圆锥、圆柱与圆台的侧面展开图及侧面上两点间最短路径
2020-05-16 13:42
本范例演示圆锥、圆柱和圆台的侧面展开图(即将三维旋转体侧面展开为二维平面图形),并探讨在这些曲面侧面上两点之间的最短曲线(测地线)。通过将立体曲面展开为平面,利用平面几何方法求解最短路径问题。构造中使用了参数曲线(CurveCartesian)、切线(Tangent)、向量(Vector)、线段(Segment)与曲面(Surface)等命令,直观展示立体几何中曲面展开与最短距离的数学原理,适用于高中及大学立体几何教学。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.553.0" app="classic" platform="d" id="808572dd-b010-4343-93a0-5f6a033184e1" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1536" height="824" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.19596354166666666" orientation="1" />
<pane location="1" divider="0.2721354166666667" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="418,59,700,705" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1,3" size="418" 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="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="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="780" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="301" 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="true" stylebar="true" location="1" size="150" 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" />
</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 | 100002 100001" position="1" help="false" />
<input show="true" cmd="true" top="true" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="418" height="631"/>
<coordSystem xZero="535.0013269166716" yZero="92.0561257396846" scale="21.69069743403687" yscale="21.690693598591835"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<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"/>
<labelStyle axes="0" serif="true"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" 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="-0.21819712542835967" yZero="-1.949645132206545" zZero="-5.151176833897857" scale="14.114668502887078" xAngle="39.0" zAngle="2256.4290793387695"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<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"/>
<light val="false"/>
<clipping use="false" show="false" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="angle" label="α">
<angleStyle val="3"/>
<value val="6.283185307179586"/>
<slider min="0" max="6.283185307179586" absoluteScreenLocation="true" width="180.0" x="55.0" y="158.0" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="10" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="R">
<value val="5.0"/>
<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="74.0" y="88.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="r">
<value val="0.0"/>
<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="R" absoluteScreenLocation="true" width="200.0" x="83.0" y="43.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="h_1">
<value val="10.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="3"/>
<slider min="0" max="10" absoluteScreenLocation="true" width="200.0" x="60.0" y="121.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation speed="1" type="3" playing="false"/>
<caption val="高=%v"/>
</element>
<expression label="a" exp="r / R" />
<element type="numeric" label="a">
<value val="0.0"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="CurveCartesian">
<input a0="(R; θ)" a1="θ" a2="0" a3="(2 * π)"/>
<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="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="B" exp="b(α)" type="point" />
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.0" y="-1.2246467991473533E-15" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Tangent">
<input a0="B" a1="b"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.0" y="1.2246467991473533E-15" z="25.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
</element>
<expression label="e" exp="1"/>
<element type="numeric" label="e">
<value val="1.0"/>
<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>
<expression label="h" exp="(h_1 * R / (R - r))" />
<element type="numeric" label="h">
<value val="10.0"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="A" exp="(0, 0, h)" type="point" />
<element type="point3d" label="A">
<show object="false" 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="10.0" w="1.0"/>
<pointSize val="4"/>
</element>
<command name="Vector">
<input a0="A" a1="B"/>
<output a0="u"/>
</command>
<element type="vector3d" label="u">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5.0" y="-1.2246467991473533E-15" z="-10.0" w="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<startPoint exp="A"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="f"/>
</command>
<element type="segment3d" label="f">
<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"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="v" exp="(f * UnitVector[g])" type="vector" />
<element type="vector" label="v">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.7383934913210135E-15" y="11.180339887498949" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
</element>
<expression label="c" exp="R / f" />
<element type="numeric" label="c">
<value val="0.4472135954999579"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Surface">
<input a0="((1 - t) * A) + (t * If[θ < α, b(θ), A + (cos((c * (θ - α))) * u) + (sin((c * (θ - α))) * v)])" a1="θ" a2="0" a3="(2 * π)" a4="t" a5="a" a6="1"/>
<output a0="d"/>
</command>
<element type="surfacecartesian3d" label="d">
<show object="false" label="false" ev="4"/>
<condition showObject="r < R"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="If[θ < α, b(θ), b(α) + ((UnitVector[g] * R) * (θ - α))] + (0, 0, s)" a1="θ" a2="0" a3="(2 * π)" a4="s" a5="0" a6="h_1"/>
<output a0="i"/>
</command>
<element type="surfacecartesian3d" label="i">
<show object="true" label="false" ev="4"/>
<condition showObject="r ≟ R"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<levelOfDetailQuality val="true"/>
</element>
<command name="CurveCartesian">
<input a0="d(θ, cos((c * π)) / cos((c * (π - θ))))" a1="θ" a2="0" a3="(2 * π)"/>
<output a0="j"/>
</command>
<element type="curvecartesian3d" label="j">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<command name="CurveCartesian">
<input a0="i(θ, (θ / 2 / π * 10))" a1="θ" a2="0" a3="(2 * π)"/>
<output a0="k"/>
</command>
<element type="curvecartesian3d" label="k">
<show object="true" label="false" ev="4"/>
<condition showObject="r ≟ R"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
</construction>
</geogebra>