手电筒的光线模拟
2022-03-08 02:15
本范例利用抛物线的光学性质，模拟手电筒的光线照射方式。通过构建几何模型，直观展示了从焦点发出的光线经抛物线反射后平行于对称轴的路径。该演示不仅涉及圆锥曲线与向量的应用，还能帮助观察者探讨并理解为何多数手电筒照射时，常常呈现外圈较亮而中间偏暗的光学现象。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.691.0" app="classic" platform="w" id="eb647bbf-7dff-4830-bde1-adcb287c987d" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="882" height="535" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.29365079365079366" 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="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="true" location="1,1" size="329" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3,3" size="140" tab="ALGEBRA" 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="390" window="247,241,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3" size="259" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="1" size="614" 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="false" 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" />
</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="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="258" height="534"/>
<coordSystem xZero="130.6342667106999" yZero="456.41732710558585" scale="0.5294314278985249" yscale="0.5294314278985256"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="0" g="0" b="0"/>
<axesColor r="160" g="160" b="160"/>
<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="false" label="" unitLabel="" tickStyle="1" showNumbers="true" positiveAxis="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="613" height="534"/>
<coordSystem xZero="306.2241119894084" yZero="450.6105549566728" scale="73.26270042733736" yscale="73.26270042733762"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="0" g="0" b="0"/>
<axesColor r="128" g="128" b="128"/>
<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" positiveAxis="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="n">
<value val="24"/>
<show object="true" label="false" ev="3"/>
<objColor r="255" g="255" b="0" alpha="0.10000000149011612"/>
<layer val="1"/>
<labelOffset x="0" y="-5"/>
<labelMode val="9"/>
<slider min="1" max="108" absoluteScreenLocation="true" width="120" x="18" y="427" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<caption val="光線數目"/>
</element>
<command name="PolyLine">
<input a0="{Corner[1], Corner[2], Corner[3], Corner[4], Corner[1]}"/>
<output a0="osq"/>
</command>
<element type="polyline" label="osq">
<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="1"/>
<labelMode val="0"/>
</element>
<expression label="f" exp="f(x) = If[-4 ≤ x ≤ 4, (0.25 * x^(2))]" type="function"/>
<element type="function" label="f">
<show object="true" label="false" ev="6"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="PointIn">
<input a0="y ≥ (0.25 * x^(2)) + 0.1"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="11"/>
<objColor r="255" g="153" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.7749630805228406" y="1.0866451071596945" z="1"/>
<pointSize val="6"/>
<pointStyle val="0"/>
<caption val="光源"/>
</element>
<command name="Sequence">
<input a0="A + (cos(π / 2 + ((2 * k) * π / n)), sin(π / 2 + ((2 * k) * π / n)))" a1="k" a2="1" a3="n - 1"/>
<output a0="B1" randomResult="{(0.51614403542032, 2.052570933448763), (0.2749630805228408, 1.9526705109441331), (0.06785629933629311, 1.7937518883462422), (-0.09106232326159791, 1.5866451071596948), (-0.19096274576622763, 1.3454641522622155), (-0.22503691947715943, 1.0866451071596945), (-0.19096274576622785, 0.8278260620571742), (-0.09106232326159824, 0.5866451071596948), (0.06785629933629289, 0.3795383259731471), (0.27496308052284013, 0.22061970337525605), (0.51614403542032, 0.12071928087062622), (0.7749630805228406, 0.08664510715969453), (1.0337821256253608, 0.12071928087062611), (1.27496308052284, 0.2206197033752555), (1.4820698617093875, 0.3795383259731462), (1.6409884843072788, 0.5866451071596941), (1.7408889068119087, 0.8278260620571729), (1.7749630805228405, 1.0866451071596945), (1.740888906811909, 1.3454641522622148), (1.6409884843072797, 1.5866451071596939), (1.4820698617093884, 1.7937518883462418), (1.2749630805228411, 1.952670510944133), (1.033782125625362, 2.0525709334487625)}"/>
