First Definition of an Ellipse
2019-07-31 02:57
This example demonstrates the first definition of an ellipse: the locus of a point whose sum of distances to two fixed foci is constant. Using GeoGebra tools like circles, segments, and perpendicular bisectors, it dynamically illustrates the distance relationships, helping users intuitively understand the geometric properties and locus generation of ellipses in conic sections.
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.549.0" app="classic" platform="d" id="5141b1d4-e7bb-42a0-bd69-8b0667701d94" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1360" height="728" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.14705882352941177" orientation="1" />
<pane location="1" divider="0.7595588235294117" 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="1" visible="true" inframe="false" stylebar="true" location="1,3" size="1128" 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="2" visible="true" inframe="false" stylebar="false" location="3" size="200" 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="4097" visible="false" inframe="false" stylebar="true" location="1" size="380" window="100,100,700,550" />
<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="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="true" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1128" height="521"/>
<coordSystem xZero="544.8080581705993" yZero="208.58665275903405" scale="18.9242829817146" yscale="18.92428298171463"/>
<evSettings axes="true" 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="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="text1" exp=""椭圆定义关键信息 : 
动点P到两焦点F_1和F_2的\\
距离和为定值,
即\[\overline {P{F_1}} + \overline {P{F_2}} = 2a\]""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="224" g="176" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="8.159130683987987" y="7.8509337307591345" z="1.0"/>
</element>
<expression label="text2" exp=""例~椭圆~\[\frac{{x^2}}{25} + \frac{y^2}{16} = 1\]，\\
其中~\[\left\{ {\begin{array}{*{20}{c}}
{a = 5}\\
{b = 4}\\
{c = 3}
\end{array}} \right.\]，
\[\overline {P{F_1}} + \overline {P{F_2}} = 10\]．""/>
<element type="text" label="text2">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="224" g="176" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="8.819162242048467" y="3.8070671885475624" z="1.0"/>
</element>
<expression label="text3" exp=""绘图步骤\\
1.绘制两焦点F_1，F_2分别为(-3，0)和(3,0)\\
2.以F_1为圆心，半径为2a=10作圆\\
3.圆周上任选一点，标示为A\\
4.连接AF_1，AF_2\\
5.作AF_2的中垂线 L\\
6.作AF_1，L两线的交点P\\
7.点P的轨迹 ""/>
<element type="text" label="text3">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="224" g="176" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint x="9.018775127546377" y="-2.6997002580514016" z="1.0"/>
</element>
<expression label="F_1" exp="(-3, 0)" type="point"/>
<element type="point" label="F_1">
<show object="true" label="true"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-27" y="-4"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-3.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="F_2" exp="(3, 0)" type="point"/>
<element type="point" label="F_2">
<show object="true" label="true"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="3.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="F_1" a1="10"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-91.0" A3="0.0" A4="3.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="true"/>
<coords x="0.3267017578858078" y="9.43043240864805" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="F_1"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.43043240864805" y="-3.326701757885808" z="28.29129722594415"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="F_2"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.43043240864805" y="2.673298242114192" z="-28.29129722594415"/>
<lineStyle thickness="5" type="0" 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"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.673298242114192" y="9.43043240864805" z="-40.01989472634257"/>
<lineStyle thickness="5" type="0" 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="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-27" y="1"/>
<labelMode val="0"/>
<trace val="true"/>
<coords x="-133.66491210570956" y="301.77383707673755" z="80.03978945268514"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="P" a1="F_2"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="2"/>
<coords x="3.7702977374163216" y="4.66998080604303" z="-11.310893212248965"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="P"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="2"/>
<coords x="5.660134671231729" y="-1.996682563928838" z="16.980404013695182"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="P" a1="F_1"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.7702977374163216" y="-1.3300191939569699" z="11.310893212248965"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
</construction>
</geogebra>