Light Reflection Inside an Ellipse
2019-02-10 16:24
This applet demonstrates light reflection inside an ellipse. Users can drag the emission and reflection points to observe the trajectory. A focus option verifies the ellipse's optical property: light from one focus reflects to the other. It uses vectors and sequences to illustrate conic section optics and geometric construction.
In collections
Conic Sections | GeoGebra Book Collection
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.527.0" app="classic" platform="w" id="d6f589c2-2701-4c57-be52-2fc99dd24284" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="772" height="498" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.6360103626943006" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1,1,1" size="400" window="534,152,854,727" />
<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="224" 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="1" visible="true" inframe="false" stylebar="false" location="3" size="491" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="1" size="271" 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="false" stylebar="true" location="1" size="150" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" 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="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="2"/>
<font size="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="490" height="496"/>
<coordSystem xZero="271.51463979231033" yZero="257.17863226432013" scale="47.77582153376667" yscale="47.77582153376674"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="64" g="64" b="64"/>
<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="270" height="496"/>
<coordSystem xZero="112.99999999999999" yZero="392.99999999999955" scale="49.99999999999995" yscale="49.99999999999994"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="64" g="64" b="64"/>
<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>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="k">
<value val="8"/>
<show object="true" label="true" ev="3"/>
<objColor r="224" g="224" b="224" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="2"/>
<slider min="2" max="If[Large, 300, 30]" absoluteScreenLocation="true" width="170" x="16" y="45" fixed="false" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="If(k>30,SetValue[SP,false])"/>
<caption val="反射點個數"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="F_2"/>
</command>
<element type="point" label="F_2">
<show object="false" label="false" ev="4"/>
<condition showObject="ELL ≠ 0"/>
<objColor r="102" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-3.1295043181337516" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="F_1"/>
</command>
<element type="point" label="F_1">
<show object="false" label="false" ev="4"/>
<condition showObject="ELL ≠ 0"/>
<objColor r="102" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2.2916478815609356" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="false" label="true" ev="4"/>
<condition showObject="ELL ≠ 0"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="4.133583764082681" y="1.2805354319460678" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Ellipse">
<input a0="F_2" a1="F_1" a2="C"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="255" b="153" 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="252.50131468984975" A1="370.0568793788679" A2="-5795.676338695668" A3="0" A4="105.77992587799437" A5="0"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="P_0"/>
</command>
<element type="point" label="P_0">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-29" y="-8"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="StartAnimation[t,false]
SetValue[i,1]
SetValue[t,0]
If(F1,SetValue[I_0,Intersect(Ray(P_0+0.001*Vector(P_0, F_1), F_1), c)])
If(F2,SetValue[I_0,Intersect(Ray(P_0+0.001*Vector(P_0, F_2), F_2), c)])
If(start&&k>30,SetValue[SP,false])
If(start,StartAnimation[t,true])"/>
<coords x="3.3072704714130907" y="2.5114671901104635" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="I_0"/>
</command>
<element type="point" label="I_0">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="33"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="StartAnimation[t,false]
SetValue[i,1]
SetValue[t,0]
SetValue[FF,false]
SetValue[F1,false]
SetValue[F2,false]
If(start,StartAnimation[t,true])"/>
<coords x="0.731831142995379" y="-3.8571695831054513" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="button" label="button1">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="2" y="135"/>
<labelMode val="3"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="StartAnimation[t,false]
S_P={}
S_I={}
S_T={}
S_V={}
SetValue[i,1]
SetValue[t,0]
SetValue[start,false]
SetValue[SP,false]"/>