</command>
<element type="list" label="B1">
<show object="false" label="false" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="A" a1="k" a2="1" a3="n - 1"/>
<output a0="A1" randomResult="{(0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945), (0.7749630805228406, 1.0866451071596945)}"/>
</command>
<element type="list" label="A1">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Ray[(0.99 * A1(k)) + (0.01 * B1(k)), B1(k)]" a1="k" a2="1" a3="n - 1"/>
<output a0="C1" randomResult="{-0.9562665680261777x - 0.2562308546514954y = -1.0195032898689655, -0.8573651497465942x - 0.4949999999999998y = -1.2023156656245957, -0.7000357133746822x - 0.700035713374682y = -1.3031922156884925, -0.49500000000000033x - 0.8573651497465942y = -1.3152583698801816, -0.25623085465149575x - 0.9562665680261775y = -1.2376918397317624, y = 1.0866451071596945, 0.2562308546514951x - 0.9562665680261777y = -0.8405529348403163, 0.49499999999999966x - 0.8573651497465944y = -0.5480449201625697, 0.700035713374682x - 0.7000357133746822y = -0.21818854986279695, 0.8573651497465941x - 0.49500000000000044y = 0.12653700953649777, 0.9562665680261776x - 0.2562308546514954y = 0.46263928084817674, x = 0.7749630805228406, 0.9562665680261777x + 0.256230854651495y = 1.019503289868965, 0.8573651497465946x + 0.49499999999999944y = 1.2023156656245957, 0.7000357133746828x + 0.7000357133746814y = 1.3031922156884923, 0.49500000000000033x + 0.8573651497465938y = 1.3152583698801812, 0.2562308546514964x + 0.9562665680261774y = 1.2376918397317627, y = 1.0866451071596945, -0.2562308546514951x + 0.9562665680261778y = 0.8405529348403165, -0.49499999999999944x + 0.8573651497465947y = 0.5480449201625703, -0.7000357133746817x + 0.7000357133746823y = 0.21818854986279734, -0.857365149746594x + 0.49500000000000055y = -0.1265370095364975, -0.9562665680261773x + 0.2562308546514963y = -0.4626392808481754}"/>
</command>
<element type="list" label="C1">
<show object="false" label="true" ev="7"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Intersect[f, Element[C1, k]]" a1="k" a2="1" a3="n - 1"/>
<output a0="A2" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (-3.8877600574127564, 3.7786695660035097), (-2.8734976734266624, 2.0642472197971102), (-2.0848454208019302, 1.0866451071596943), (-1.4142720131226652, 0.500041331775509), (-0.8176617988611905, 0.16714270432922948), (-0.29057375051479567, 0.021108276122058678), (0.15087377648612038, 0.005690724107795953), (0.5005833242351246, 0.06264591612557198), (0.7749630805228406, 0.15014194404336267), (0.9992428604077219, 0.2496215735189515), (1.1959071988519119, 0.3575485070664566), (1.383257712727503, 0.4783504749550308), (1.5783589806542515, 0.6228042679529819), (1.8017009037021692, 0.8115315366003033), (2.0848454208019302, 1.0866451071596943), (2.4860687828471537, 1.545134498261782), (3.127062875619691, 2.444630557019723), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="A2">
<show object="false" label="false" ev="7"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="If[A2(k) ≠ A1(k), Mirror[A1(k), OrthogonalLine[A2(k), Tangent[A2(k), f]]]]" a1="k" a2="1" a3="n - 1"/>
<output a0="B2" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (-3.3666481288695427, 9.137440423764017), (-2.5230929114374794, 5.825123217024115), (-1.9660957526368557, 3.943987085390532), (-1.59087365330402, 2.759613447294127), (-1.3099642903859503, 1.939027839455802), (-1.008840529106593, 1.345808359675622), (-0.6283175148169617, 0.9807859857153908), (-0.22382345366636325, 0.8366571654168392), (0.1439493348136811, 0.8421389290461812), (0.464604938159903, 0.93158352819691), (0.7528322793798774, 1.0734119149580796), (1.0287330959265943, 1.2621597726928042), (1.313958733529125, 1.5120094218877305), (1.6349002365629715, 1.861319882741983), (2.0304543337164214, 2.395397702198452), (2.5674214241780815, 3.3147324735172994), (3.373414591626779, 5.149405734395203), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="B2">
