抛物线的定义与轨迹动画演示
2020-10-31 12:45
本 GeoGebra 范例通过动态几何作图直观演示了抛物线的严格数学定义。抛物线是平面内到一个定点（焦点）和一条定直线（准线）距离相等的点的轨迹。动画中，通过构造线段、交点和角度，实时追踪动点到焦点与准线的距离，并利用轨迹命令生成抛物线。该范例结合 LaTeX 文本标注，清晰展示了圆锥曲线中抛物线的几何性质与轨迹形成过程，适合高中解析几何教学。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.609.0" app="classic" platform="d" id="d7b6d5d3-203a-4bb4-9789-1caa65c4668b" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1200" height="900" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.25" 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="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 || 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="360" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="862" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="300" tab="ALGEBRA" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" 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" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="70" visible="false" inframe="false" 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="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="862" height="651"/>
<coordSystem xZero="67.71918187867601" yZero="411.98485739442424" scale="35.34293111839274" yscale="35.34293111839279"/>
<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>
<algebraView>
<mode val="1"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="2" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="F">
<show object="true" label="true"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="1.022610101748066" y="0.8809041939070859" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="(0, y(Corner[4]))" a1="(0, y(Corner[1]))"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="1" y="28"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="18.476113308558404" y="0.0" z="-0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="numeric" label="k">
<value val="0.8"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="k'" absoluteScreenLocation="true" width="200.0" x="376.0" y="46.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="true"/>
<ggbscript onUpdate=""/>
</element>
<command name="Point">
<input a0="l" a1="k"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="-3.095814046635059" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="OrthogonalLine[A, l]" a1="LineBisector[F, A]"/>
<output a0="M"/>
</command>
<element type="point" label="M">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="155.75381426956199" y="-58.49187752485275" z="18.893860110373705"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F" a1="M"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-4" y="-23"/>
<labelMode val="3"/>
<coords x="3.9767182405421453" y="7.221010490321687" z="-10.427650669755629"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="|FM|"/>
</element>
<command name="Segment">
<input a0="A" a1="M"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-4" y="7"/>
<labelMode val="1"/>
<coords x="0.0" y="8.243620592069753" z="25.520716424059565"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="l" a1="g"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="1.5707963267948966"/>
<show object="true" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Locus">
<input a0="M" a1="k"/>
<output a0="loc1"/>
</command>
<element type="locus" label="loc1">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="54" y="59"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="LaTeX">
<input a0=""|FM|=" + f"/>
<output a0="text1"/>
</command>
<element type="text" label="text1">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<startPoint x="4.4369864538025166" y="-4.123260138887175" z="1.0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<command name="LaTeX">
<input a0=""d=" + g"/>
<output a0="text2"/>
</command>
<element type="text" label="text2">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1.4" size="8" style="0"/>
<startPoint x="5.422983443536409" y="-4.9972120161513045" z="1.0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<element type="numeric" label="k'">
<value val="0.8"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200.0" x="380.0" y="102.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
<ggbscript onUpdate="SetValue(k,k')
"/>
</element>
</construction>
</geogebra>