圆锥的三维构造与演示
2016-04-17 07:30
本GeoGebra范例演示了如何通过几何作图与三维曲面命令构造一个圆锥体。通过定义底面圆、顶点以及高,利用扇形、圆弧和曲面等工具生成圆锥的侧面与底面。该模型结合了立体几何与平面几何的知识,直观展示了圆锥的生成过程及其几何特征,适合用于立体几何教学与三维图形探索。
所属专题
S3数学:面积与体积
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.228.0" id="a7462b29-dca9-41c7-988f-0534cd3f6492" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.17666666666666667" orientation="0" />
<pane location="1" divider="0.37" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="1395,243,910,764" />
<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="209" window="100,100,250,400" />
<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="16" visible="true" inframe="false" stylebar="false" location="2,3" size="296" window="50,50,500,500" />
<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="true" inframe="false" stylebar="false" location="2,1" size="494" window="583,437,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="0" size="106" window="100,100,600,400" />
<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="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="true" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 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="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="798" height="106"/>
<coordSystem xZero="539.8995391454745" yZero="188.44843517541557" scale="24.984818167088893" yscale="24.984818167088772"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="239" b="213"/>
<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>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="296" height="484"/>
<coordSystem xZero="141.1361959035173" yZero="158.44198600390837" scale="129.6203344859984" yscale="129.62033448599928"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="224" g="224" b="224"/>
<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"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.03150632967986533" yZero="0.06594671540385111" zZero="-0.29223183979000505" scale="149.90865136857502" xAngle="16" zAngle="-62.57622975086113"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="2" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<plate show="true"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="true" show="false" size="2"/>
<projection type="1"/>
</euclidianView3D>
<construction title="" author="" date="">
<expression label ="A" exp="(1, 0)" type="point"/>
<element type="point" label="A">
<show object="true" label="true" ev="11"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="3" y="26"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="1" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label ="O" exp="(0, 0)" type="point"/>
<element type="point" label="O">
<show object="true" label="true" ev="7"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-2" y="35"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="O" a1="1"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false" ev="11"/>
<objColor r="128" g="128" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="3" type="15" typeHidden="1" opacity="191"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-1" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="11"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-22" y="31"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-0.4995710587845122" y="-0.8662729115151422" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="A" a1="O" a2="B"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="4.189285432661327"/>
<show object="true" label="false" ev="11"/>
<condition showObject="l ≟ 1"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-32" y="-25"/>
<labelMode val="2"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<expression label="r" exp="α / (360°)" />
<element type="numeric" label="r">
<value val="0.6667454846309196"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="e" exp="((2 * π) * r)" />
<element type="numeric" label="e">
<value val="4.189285432661327"/>
<objColor r="0" g="0" b="0" alpha="0"/>
</element>
<expression label="h" exp="sqrt(1 - r^(2))" />
<element type="numeric" label="h">
<value val="0.7452854880676801"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="P" exp="(0, 0, h)" type="point" />
<element type="point3d" label="P">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="0.7452854880676801" w="1"/>
<coordStyle style="cartesian"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="O" a1="P"/>
<output a0="a"/>
</command>
<element type="segment3d" label="a">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="a"/>
<output a0="V"/>
</command>
<element type="point3d" label="V">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-25" y="-4"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0" y="0" z="0.5288817865568483" w="1"/>
<coordStyle style="cartesian"/>
<pointSize val="5"/>
</element>
<expression label="hz" exp="z(V)" />
<element type="numeric" label="hz">
<value val="0.5288817865568483"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="rr" exp="sqrt(1 - hz^(2))" />
<element type="numeric" label="rr">
<value val="0.8486955024320775"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="t" exp="e / (((2 * rr) * π))" />
<element type="numeric" label="t">
<value val="0.7856121338221425"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="β" exp="((2 * π) * t)" />
<element type="numeric" label="β">
<value val="4.936146616373288"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="f" exp="f(x) = (hz * (1 - x / rr))" />
<element type="function" label="f">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
</element>
<expression label="C" exp="(rr, 0, 0)" type="point" />
<element type="point3d" label="C">
<show object="true" label="true" ev="5"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="8" y="21"/>
<labelMode val="3"/>
<coords x="0.8486955024320775" y="0" z="0" w="1"/>