<show object="false" label="false" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Ray[(0.99 * A2(k)) + (0.01 * B2(k)), B2(k)]" a1="k" a2="1" a3="n - 1"/>
<output a0="C2" randomResult="{?, ?, ?, -5.305183149182902x + 0.5159008092577815y = 22.57469783167147, -3.7232672372547344x + 0.34690071436929104y = 11.414888578979635, -2.8287685584485294x + 0.11756217148342385y = 6.025293534019426, -2.236976394363432x - 0.17483562377954143y = 3.0762680704077274, -1.754166283775307x - 0.48737946660951226y = 1.3528528370097155, -1.3114530827180277x - 0.7110841108058793y = 0.3660640811126688, -0.9653443089915189x - 0.7713993783900512y = -0.15003496254627782, -0.7662711367983546x - 0.717162710122473y = -0.4285098679106686, -0.6850770151527903x - 0.6247036082520678y = -0.6247036082520678, -0.6751423351311789x - 0.5292915430253408y = -0.8067537459590874, -0.7087047738127068x - 0.43864417027731406y = -1.0043817090793903, -0.7759712047603957x - 0.3509793706328994y = -1.2412593024809349, -0.8803131023954011x - 0.26175624465387526y = -1.5524729972871656, -1.0392904626802628x - 0.16513266046780561y = -2.0065009275124095, -1.29566506908837x - 0.053847176214653736y = -2.759774156749924, -1.7519019955029622x + 0.08053911491761845y = -4.230905096708867, -2.6777274256017254x + 0.24388819884701718y = -7.777205480229924, ?, ?, ?}"/>
</command>
<element type="list" label="C2">
<show object="false" label="false" ev="10"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Intersect[f, Element[C2, k]]" a1="k" a2="1" a3="n - 1"/>
<output a0="A3" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="A3">
<show object="false" label="false" ev="7"/>
<objColor r="0" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<expression label="W" exp="x = x(Corner[1])" type="line" />
<element type="line" label="W">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="1" y="0" z="248.63326915290335"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<expression label="E" exp="x = x(Corner[2])" type="line" />
<element type="line" label="E">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="1" y="0" z="-242.45960199004978"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<expression label="N" exp="y = y(Corner[3])" type="line" />
<element type="line" label="N">
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0" y="1" z="-863.9784172262202"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
</element>
<command name="Max">
<input a0="Sequence[Distance[(0, 0), Corner[k]], k, 1, 4]"/>
<output a0="r"/>
</command>
<element type="numeric" label="r">
<value val="899.0423838520544"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<element type="numeric" label="a">
<value val="0.7"/>
<show object="true" label="false" ev="3"/>
<objColor r="255" g="255" b="0" alpha="0.10000000149011612"/>
<layer val="1"/>
<labelOffset x="63" y="-49"/>
<labelMode val="3"/>
<slider min="0.2" max="4" absoluteScreenLocation="true" width="120" x="419" y="514" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.02" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="SetValue[A,(0,a)]"/>
<caption val="光源在軸上的位置"/>
</element>
<command name="Vector">
<input a0="A" a1="(x(A), y(Corner[3]))"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="6"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="862.8917721190605" z="0"/>
<lineStyle thickness="4" type="0" typeHidden="1" opacity="63"/>
<coordStyle style="cartesian"/>
<startPoint exp="A"/>
</element>
<command name="Sequence">
<input a0="If[A3(k) ≠ A2(k), Mirror[A2(k), OrthogonalLine[A3(k), Tangent[A3(k), f]]]]" a1="k" a2="1" a3="n - 1"/>
<output a0="B3" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="B3">
<show object="false" label="false" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Ray[(0.99 * A3(k)) + (0.01 * B3(k)), B3(k)]" a1="k" a2="1" a3="n - 1"/>