<font serif="false" sizeM="2" size="20" style="0"/>
<caption val="Reset"/>
</element>
<expression label="S_P" exp="{(-3.0794984632240316, 0.021721154318748048)}"/>
<element type="list" label="S_P">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<expression label="S_I" exp="{(-5.113604516424597, -0.8618380121434712)}"/>
<element type="list" label="S_I">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<expression label="S_T" exp="{(888.8905435985671 * x) + (1001.5726306068423 * y) = 5467.609202892821}"/>
<element type="list" label="S_T">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<expression label="S_V" exp="{(-1001.5726306068423 * x) + (888.8905435985671 * y) = -700.4742844930611}"/>
<element type="list" label="S_V">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<element type="numeric" label="i">
<value val="1"/>
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="2"/>
<slider min="1" max="k" absoluteScreenLocation="true" width="200" x="30" y="626" fixed="false" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Vector">
<input a0="P_0" a1="I_0"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="4"/>
<condition showObject="AIM ≠ 0 ∧ t ≟ 0"/>
<objColor r="255" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.575439328417712" y="-6.368636773215915" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="P_0"/>
</element>
<element type="numeric" label="t">
<value val="0"/>
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="k" absoluteScreenLocation="true" width="200" x="23" y="677" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.02" speed="(speed * If[k ≥ 150, 0.3, 1])" type="3" playing="false"/>
<ggbscript onUpdate="If(i<=k,SetValue[S_T,i,Tangent(If(i==1,I_0,Element(S_I,i-1)),c)])
If(i<=k,SetValue[S_V,i,OrthogonalLine(If(i==1,I_0,Element(S_I,i-1)),Element(S_T,i))])
If(i<=k,SetValue[S_P,i,Mirror(If(i==1,P_0,If(i==2,I_0,Element(S_I,i-2))),Element(S_V,i))])
If(i<=k,SetValue[S_I,i,Intersect(Ray(If(i==1,I_0,Element(S_I,i-1))+0.01*Vector(If(i==1,I_0,Element(S_I,i-1)),Element(S_P,i)),Element(S_P,i)),c)])
If(i<=k,SetValue[i,i+1])"/>
</element>
<command name="Union">
<input a0="{P_0, I_0}" a1="S_I"/>
<output a0="I"/>
</command>
<element type="list" label="I">
<show object="false" label="false" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<expression label="P" exp="((t - floor(t)) * Element[I, floor(t) + 2]) + ((1 - t + floor(t)) * Element[I, floor(t) + 1])" type="point" />
<element type="point" label="P">
<show object="true" label="false" ev="4"/>
<condition showObject="t ≠ 0 ∧ t ≠ k"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="0"/>
<labelOffset x="78" y="17"/>
<labelMode val="0"/>
<coords x="3.3072704714130907" y="2.5114671901104635" z="1"/>
<pointSize val="4"/>
<pointStyle val="5"/>
</element>
<element type="boolean" label="SP">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="k ≤ 30 ∨ i ≟ k"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-3" y="202"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(SP,SetValue[speed,10],SetValue[speed,1])"/>
<caption val="快速畫線"/>
</element>
<element type="numeric" label="speed">
<value val="1"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="2"/>
<slider min="1" max="10" absoluteScreenLocation="true" width="200" x="30" y="130" fixed="false" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="boolean" label="FF">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-3" y="233"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[F2,false]
If(FF,SetValue[F1,true],SetValue[F1,false])"/>
<caption val="通過焦點"/>
</element>
<element type="boolean" label="F1">
<value val="true"/>
<show object="false" label="true" ev="3"/>
<condition showObject="FF ≠ 0"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="8" y="263"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(FF,If(F1,SetValue[F2,false],SetValue[F2,true]),SetValue[F1,false])
If(F1,SetValue[I_0,Intersect(Ray(P_0+0.001*Vector(P_0, F_1), F_1), c)])"/>
<caption val="通過焦點F_1"/>
</element>
<element type="boolean" label="F2">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="FF ≠ 0"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="8" y="292"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(FF,If(F2,SetValue[F1,false],SetValue[F1,true]),SetValue[F2,false])
If(F2,SetValue[I_0,Intersect(Ray(P_0+0.001*Vector(P_0, F_2), F_2), c)])"/>
<caption val="通過焦點F_2"/>
</element>
<element type="button" label="button2">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="3" y="91"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="StartAnimation[t,false]
S_P={}
S_I={}
S_T={}
S_V={}
SetValue[i,1]
SetValue[t,0]
StartAnimation[t,true]
SetValue[start,true]"/>
<font serif="false" sizeM="2" size="20" style="0"/>
<caption val="Start"/>
