面动成体:直角三角形旋转生成圆锥的三维演示
2020-07-19 05:55
本GeoGebra范例生动演示了“面动成体”的几何原理,具体展示直角三角形绕其一条直角边旋转一周形成圆锥的过程。通过滑块控制旋转角度与底面半径,结合多边形、圆弧与曲面命令,直观呈现圆锥的生成机制、空间结构及高与底面半径的关系,帮助学生建立立体几何的空间想象力。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.593.0" app="classic" platform="d" id="7b088cf1-a71e-40eb-9fe6-8dbc89955c1e" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1366" height="728" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.1800878477306003" orientation="1" />
<pane location="1" divider="0.328696925329429" orientation="1" />
</panes>
<views>
<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="3,1,1" size="372" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1,3" size="449" 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="16" visible="false" inframe="false" stylebar="true" location="3,1" size="544" 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 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 , 538 | 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="1,1" size="634" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="246" 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="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="239" 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="3"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="449" height="497"/>
<coordSystem xZero="132.53515746926175" yZero="347.38035079675655" scale="66.87350179509912" yscale="66.87350179509896"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="0" g="51" b="0"/>
<axesColor r="255" g="255" b="255"/>
<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="1"/>
</algebraView>
<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="0.778013859063017" yZero="-0.5458393541789994" zZero="-1.2386854072851121" scale="61.89014038897301" xAngle="24.0" zAngle="-43.0"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<plate show="true"/>
<bgColor r="0" g="51" b="0"/>
<clipping use="false" show="false" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="numeric" label="r">
<value val="1.29"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="120.0" x="67.0" y="42.0" fixed="false" horizontal="true" showAlgebra="true"/>
<pointSize val="3.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<ggbscript onUpdate=""/>
</element>
<element type="numeric" label="h">
<value val="2.83"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="255" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="120.0" x="65.0" y="77.0" fixed="false" horizontal="true" showAlgebra="true"/>
<pointSize val="3.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<ggbscript onUpdate=""/>
</element>
<element type="angle" label="α">
<angleStyle val="3"/>
<value val="6.283185307179587"/>
<slider min="0" max="6.283185307179586" absoluteScreenLocation="true" width="120.0" x="66.0" y="119.0" fixed="false" horizontal="true" showAlgebra="true"/>
<pointSize val="3.0"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.10000000149011612"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="6" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation step="0.01°" speed="1" type="0" playing="true"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-10" y="1"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="0.0" y="-0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="A" exp="(r, 0, 0)" type="point" />
<element type="point3d" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="6" y="9"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="1.29" y="0.0" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<expression label="B" exp="(0, 0, h)" type="point" />
<element type="point3d" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="0.0" y="0.0" z="2.83" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Polygon">
<input a0="B" a1="O" a2="A"/>
<output a0="t1" a1="a" a2="b" a3="o"/>
</command>
<element type="polygon3d" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.15000000596046448"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<element type="segment3d" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="o">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Rotate">
<input a0="Polygon[B, O, A]" a1="α" a2="O"/>
<output a0="t2"/>
</command>
<element type="polygon3d" label="t2">
<lineStyle thickness="3" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<command name="CircleArc">
<input a0="O" a1="A" a2="Rotate[A, α, O]" a3="xOyPlane"/>
<output a0="d"/>
</command>
<element type="conicpart" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<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="1.0" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="parametric"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Circle">
<input a0="O" a1="r"/>
<output a0="e"/>
</command>
<element type="conic" label="e">
<show object="true" label="false" ev="4"/>
<condition showObject="α ≟ 360°"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<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="-1.6641000000000001" A3="0.0" A4="-0.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Surface">
<input a0="((r - (u * r / h)) * cos(θ))" a1="((r - (u * r / h)) * sin(θ))" a2="u" a3="θ" a4="0" a5="α" a6="u" a7="0" a8="h"/>
<output a0="c"/>
</command>
<element type="surfacecartesian3d" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
</construction>
</geogebra>