<output a0="C3" randomResult="{?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?}"/>
</command>
<element type="list" label="C3">
<show object="false" label="false" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Intersect[f, Element[C3, k]]" a1="k" a2="1" a3="n - 1"/>
<output a0="A4" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="A4">
<show object="false" label="false" ev="7"/>
<objColor r="0" g="204" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="If[A4(k) ≠ A3(k), Mirror[A3(k), OrthogonalLine[A4(k), Tangent[A4(k), f]]]]" a1="k" a2="1" a3="n - 1"/>
<output a0="B4" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="B4">
<show object="false" label="false" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Ray[(0.99 * A4(k)) + (0.01 * B4(k)), B4(k)]" a1="k" a2="1" a3="n - 1"/>
<output a0="C4" randomResult="{?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?}"/>
</command>
<element type="list" label="C4">
<show object="false" label="false" ev="11"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Intersect[f, Element[C4, k]]" a1="k" a2="1" a3="n - 1"/>
<output a0="A5" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="A5">
<show object="false" label="false" ev="7"/>
<objColor r="51" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<element type="numeric" label="step">
<value val="5"/>
<show object="false" label="true"/>
<objColor r="192" g="192" b="192" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="3"/>
<slider min="1" max="5" absoluteScreenLocation="true" width="200" x="387" y="738" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<caption val="製作by宋政蒲"/>
</element>
<expression label="F" exp="(0, 1)" type="point"/>
<element type="point" label="F">
<show object="true" label="true" ev="7"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-68" y="39"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0" y="1" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="焦點"/>
</element>
<expression label="text1" exp=""光杯、光源附近的模擬""/>
<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"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.2" size="5" style="0"/>
<absoluteScreenLocation x="5" y="485"/>
<ggbscript val="If(n<-500,SetValue[text1,"製作by宋政蒲"])"/>
</element>
<expression label="text2" exp=""射向遠處之模擬""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.2" size="5" style="0"/>
<absoluteScreenLocation x="16" y="482"/>
</element>
<expression label="text3" exp=""光源在軸上的位置=" + (LaTeX[a]) + """ />
<element type="text" label="text3">
<show object="true" label="false" ev="43"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.8" size="-5" style="0"/>
<absoluteScreenLocation x="318" y="470"/>
</element>
<expression label="text4" exp=""光線數目=" + (LaTeX[n]) + """ />
<element type="text" label="text4">
<show object="true" label="false" ev="43"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.8" size="-5" style="0"/>
<absoluteScreenLocation x="3" y="374"/>
</element>
<command name="Sequence">
<input a0="Vector[A2(k), If[Defined[A3(k)] ∧ A3(k) ≠ A2(k), A3(k), Intersect[Element[C2, k], osq]]]" a1="k" a2="1" a3="n - 1"/>
<output a0="D3" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (83.64984458822978, 860.1997476602166), (80.30544740610736, 861.9141700064231), (35.86133979838007, 862.8917721190604), (-67.48697967937684, 863.4783758944448), (-240.002263252763, 863.8112745218909), (-248.34269540238853, 458.01866264014654), (-248.78414292938947, 311.33335503257666), (-249.13385247713848, 266.19353970594614), (-249.40823223342616, 273.51186232314194), (-249.63251201331107, 318.4208765588758), (-249.82917635175528, 403.6418170250616), (-250.01652686563085, 552.755067091536), (-250.21162813355764, 841.4873727613112), (-137.14841940725924, 863.1668856896199), (-35.86133979838027, 862.8917721190605), (39.64811584478186, 862.4332827279585), (78.46874983154234, 861.5337866692005), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="D3">
