A5="-1.533869025028689"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="d"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-78" y="-16"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-59.24539811109533" y="-8.22997871742738" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="调整击球角度"/>
</element>
<command name="Ray">
<input a0="A" a1="E"/>
<output a0="f"/>
</command>
<element type="ray" label="f">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.76384774245607" y="-18.531384490784433" z="425.9501466343095"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Ray">
<input a0="A" a1="E"/>
<output a0="g"/>
</command>
<element type="ray" label="g">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.76384774245607" y="-18.531384490784433" z="425.9501466343095"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Ray">
<input a0="E" a1="A"/>
<output a0="h"/>
</command>
<element type="ray" label="h">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-9.76384774245607" y="18.531384490784433" z="-425.9501466343095"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="c" a1="h" a2="2"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="false"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.1562294199261625" y="22.3761424209298" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Tangent">
<input a0="F" a1="c"/>
<output a0="i" a1=""/>
</command>
<element type="line" label="i">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="69391.30396911815" y="565192.3840943306" z="-12566593.014583465"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="F" a1="i"/>
<output a0="j"/>
</command>
<element type="line" label="j">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-565192.3840943306" y="69391.30396911815" z="-2206201.7627950916"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Mirror">
<input a0="A" a1="j"/>
<output a0="A'"/>
</command>
<element type="point" label="A'">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="32.18489830851994" y="-7.41627025325854" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="F" a1="A'"/>
<output a0="k"/>
</command>
<element type="ray" label="k">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="29.79241267418834" y="33.341127728446104" z="-711.598958501644"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="j"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-3.6421467722293075" y="2.128337946251385" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="F"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="true" label="false"/>
<objColor r="204" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-20.84227339590111" y="39.55778420038473" z="-909.2490628027006"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F" a1="G"/>
<output a0="m"/>
</command>
<element type="segment" label="m">
<show object="true" label="false"/>
<objColor r="204" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="20.247804474678414" y="-2.485917352303145" z="79.03634794433167"/>
<lineStyle thickness="8" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F" a1="B"/>
<output a0="n"/>
</command>
<element type="segment" label="n">
<show object="true" label="false"/>
<objColor r="204" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="23.324278929593163" y="26.102544009106705" z="-557.1060247994"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="A"/>
<output a0="p"/>
</command>
<element type="segment" label="p">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-9.76384774245607" y="18.531384490784433" z="-425.9501466343095"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="A" a1="F" a2="G"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="1"/>
<value val="0.9637120359432996"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.52"/>
<layer val="0"/>
<labelOffset x="-71" y="36"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="G" a1="F" a2="B"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="1"/>
<value val="0.9637120359432996"/>
<show object="true" label="false"/>
<objColor r="255" g="255" b="102" alpha="1"/>
<layer val="0"/>
<labelOffset x="13" y="24"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<expression label="text1" exp=""人人都是神球手""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="2" size="24" style="0"/>
<absoluteScreenLocation x="57" y="82"/>
</element>
<expression label="text2" exp=""入射角\beta=" + (LaTeX[β]) + """ />
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="102" g="204" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="50" y="172"/>
</element>
<expression label="text3" exp=""反射角\alpha=" + (LaTeX[α]) + """ />
<element type="text" label="text3">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="102" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="50" y="126"/>
</element>
<expression label="text4" exp=""无论从哪个角度击打8号球,它都能在一次触壁反弹后进洞""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="0.7" size="-8" style="0"/>
<absoluteScreenLocation x="59" y="110"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="316" y="128"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[t,!t]
StartAnimation[H]"/>
<font serif="false" sizeM="2" size="24" style="0"/>
<caption val="开始击球"/>
</element>
<command name="PolyLine">
<input a0="A" a1="F" a2="B"/>
<output a0="q"/>
</command>
<element type="polyline" label="q">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="Point">
<input a0="q"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-15" y="9"/>
<labelMode val="3"/>
<animation step="0.1" speed="2" type="3" playing="false"/>
<coords x="-40.714013620310894" y="1.533869025028689" z="1"/>
<pointSize val="1"/>
<pointStyle val="0"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="318" y="173"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[H,A]
SetValue[t,false]"/>
<font serif="false" sizeM="2" size="24" style="0"/>
<caption val="复位"/>
</element>
<element type="boolean" label="t">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="image" label="pic1">
<file name="9812ece4e00c0f1c376aa5043efd18f2/9812ece4e00c0f1c376aa5043efd18f2.png"/>
<inBackground val="false"/>
<startPoint number="0" exp="H + (-2.5, -2.5)"/>
<startPoint number="1" exp="H + (2.5, -2.5)"/>
<startPoint number="2" exp="H + (-2.5, 2.5)"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="2"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="image" label="pic2">
<file name="3546ed8b53e645c42295c19948baab82/3546ed8b53e645c42295c19948baab82.png"/>
<inBackground val="false"/>
<startPoint number="0" exp="B + (-2.5, -2.5)"/>
<startPoint number="1" exp="B + (2.5, -2.5)"/>
<startPoint number="2" exp="B + (-2.5, 2.5)"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="2"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
</construction>
</geogebra>