Geometric Definition and Locus Construction of a Parabola
2021-01-07 01:07
This GeoGebra applet dynamically demonstrates the geometric definition of a parabola. Using a focus and a directrix, it constructs the locus of a moving point that is equidistant from the focus and the directrix. Through geometric construction commands like orthogonal lines, intersections, and segments, it visually illustrates the formation of a parabola, helping learners deeply understand the cor
In collections
Conic Sections - Interactive GeoGebra Book
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.620.0" app="classic" platform="w" id="d36c84df-aa07-4561-9a87-ebd572f3541f" xsi:noNamespaceSchemaLocation="https://ggb123.cn/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1334" height="536" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.14992503748125938" 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 , 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="305" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1124" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" 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" 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="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1124" height="534"/>
<coordSystem xZero="468.52890322461735" yZero="303.7191672942387" scale="43.081611699339874" yscale="43.08161169934002"/>
<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="3"/>
<auxiliary show="true"/>
</algebraView>
<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="" author="" date="">
<element type="point" label="F">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="3.6551811913247305" y="0.7826811942310696" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\large{F}$"/>
</element>
<element type="point" label="B">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-1.62" y="2.6" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-1.62" y="-0.88" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="B" a1="C"/>
<output a0="l"/>
</command>
<element type="line" label="l">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-25" y="76"/>
<labelMode val="0"/>
<coords x="3.48" y="0" z="5.637600000000001"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Parabola">
<input a0="F" a1="l"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="true" ev="4"/>
<condition showObject="t ≟ true"/>
<objColor r="255" g="0" b="0" 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="0" A1="12.1104" A2="137.43535146761573" A3="0" A4="-63.88455429941902" A5="-9.478582334615945"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="-20" y="-5"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="true"/>
<coords x="700.5156291300316" y="86.68936191517439" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="$\large{P}$"/>
</element>
<command name="OrthogonalLine">
<input a0="P" a1="l"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="3.48" z="-301.67897946480684"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="l" a1="g"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="-19.618848000000003" y="1049.8428485375277" z="12.1104"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large{H}$"/>
</element>
<command name="Segment">
<input a0="H" a1="P"/>
<output a0="d"/>
</command>
<element type="segment" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.4210854715202004e-14" y="702.1356291300316" z="-60867.68966719196"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="P" a1="F"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="85.90668072094331" y="-696.8604479387069" z="231.41508402473437"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""PF=" + i + """ />
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="0" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<absoluteScreenLocation x="12" y="132"/>
</element>
<expression label="text2" exp=""PH=" + d + """ />
<element type="text" label="text2">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="0" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<absoluteScreenLocation x="12" y="91"/>
</element>
<element type="image" label="pic1">
<file name="42291f95ac7d66a1dd5154e0544aa6c0/J`H0[6%}{0OC50FU2G(6~RU.png"/>
<inBackground val="false"/>
<startPoint number="0" exp="P"/>
<startPoint number="1" exp="P + (1, 0)"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="160"/>
<labelMode val="3"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(t,!t)
If(t==true,SetCaption(button1,"显示踪迹"),SetCaption(button1,"显示轨迹"))
If(t==true,SetTrace(P,false),SetTrace(P,true))
ZoomIn(1)"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<caption val="显示踪迹"/>
</element>
<expression label="text3" exp=""抛物线的定义""/>
<element type="text" label="text3">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="0" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="0"/>
<absoluteScreenLocation x="6" y="9"/>
</element>
<command name="OrthogonalLine">
<input a0="F" a1="l"/>
<output a0="f"/>
</command>
<element type="line" label="f">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="3.48" z="-2.7237305559241225"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="l" a1="f"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-19.618848000000003" y="9.478582334615947" z="12.1104"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="c" a1="f" a2="1"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<condition showObject="a"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-29" y="37"/>
<labelMode val="3"/>
<coords x="1.017590595662365" y="0.7826811942310696" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="$\large{O}$"/>
</element>
<expression label="E" exp="A + A - D" type="point" />
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.257590595662366" y="0.7826811942310699" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="G" exp="F + D - A" type="point" />
<element type="point" label="G">
<show object="true" label="true" ev="4"/>
<condition showObject="a"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-29" y="35"/>
<labelMode val="3"/>
<coords x="6.292771786987095" y="0.7826811942310695" z="1"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<caption val="$\large{x}$"/>
</element>
<command name="Rotate">
<input a0="G" a1="90°" a2="D"/>
<output a0="G'"/>
</command>
<element type="point" label="G'">
<show object="true" label="true" ev="4"/>
<condition showObject="a"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-29" y="29"/>
<labelMode val="3"/>
<coords x="1.017590595662365" y="6.0578623855558" z="1"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<caption val="$\large{y}$"/>
</element>
<command name="Rotate">
<input a0="E" a1="90°" a2="D"/>
<output a0="E'"/>
</command>
<element type="point" label="E'">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.0175905956623648" y="-4.492499997093661" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="E" a1="G"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="4"/>
<condition showObject="a"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="10.55036238264946" y="-3.3306690738754696e-16" z="0"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="E"/>
</element>
<command name="Vector">
<input a0="E'" a1="G'"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="4"/>
<condition showObject="a ≟ true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.220446049250313e-16" y="10.55036238264946" z="0"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="E'"/>
</element>
<element type="boolean" label="t">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="a">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="3" y="253"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="$\large{建系}$"/>
</element>
<element type="boolean" label="b">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="19" y="332"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="4" y="208"/>
<labelMode val="3"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="b=!b
If(b==true,StartAnimation(P,true),StartAnimation(P,false))
If(b==true,SetCaption(button2,"暂停动画“P”"),SetCaption(button2,"启动动画“P”"))"/>
<font serif="false" sizeM="2" size="12" style="0"/>
<caption val="暂停动画“P”"/>
</element>
</construction>
</geogebra>