圆柱截方：方孔圆柱的三维几何构造
2018-08-01 13:57
本GeoGebra范例演示了带有方孔的圆柱体（圆柱截方）的三维几何构造。通过参数方程、曲面和平移等命令，构建了圆柱体及其内部的方形孔洞结构。利用序列和相交命令，展示了截面变化及几何特征。该模型结合了立体几何与解析几何，直观呈现了复杂曲面与多边形相交的空间关系，适用于立体图形和曲面截线的教学与探索。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.481.0" app="classic" platform="d" id="0653cf2a-687f-4d71-a32f-2962ed77676c" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1304" height="768" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.15567484662576686" orientation="1" />
<pane location="1" divider="0.23542944785276074" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="133,86,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1,3" size="307" 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="3,1" 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="3,1" size="378" 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 | 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="1,1" size="757" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="203" window="100,100,250,400" />
<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="263" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="422,109,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</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 | 100007 100001 100004 100002" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="307" height="575"/>
<coordSystem xZero="336.8491883626999" yZero="300.3909718965692" scale="33.2067989088051" yscale="33.20679890880537"/>
<evSettings axes="false" grid="false" gridIsBold="true" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="9" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="0" 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="3"/>
<angleUnit val="radiant"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.0" yZero="0.0" zZero="-1.5" scale="50.0" xAngle="29.0" zAngle="-92.86037912466546"/>
<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="false"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="false" show="false" size="2"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="numeric" label="n">
<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="3" max="30" absoluteScreenLocation="true" width="200.0" x="77.0" y="66.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="a">
<value val="1.93"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="200.0" x="77.0" y="108.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<command name="CurveCartesian">
<input a0="(a / cos(θ - ((floor((θ * n / ((2 * π)))) * 2) * π / n) - π / n); θ)" a1="θ" a2="0" a3="(2 * 3.141592653589793)"/>
<output a0="b"/>
</command>
<element type="curvecartesian" label="b">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<expression label="A" exp="(5; π / 3; 0)" type="point"/>
<element type="point3d" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="2.5000000000000004" y="4.330127018922193" z="0.0" w="1.0"/>
<coordStyle style="spherical"/>
<pointSize val="3"/>
</element>
<element type="numeric" label="r">
<value val="2.9"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="5" absoluteScreenLocation="true" width="200.0" x="78.0" y="147.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<expression label="c" exp="x^(2) + y^(2) = r^(2)" type="conic" />
<element type="conic" label="c">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<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="-8.41" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Surface">
<input a0="(r + (t * (a / cos(θ - ((floor((θ * n / ((2 * π)))) * 2) * π / n) - π / n) - r)); θ; 0)" a1="θ" a2="0" a3="(2 * π)" a4="t" a5="0" a6="1"/>
<output a0="d"/>
</command>
<element type="surfacecartesian3d" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="Translate">
<input a0="d" a1="Vector[(0, 0, 3)]"/>
<output a0="d'"/>
</command>
<element type="surfacecartesian3d" label="d'">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="B" exp="(r, 0, 0)" type="point" />
<element type="point3d" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.9" y="0.0" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Intersect">
<input a0="zAxis" a1="xAxis"/>
<output a0="C"/>
</command>
<element type="point3d" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Point">
<input a0="zAxis"/>
<output a0="D"/>
</command>
<element type="point3d" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="0.0" z="3.0" w="1.0"/>
<pointSize val="5"/>
</element>
<command name="Surface">
<input a0="(r * cos(θ))" a1="(r * sin(θ))" a2="t" a3="θ" a4="0" a5="(2 * π)" a6="t" a7="0" a8="3"/>
<output a0="e"/>
</command>
<element type="surfacecartesian3d" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="E" exp="(a / cos(π / n), 0)" type="point" />
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.2285720390719552" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Rotate[E, ((2 * π / n) * i)]" a1="i" a2="0" a3="n"/>
<output a0="list1"/>
</command>
<element type="list" label="list1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Polygon[list1(i), list1(i + 1), list1(i + 1) + (0, 0, 3), list1(i) + (0, 0, 3)]" a1="i" a2="1" a3="n + 1"/>
<output a0="list2"/>
</command>
<element type="list" label="list2">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="1.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1" opacity="204"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
</construction>
</geogebra>