</element>
<element type="boolean" label="AIM">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-3" y="173"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="瞄準線"/>
</element>
<expression label="text1" exp=""反射點個數""/>
<element type="text" label="text1">
<show object="true" label="false" ev="43"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-1.340000000000001" y="6.379999999999999" z="1"/>
</element>
<element type="boolean" label="PRO">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-2" y="324"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[SEG,false]
SetValue[arrow,true]
SetValue[ELL,false]
SetValue[Large,false]"/>
<caption val="進階選項"/>
</element>
<element type="boolean" label="arrow">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<condition showObject="PRO ≠ 0"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="7" y="357"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(PRO,If(arrow,SetValue[SEG,false],SetValue[SEG,true]))"/>
<caption val="有向光線"/>
</element>
<element type="boolean" label="SEG">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="PRO ≠ 0"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="122" y="359"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(PRO,If(SEG,SetValue[arrow,false],SetValue[arrow,true]),SetValue[SEG,false])"/>
<caption val="線段"/>
</element>
<element type="boolean" label="ELL">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="PRO ≠ 0"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="7" y="388"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="調整橢圓形狀位置"/>
</element>
<expression label="FF_1" exp="F_1" type="point" />
<element type="point" label="FF_1">
<show object="true" label="false" ev="8"/>
<condition showObject="¬ELL"/>
<objColor r="153" g="255" b="153" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="2.2916478815609356" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<caption val="F_1"/>
</element>
<expression label="FF_2" exp="F_2" type="point" />
<element type="point" label="FF_2">
<show object="true" label="false" ev="8"/>
<condition showObject="¬ELL"/>
<objColor r="153" g="255" b="153" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="-3.1295043181337516" y="0" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<caption val="F_2"/>
</element>
<element type="boolean" label="Large">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<condition showObject="PRO ≠ 0"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="435"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="巨量反射點(用於包絡線)"/>
</element>
<command name="Sequence">
<input a0="Vector[Element[I, n], If[t ≤ n, P, Element[I, n + 1]]]" a1="n" a2="1" a3="floor(t) + 1"/>
<output a0="V"/>
</command>
<element type="list" label="V">
<show object="false" label="false" ev="4"/>
<condition showObject="arrow ≠ 0"/>
<objColor r="255" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="3" type="0" typeHidden="1" opacity="127"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Segment[Element[I, n], If[t ≤ n, P, Element[I, n + 1]]]" a1="n" a2="1" a3="floor(t) + 1"/>
<output a0="SE"/>
</command>
<element type="list" label="SE">
<show object="false" label="false" ev="4"/>
<condition showObject="SEG ≠ 0"/>
<objColor r="255" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="3" type="0" typeHidden="1" opacity="127"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<element type="boolean" label="start">
<value val="false"/>
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="79" y="91"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="text3" exp=""調整焦點F_1,F_2及過點C""/>
<element type="text" label="text3">
<show object="true" label="false" ev="43"/>
<condition showObject="PRO ≠ 0"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.7" size="-6" style="0"/>
<startPoint x="-1.4000000000000004" y="-0.47999999999999854" z="1"/>
</element>
<expression label="text4" exp=""F_2""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<objColor r="153" g="255" b="153" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="F_2"/>
<labelOffset x="5" y="3"/>
</element>
<expression label="text5" exp=""F_1""/>
<element type="text" label="text5">
<show object="true" label="false" ev="40"/>
<objColor r="153" g="255" b="153" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="F_1"/>
<labelOffset x="-2" y="5"/>
</element>
<expression label="text2" exp=""由P_0為發光點，射向I_0""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="-5.578441798304707" y="4.901613928267225" z="1"/>
</element>
<expression label="text6" exp=""可調整P_0和I_0之位置""/>
<element type="text" label="text6">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<startPoint x="-5.557510708730601" y="-4.747618365397826" z="1"/>
</element>
</construction>
</geogebra>