Demonstration of the Optical Reflection Property of an Ellipse
2017-02-03 13:51
This GeoGebra applet demonstrates the optical reflection property of an ellipse. By constructing the foci, tangent, and normal lines, it visually illustrates the geometric principle that a light ray emitted from one focus of an ellipse will always pass through the other focus after reflecting off the boundary. Utilizing geometric objects such as vectors, tangents, orthogonal lines, and segments, t
In collections
圓錐曲線
<?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>