<coordStyle style="cartesian"/>
<pointSize val="3"/>
<caption val="A"/>
</element>
<command name="Segment">
<input a0="V" a1="C"/>
<output a0="b"/>
</command>
<element type="segment3d" label="b">
<show object="true" label="false" ev="5"/>
<objColor r="255" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="V" a1="O"/>
<output a0="g"/>
</command>
<element type="segment3d" label="g">
<show object="true" label="false" ev="5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="O" a1="C"/>
<output a0="i"/>
</command>
<element type="segment3d" label="i">
<show object="true" label="false" ev="5"/>
<objColor r="0" g="204" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="3" type="15" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="O" a1="C" a2="V"/>
<output a0="poly1" a1="v" a2="o" a3="c_1"/>
</command>
<element type="polygon3d" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="5"/>
<condition showObject="hz ≟ h"/>
<objColor r="0" g="0" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment3d" label="v">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="o">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="c_1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Circle">
<input a0="zAxis" a1="C"/>
<output a0="p"/>
</command>
<element type="conic3d" label="p">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-0.7202840558484365" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
</element>
<expression label="E" exp="((rr * cos(β / 2)), (rr * sin(β / 2)), 0)" type="point" />
<element type="point3d" label="E">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.6633670130806597" y="0.5293659054046456" z="0" w="1"/>
<coordStyle style="cartesian"/>
<pointSize val="3"/>
</element>
<expression label="D" exp="((rr * cos(β)), (rr * sin(β)), 0)" type="point" />
<element type="point3d" label="D">
<show object="true" label="true" ev="5"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="2" y="23"/>
<labelMode val="3"/>
<coords x="0.1883214083032886" y="-0.8275379767860219" z="0" w="1"/>
<coordStyle style="cartesian"/>
<pointSize val="3"/>
<caption val="B"/>
</element>
<command name="CircumcircleArc">
<input a0="D" a1="E" a2="C"/>
<output a0="q"/>
</command>
<element type="conicpart" label="q">
<show object="true" label="false" ev="5"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-8.326672684688674e-17" A3="0" A4="-0.801024549034483" A5="-0.28043489029102586"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Sector">
<input a0="c" a1="A" a2="B"/>
<output a0="d"/>
</command>
<element type="conicpart" label="d">
<show object="true" label="false" ev="11"/>
<objColor r="0" g="153" b="255" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-1" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CircleArc">
<input a0="O" a1="A" a2="B"/>
<output a0="k"/>
</command>
<element type="conicpart" label="k">
<show object="true" label="false" ev="11"/>
<objColor r="255" g="215" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-1" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Surface">
<input a0="(u * cos(v))" a1="(u * sin(v))" a2="f(u)" a3="u" a4="0" a5="rr" a6="v" a7="0" a8="β"/>
<output a0="S"/>
</command>
<element type="surfacecartesian3d" label="S">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="255" b="204" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="Segment">
<input a0="O" a1="A"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false" ev="11"/>
<objColor r="255" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="8" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="l">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="38" y="21"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="θ="/>
</element>
<element type="numeric" label="sl">
<value val="12"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="2"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Textfield">
<input a0="sl"/>
<output a0="InputBox1"/>
</command>
<element type="textfield" label="InputBox1">
<show object="true" label="true"/>
<objColor r="255" g="102" b="153" alpha="1"/>
<layer val="0"/>
<labelOffset x="290" y="13"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<fixed val="true"/>
<auxiliary val="true"/>
<font serif="false" sizeM="2" size="18" style="0"/>
<linkedGeo exp="sl"/>
<length val="3"/>
<caption val="l="/>
</element>
<expression label="text1" exp=""" + (LaTeX[round(α)]) + """ />
<element type="text" label="text1">
<show object="true" label="false" ev="8"/>
<condition showObject="l"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="2" size="18" style="0"/>
<startPoint x="-17.74535007411515" y="7.009968910589441" z="1"/>
</element>
<expression label="br" exp="(r * sl)" />
<element type="numeric" label="br">
<value val="8.000945815571036"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="ro" exp="(r * o)" />
<element type="numeric" label="ro">
<value val="0.6667454846309196"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<auxiliary val="true"/>
</element>
<expression label="text2" exp=""" + (LaTeX[round(br, 2)]) + """ />
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<condition showObject="m"/>
<objColor r="0" g="204" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="2" size="18" style="0"/>
<startPoint x="-0.032364856496358896" y="7.0949732676902855" z="1"/>
</element>
<element type="boolean" label="m">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="477" y="20"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="r="/>
</element>
<command name="Circle">
<input a0="O" a1="r"/>
<output a0="s"/>
</command>
<element type="conic" label="s">
<show object="true" label="false" ev="5"/>
<condition showObject="hz ≟ h"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-0.4445495412757199" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
</element>
</construction>
</geogebra>