Origin of the Name Conic Sections: Geometric Demonstration of Cross-Sections
2020-04-10 17:32
This GeoGebra applet visually demonstrates the origin of the name 'conic sections' (ellipse, parabola, hyperbola) through 3D geometric construction. By intersecting a plane with a cone, it shows how the shape of the cross-section changes with different angles of inclination. Using commands like IntersectPath, it dynamically presents the intersection curves between the plane and the cone. This reso
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="d" id="acd188e9-fd20-454c-99ac-2f4751508134" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1266" height="842" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.669826224328594" orientation="1" />
<pane location="0" divider="0.2401263823064771" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="395" window="966,246,880,683" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3,3" size="304" 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="true" location="1,1" size="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3,3" size="254" 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="true" location="1,3" size="300" window="100,100,600,400" />
<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="true" inframe="false" stylebar="true" location="3,1" size="539" 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="true" location="1" size="150" 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" />
<view id="1024" toolbar="0 | 1 501 67 , 5 19 | 2 15 45 , 18 , 7 37 | 4 3 8 9 , 13 44 | 16 51 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 | 30 29 32 31 33 | 17 | 40 41 42 , 27 28 35 , 6" visible="true" inframe="false" stylebar="true" location="1" size="397" window="660,247,973,545" plane="d" />
</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="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="304" height="721"/>
<coordSystem xZero="115.00000000000016" yZero="394.500000000001" scale="63.56281250000019" yscale="63.56281249999999"/>
<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>
<uses3D val="true"/>
<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"/>
<euclidianView3D>
<coordSystem xZero="-0.0" yZero="-0.0" zZero="-0.0" scale="50.0" xAngle="10.0" zAngle="-1083.933869537222"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<plate show="false"/>
<bgColor r="0" g="0" b="0"/>
<clipping use="true" show="false" size="1"/>
<projection type="0"/>
</euclidianView3D>
<euclidianView>
<viewId plane="d"/>
<size width="397" height="688"/>
<coordSystem xZero="200.5" yZero="344.0" scale="24.08491243410423" yscale="24.084912434104247"/>
<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="false"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="false"/>
<transformForPlane mirror="false" rotate="0"/>
</euclidianView>
<construction title="" author="" date="">
<element type="numeric" label="a">
<value val="3.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-8" y="-3"/>
<labelMode val="1"/>
<slider min="1" max="5" absoluteScreenLocation="true" width="200.0" x="46.0" y="172.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="If[n==1,SetValue[f,0]]
If[n==1,SetValue[g,0]]
If[n==2,SetValue[f,0.1]]
If[n==2,SetValue[g,a-0.1]]
If[n==3,SetValue[f,a]]
If[n==3,SetValue[g,a]]
If[n==4,SetValue[f,a+0.1]]
If[n==4,SetValue[g,10]]
SetActiveView("T")
CenterView((0,0,0))
SetActiveView(d)
CenterView((0,0))"/>
</element>
<expression label="eq1" exp="(a * x^(2)) + (a * y^(2)) = z^(2)" type="quadric" />
<element type="quadric" label="eq1">
<show object="true" label="false" ev="37"/>
<objColor r="255" g="0" b="255" alpha="0.8500000238418579"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="0.0" x1="-0.0" y1="1.0" z1="-0.0" x2="-0.0" y2="-0.0" z2="1.0"/>
<matrix A0="3.0" A1="3.0" A2="-1.0" A3="0.0" A4="0.0" A5="0.0" A6="0.0" A7="0.0" A8="0.0" A9="0.0"/>
<eqnStyle style="implicit"/>
</element>
<element type="numeric" label="b">
<value val="6.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="f" max="g" absoluteScreenLocation="true" width="200.0" x="46.0" y="236.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="SetActiveView("T")
CenterView((0,0,0))
SetActiveView(d)
CenterView((0,0))"/>
</element>
<element type="numeric" label="c">
<value val="1.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-10" max="10" absoluteScreenLocation="true" width="200.0" x="46.0" y="299.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="SetActiveView("T")
CenterView((0,0,0))
SetActiveView(d)
CenterView((0,0))"/>
</element>
<expression label="p" exp="(sqrt(b) * x) + z = c" type="plane" />
<element type="plane3d" label="p">
<show object="true" label="false" ev="21"/>
<objColor r="0" g="255" b="255" alpha="0.699999988079071"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="2.449489742783178" y="0.0" z="1.0" w="-1.0"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="IntersectPath">
<input a0="p" a1="eq1"/>
<output a0="d"/>
</command>
<element type="conic3d" label="d">
<show object="true" label="false" ev="21"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="-0.0" y0="1.0" z0="1.0" x1="-1.0" y1="-0.0" z1="1.0"/>
<matrix A0="3.0" A1="-0.4285714285714285" A2="0.3469387755102042" A3="0.0" A4="0.0" A5="0.5290400570128867"/>
<eqnStyle style="parametric"/>
</element>
<element type="numeric" label="f">
<value val="3.1"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="g">
<value val="10.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation 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="1.0" dynamicr="n ≟ 1" dynamicg="0" dynamicb="0" dynamica="1" colorSpace="0"/>
<layer val="0"/>
<labelOffset x="53" y="394"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[f,0]
SetValue[g,0]
SetValue[n,1]
CenterView((0,0,0))"/>
<font serif="true" sizeM="2.0" size="24" style="1"/>
<caption val="圆形"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="1.0" dynamicr="n ≟ 2" dynamicg="0" dynamicb="0" dynamica="1" colorSpace="0"/>
<layer val="0"/>
<labelOffset x="53" y="457"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[f,0.1]
SetValue[g,a-0.1]
SetValue[n,2]"/>
<font serif="true" sizeM="2.0" size="24" style="1"/>
<caption val="椭圆"/>
</element>
<element type="button" label="button3">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="1.0" dynamicr="n ≟ 3" dynamicg="0" dynamicb="0" dynamica="1" colorSpace="0"/>
<layer val="0"/>
<labelOffset x="145" y="395"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[f,a]
SetValue[g,a]
SetValue[n,3]"/>
<font serif="true" sizeM="2.0" size="24" style="1"/>
<caption val="抛物线"/>
</element>
<element type="button" label="button4">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="1.0" dynamicr="n ≟ 4" dynamicg="0" dynamicb="0" dynamica="1" colorSpace="0"/>
<layer val="0"/>
<labelOffset x="145" y="458"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[f,a+0.1]
SetValue[g,10]
SetValue[n,4]"/>
<font serif="true" sizeM="2.0" size="24" style="1"/>
<caption val="双曲线"/>
</element>
<element type="numeric" label="n">
<value val="4.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="4" absoluteScreenLocation="true" width="200.0" x="193.0" y="58.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="text1" exp=""制作：孟宝兴
QQ：319774699
QQ群：217440999""/>
<element type="text" label="text1">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="224" g="224" b="224" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<font serif="true" sizeM="0.7" size="-8" style="1"/>
<absoluteScreenLocation x="49" y="581"/>
</element>
</construction>
</geogebra>