<show object="false" label="false" ev="10"/>
<condition showObject="step ≥ 3"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="4" type="0" typeHidden="1" opacity="63"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Segment">
<input a0="(x(Corner[1]) + 5, 0)" a1="(x(Corner[1]) + 5, y(Corner[3]))"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="8"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-863.9784172262202" y="0" z="-210493.88626637514"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="229"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Sequence">
<input a0="(x(Corner[1]) + 5, (k * 200))" a1="k" a2="1" a3="10"/>
<output a0="l2" randomResult="{(-243.63326915290335, 200), (-243.63326915290335, 400), (-243.63326915290335, 600), (-243.63326915290335, 800), (-243.63326915290335, 1000), (-243.63326915290335, 1200), (-243.63326915290335, 1400), (-243.63326915290335, 1600), (-243.63326915290335, 1800), (-243.63326915290335, 2000)}"/>
</command>
<element type="list" label="l2">
<show object="true" label="false" ev="4"/>
<objColor r="96" g="96" b="96" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="3"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text[(k * 200), (x(Corner[1]) + 20, (k * 200) - 12)]" a1="k" a2="1" a3="10"/>
<output a0="l1" randomResult="{"200", "400", "600", "800", "1000", "1200", "1400", "1600", "1800", "2000"}"/>
</command>
<element type="list" label="l1">
<show object="true" label="false" ev="4"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<font serif="false" sizeM="0.75" size="-6" style="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Vector[A1(k), If[Defined[A2(k)] ∧ A2(k) ≠ A1(k), A2(k), Intersect[Element[C1, k], osq]]]" a1="k" a2="1" a3="n - 1"/>
<output a0="D2" randomResult="{(-231.21115349476244, 862.8917721190604), (-249.40823223342622, 431.9877300542321), (-249.40823223342616, 249.40823223342625), (-4.662723137935597, 2.6920244588438154), (-3.648460753949503, 0.9776021126374157), (-2.8598085013247707, -2.220446049250313E-16), (-2.1892350936455056, -0.5866037753841855), (-1.5926248793840312, -0.919502402830465), (-1.0655368310376363, -1.0655368310376359), (-0.6240893040367201, -1.0809543830518986), (-0.27437975628771594, -1.0239991910341226), (0, -0.9365031631163319), (0.22427977988488135, -0.837023533640743), (0.4209441183290713, -0.729096600093238), (0.6082946322046624, -0.6082946322046637), (0.8033959001314109, -0.4638408392067126), (1.0267378231793285, -0.27511357055939123), (1.3098823402790898, -2.220446049250313E-16), (1.7111057023243132, 0.45848939110208753), (2.3520997950968505, 1.3579854498600283), (241.6846389095269, 241.6846389095267), (241.68463890952694, 418.610074000238), (231.2111534947634, 862.8917721190605)}"/>
</command>
<element type="list" label="D2">
<show object="false" label="false" ev="10"/>
<condition showObject="step ≥ 2"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="4" type="0" typeHidden="1" opacity="63"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Vector[A3(k), If[Defined[A4(k)] ∧ A4(k) ≠ A3(k), A4(k), Intersect[Element[C3, k], osq]]]" a1="k" a2="1" a3="n - 1"/>
<output a0="D4" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="D4">
<show object="false" label="false" ev="11"/>
<condition showObject="step ≥ 3"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="4" type="0" typeHidden="1" opacity="63"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Vector[A4(k), If[Defined[A5(k)] ∧ A5(k) ≠ A4(k), A5(k), Intersect[Element[C4, k], osq]]]" a1="k" a2="1" a3="n - 1"/>
<output a0="D5" randomResult="{(NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN), (NaN, NaN)}"/>
</command>
<element type="list" label="D5">
<show object="false" label="false" ev="11"/>
<condition showObject="step ≥ 3"/>
<objColor r="255" g="204" b="51" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="4" type="0" typeHidden="1" opacity="63"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
</construction>
</geogebra>