椭圆的光学反射性质演示
2017-02-03 13:51
本GeoGebra范例演示了椭圆的光学反射性质。通过构造椭圆的焦点、切线与法线,展示从椭圆一个焦点发出的光线经椭圆边界反射后,必定经过另一个焦点的几何原理。范例结合了向量、切线、正交线及线段等几何对象,直观呈现了入射角与反射角相等的物理与几何规律,适用于解析几何与圆锥曲线光学性质的教学与探索。
所属专题
圆锥曲线
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.307.0" id="0b543706-2be3-4f01-ab70-9bb0d2868488" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="980" height="673" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.503061224489796" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="592,89,700,550" />
<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="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="2" visible="false" inframe="false" stylebar="false" location="3,3" size="173" 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="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="3" size="493" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="true" location="1" size="467" 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="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</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="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="493" height="469"/>
<coordSystem xZero="252.4752190081224" yZero="247.56889187915823" scale="68.84079615304555" yscale="68.84079615304564"/>
<evSettings axes="true" grid="true" 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="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="467" height="469"/>
<coordSystem xZero="181.9999999999999" yZero="336.0000000000001" scale="49.99999999999997" yscale="50.00000000000002"/>
<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>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="a">
<value val="3.0"/>
<symbolic val="true" />
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-46" y="-1"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="200.0" x="19.0" y="101.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="b">
<value val="2.0"/>
<symbolic val="true" />
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-13" y="1"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="200.0" x="250.0" y="102.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<expression label="c" exp="(a^(2) - b^(2))^(0.5)" />
<element type="numeric" label="c">
<value val="2.23606797749979"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.0"/>
</element>
<expression label="Elli" exp="(x / a)^(2) + (y / b)^(2) = 1" type="conic" />
<element type="conic" label="Elli">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="0.4444444444444444" A1="1.0" A2="-4.0" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="implicit"/>
</element>
<expression label="F_1" exp="(c, 0)" type="point" />
<element type="point" label="F_1">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="0" y="40"/>
<labelMode val="0"/>
<coords x="2.23606797749979" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="F_2" exp="((-c), 0)" type="point" />
<element type="point" label="F_2">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-22" y="28"/>
<labelMode val="0"/>
<coords x="-2.23606797749979" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="PointIn">
<input a0="Elli"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<condition showObject="blPQ"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="4" y="48"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="1.387618771571275" y="-0.13699882406756111" z="1.0"/>
<pointSize val="7"/>
<pointStyle val="2"/>
</element>
<command name="Point">
<input a0="Elli"/>
<output a0="Q"/>
</command>
<element type="point" label="Q">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.27737609398636204" y="1.9914330409682224" z="1.0"/>
<pointSize val="7"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="P" a1="Q"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="4"/>
<condition showObject="blPQ"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="16" y="-4"/>
<labelMode val="0"/>
<coords x="-1.1102426775849128" y="2.1284318650357834" z="0.0"/>
<lineStyle thickness="7" type="15" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="P"/>
</element>
<command name="Tangent">
<input a0="Q" a1="Elli"/>
<output a0="t_Q" a1=""/>
</command>
<element type="line" label="t_Q">
<show object="true" label="true"/>
<condition showObject="bltn"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.12327826399393868" y="1.9914330409682224" z="-4.0"/>
<lineStyle thickness="5" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="Q" a1="t_Q"/>
<output a0="n_Q"/>
</command>
<element type="line" label="n_Q">
<show object="true" label="true"/>
<condition showObject="bltn"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.9914330409682224" y="0.12327826399393868" z="0.30687551018841586"/>
<lineStyle thickness="5" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="Elli" a1="Line[Q, Mirror[P, n_Q]]" a2="2"/>
<output a0="R"/>
</command>
<element type="point" label="R">
<show object="true" label="true" ev="4"/>
<condition showObject="blPQ"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-34" y="29"/>
<labelMode val="0"/>
<coords x="-2.089358082072932" y="-1.4352054138502683" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="Q" a1="R"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="4"/>
<condition showObject="blPQ"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-31" y="-17"/>
<labelMode val="0"/>
<coords x="-2.366734176059294" y="-3.426638454818491" z="0.0"/>
<lineStyle thickness="6" type="15" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="Q"/>
</element>
<expression label="text1" exp=""調整 a, b 觀察橢圓形狀的變化""/>
<element type="text" label="text1">
<show object="true" label="true" ev="43"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="25" y="40"/>
</element>
<expression label="text2" exp=""橢圓方程式 (\frac{x}{" + (LaTeX[a]) + "})^2+(\frac{y}{" + (LaTeX[b]) + "})^2=1"" />
<element type="text" label="text2">
<show object="true" label="true" ev="11"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="-3.2800000000000002" y="3.9200000000000004" z="1.0"/>
</element>
<element type="boolean" label="bltn">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="32" y="258"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="顯示切線、法線"/>
</element>
<element type="boolean" label="blQF">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="33" y="304"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="顯示 Q 到兩焦點連線"/>
</element>
<command name="Segment">
<input a0="Q" a1="F_1"/>
<output a0="d1"/>
</command>
<element type="segment" label="d1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.9914330409682224" y="1.9586918835134277" z="-4.452979652244069"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="Q" a1="F_2"/>
<output a0="d2"/>
</command>
<element type="segment" label="d2">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.9914330409682224" y="-2.513444071486152" z="4.452979652244069"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="blPQ">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="32" y="212"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="P 射向 Q 的射線、反射線"/>
</element>
<expression label="d" exp="d1 + d2" />
<element type="numeric" label="d">
<value val="6.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="text3" exp=""QF_1 + QF_2 = " + d1 + "+" + d2 + "=" + d + """ />
<element type="text" label="text3">
<show object="true" label="true" ev="43"/>
<condition showObject="blQF"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-2.88" y="-0.8999999999999987" z="1.0"/>
</element>
</construction>
</geogebra>