椭圆轨迹的几何构造与光学性质演示
2019-05-22 02:15
本范例展示了椭圆的几何构造及其轨迹生成过程。通过定义两个焦点F1和F2,利用圆、线段中垂线及交点等几何作图命令,动态生成椭圆上的点P并描绘轨迹。同时,该模型探讨了椭圆的光学反射性质,直观演示了从焦点发出的光线经椭圆边界反射后必经过另一焦点的特性。适合用于解析几何与圆锥曲线的教学与探究。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.529.0" app="classic" platform="w" id="1B907E29-7F56-4120-BC14-E5E912E6B416" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="900" height="640" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.4222222222222222" 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 | 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 || 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="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 | 1009 || 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="486" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="898" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="380" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="300" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" 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 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="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="898" height="638"/>
<coordSystem xZero="444.66161895586043" yZero="314.1002108927524" scale="32.67337172908524" yscale="32.67337172908521"/>
<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="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"/>
<startAnimation val="true"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="橢圓軌跡" author="" date="">
<element type="numeric" label="t">
<value val="1"/>
<show object="true" label="false"/>
<objColor r="255" g="51" b="102" alpha="0.10000000149011612"/>
<bgColor r="192" g="192" b="192" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200" x="42" y="55" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="40" type="0" typeHidden="1"/>
<animation speed="1" type="3" playing="false"/>
</element>
<expression label="F_1" exp="(0, 0)" type="point"/>
<element type="point" label="F_1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="Segment[F_1, F_1 + (8, 0)]"/>
<output a0="F_2"/>
</command>
<element type="point" label="F_2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<selectionAllowed val="false"/>
<coords x="5.473260817476112" y="0" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="F_1" a1="8"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="20" 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="-64" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="0.3" type="1" playing="true"/>
<coords x="7.55527548012326" y="2.6301734580533354" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F_1" a1="B"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-2.6301734580533354" y="7.55527548012326" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F_2" a1="B"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<condition showObject="showReflection ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-2.6301734580533354" y="2.082014662647148" z="14.395625331128972"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="LineBisector">
<input a0="g"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="true" label="false" ev="4"/>
<condition showObject="showReflection ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-2.082014662647148" y="-2.6301734580533354" z="17.021708011940362"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="h" a1="f"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="false" ev="4"/>
<condition showObject="showReflection ≠ 0"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="128.60369317243067" y="44.770044623739345" z="22.648006749403336"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F_1" a1="P"/>
<output a0="PF1"/>
</command>
<element type="segment" label="PF1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-1.9767763723807092" y="5.678366957207779" z="0"/>
<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="P"/>
<output a0="BP"/>
</command>
<element type="segment" label="BP">
<show object="true" label="false" ev="4"/>
<condition showObject="t > 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.6533970856726261" y="-1.876908522915481" z="0"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="T" exp="((1 - t) * B) + (t * F_2)" type="point" />
<element type="point" label="T">
<show object="false" label="false" ev="4"/>
<condition showObject="0 < t < 1"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.473260817476112" y="0" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="P" a1="T"/>
<output a0="PF2"/>
</command>
<element type="segment" label="PF2">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="1.9767763723807092" y="-0.20510613973166691" z="-10.819412663863904"/>
<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Midpoint">
<input a0="g"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="6.514268148799687" y="1.3150867290266677" z="1"/>
<pointSize val="7"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="{P, D, T}"/>
<output a0="PDT"/>
</command>
<element type="polygon" label="PDT">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="0 < t < 1"/>
<objColor r="153" g="204" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<expression label="showReflection" exp="t > 0" />
<element type="boolean" label="showReflection">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<checkbox fixed="true"/>
<caption val="Show Reflection"/>
</element>
<command name="Ellipse">
<input a0="F_1" a1="F_2" a2="P"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="true" label="false" ev="4"/>
<condition showObject="showTrace ≠ 0"/>
<objColor r="102" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="136.17366409552298" A1="256.0000000000001" A2="-1158.9541745750203" A3="0" A4="-372.65699003308987" A5="0"/>
<eqnStyle style="implicit"/>
</element>
<element type="boolean" label="showTrace">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="33" y="111"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="show trace"/>
</element>
</construction>
</geogebra>