Geometric Construction and Properties of a Hyperbola
2017-11-16 03:21
This applet demonstrates the geometric construction and properties of a hyperbola. By utilizing fundamental geometric elements such as circles, points, lines, angle bisectors, and midpoints, combined with distance calculations and intersection solving, it dynamically constructs the hyperbola curve. This construction process visually illustrates the geometric definition and locus characteristics of
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.385.0" app="classic" platform="d" id="4f06b3fe-9e74-4a24-bd9c-476b133859f3" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1936" height="1056" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.10330578512396695" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="610,245,700,550" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 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="1" visible="true" inframe="false" stylebar="false" location="1" size="1704" 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" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" visible="false" inframe="true" 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="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1704" height="889"/>
<coordSystem xZero="482.78197720500344" yZero="812.3689438972726" scale="50.39439331222093" yscale="50.39439331222518"/>
<evSettings axes="true" grid="true" 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>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation speed="1" type="1" playing="false"/>
<isShape val="false"/>
<coords x="1.6367134659371803" y="5.151318848587067" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation speed="1" type="1" playing="false"/>
<isShape val="false"/>
<coords x="6.0" y="2.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="A" a1="B"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<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="0.2458369855944369" A3="0.0" A4="-1.6367134659371803" A5="-5.151318848587067"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="21" y="0"/>
<labelMode val="0"/>
<animation speed="1" type="1" playing="true"/>
<isShape val="false"/>
<coords x="-2.689675339814006" y="8.353105812965689" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation speed="1" type="1" playing="false"/>
<isShape val="false"/>
<coords x="2.0" y="10.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="LineBisector">
<input a0="C" a1="D"/>
<output a0="f"/>
</command>
<element type="line" label="f">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="-4.689675339814006" y="-1.6468941870343112" z="13.495634921897615"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="C" a1="A"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="3.2017869643786216" y="4.326388805751186" z="-27.527016041042796"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="g" a1="f"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<objColor r="224" g="176" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="13.05328114820901" y="85.88262033739827" z="15.01635455302132"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Distance">
<input a0="A" a1="E"/>
<output a0="AE"/>
</command>
<element type="numeric" label="AE">
<value val="0.9547455568674394"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<auxiliary val="true"/>
</element>
<command name="Distance">
<input a0="D" a1="E"/>
<output a0="DE"/>
</command>
<element type="numeric" label="DE">
<value val="4.427547622047477"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<auxiliary val="true"/>
</element>
<expression label="a" exp="abs(AE - DE)" />
<element type="numeric" label="a">
<value val="3.472802065180038"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Hyperbola">
<input a0="A" a1="D" a2="E"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="0.0747153916977107" y0="0.9972048988264436" z0="1.0" x1="-0.9972048988264436" y1="0.0747153916977107" z1="1.0"/>
<matrix A0="47.71350831234944" A1="-45.79741889659323" A2="-2525.0226453156265" A3="-7.045842281090106" A4="-33.383277614487284" A5="359.7575028235502"/>
<eqnStyle style="specific"/>
</element>
<command name="PointIn">
<input a0="d"/>
<output a0="點d_1"/>
</command>
<element type="point" label="點d_1">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="1.9480926162628627" y="9.307207040319593" z="1.0"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="PointIn">
<input a0="d"/>
<output a0="點d_2"/>
</command>
<element type="point" label="點d_2">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="1.9480926162628627" y="9.307207040319593" z="1.0"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="PointIn">
<input a0="d"/>
<output a0="點d_3"/>
</command>
<element type="point" label="點d_3">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="1.9480926162628627" y="9.307207040319593" z="1.0"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="PointIn">
<input a0="d"/>
<output a0="點d"/>
</command>
<element type="point" label="點d">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="1.9480926162628627" y="9.307207040319593" z="1.0"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Distance">
<input a0="C" a1="E"/>
<output a0="distanceCE"/>
</command>
<element type="numeric" label="distanceCE">
<value val="4.427547622047477"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="文字CE" exp="Name[C] + (Name[E]) + " = " + distanceCE" />
<element type="text" label="文字CE">
<show object="true" label="true" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="Midpoint[C, E]"/>
</element>
<command name="Distance">
<input a0="A" a1="E"/>
<output a0="distanceAE"/>
</command>
<element type="numeric" label="distanceAE">
<value val="0.9547455568674394"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="文字AE" exp="Name[A] + (Name[E]) + " = " + distanceAE" />
<element type="text" label="文字AE">
<show object="true" label="true" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="Midpoint[A, E]"/>
</element>
<command name="Distance">
<input a0="D" a1="E"/>
<output a0="distanceDE"/>
</command>
<element type="numeric" label="distanceDE">
<value val="4.427547622047477"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Midpoint">
<input a0="D" a1="E"/>
<output a0=""/>
</command>
<expression label="文字DE" exp="Name[D] + (Name[E]) + " = " + distanceDE" />
<element type="text" label="文字DE">
<show object="true" label="true" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="Midpoint[D, E]"/>
</element>
</construction>
</geogebra>