Shortest Path on the Surface of a Cylinder
2020-05-16 09:28
This applet demonstrates how to find the shortest path between two points on the lateral surface of a cylinder. By unrolling the cylinder into a flat rectangle, it visually illustrates the optimization process of converting a spatial curve into a straight line segment based on the principle that the shortest distance between two points is a straight line. Ideal for teaching solid geometry and surf
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.581.0" app="classic" platform="d" id="9fe27d15-5fd9-4aba-8949-d772b72d3d14" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="2578" height="1408" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.2339022498060512" 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="1,1" size="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3,3" size="297" 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="4097" visible="false" inframe="true" stylebar="true" location="1,1" size="395" window="137,418,2060,794" />
<view id="1" visible="true" inframe="false" stylebar="true" location="3" size="603" 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="false" 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="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="1" size="1941" window="100,100,600,400" />
</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="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="603" height="1129"/>
<coordSystem xZero="579.1105407113143" yZero="752.344952089994" scale="62.97703012455283" yscale="62.97703012455266"/>
<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>
<euclidianView>
<viewNumber viewNo="2"/>
<coordSystem xZero="215.0" yZero="315.0" scale="50.0" yscale="50.0"/>
<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="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<spreadsheetView>
<prefCellSize height="40"/>
</spreadsheetView>
<algebraView>
<mode val="3"/>
<auxiliary show="true"/>
</algebraView>
<kernel>
<uses3D val="true"/>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="2" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.203146682579475" yZero="-0.13803498221652077" zZero="-2.096002150039014" scale="97.35159876367536" xAngle="21.0" zAngle="-48.95675573754448"/>
<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="255" g="255" b="255"/>
<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="4.433136300065597"/>
<slider min="0" max="6.283185307179586" absoluteScreenLocation="true" width="180.0" x="139.0" y="161.0" 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="3" playing="false"/>
</element>
<command name="Cylinder">
<input a0="(0, 0, 0)" a1="(0, 0, 4)" a2="3 / π"/>
<output a0="a" a1="a_2" a2="a_3" a3="a_4"/>
</command>
<element type="quadriclimited" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="216" g="27" b="96" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="conic3d" label="a_2">
<show object="true" label="false" ev="5"/>
<objColor r="216" g="27" b="96" alpha="0.4000000059604645"/>
<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="-0.91189065278104" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="parametric"/>
</element>
<element type="conic3d" label="a_3">
<show object="true" label="false" ev="4"/>
<objColor r="216" g="27" b="96" alpha="0.4000000059604645"/>
<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="-0.91189065278104" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="parametric"/>
</element>
<element type="quadricpart" label="a_4">
<show object="true" label="true" ev="4"/>
<objColor r="216" g="27" b="96" alpha="0.4000000059604645"/>
<layer val="0"/>
<labelMode val="0"/>
<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"/>
<eqnStyle style="implicit"/>
</element>
<expression label="A" exp="(0, -3 / π, 0)" type="point"/>
<element type="point3d" label="A">
<show object="true" label="true" ev="7"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-10" y="41"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="-0.954929658551372" z="0.0" w="1.0"/>
<pointSize val="5"/>
</element>
<expression label="O" exp="A + (0, (((2 * π)) / ((2 * π) - α) * 3 / π), 0)" type="point" />
<element type="point3d" label="O">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<coords x="0.0" y="2.288227672377815" z="0.0" w="1.0"/>
<pointSize val="5"/>
</element>
<command name="Rotate">
<input a0="A" a1="(α - (2 * π)) / 2" a2="O"/>
<output a0="A'"/>
</command>
<element type="point3d" label="A'">
<show object="false" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.5901006091502485" y="0.33644686817893166" z="0.0" w="1.0"/>
<pointSize val="5"/>
</element>
<expression label="B" exp="A' + (0, 0, 4)" type="point" />
<element type="point3d" 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="-2.5901006091502485" y="0.33644686817893166" z="4.0" w="1.0"/>
<pointSize val="4"/>
</element>
<command name="Segment">
<input a0="A'" a1="B"/>
<output a0="f"/>
</command>
<element type="segment3d" label="f">
<show object="false" label="true" 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="C" exp="O + (0, 0, 4)" type="point" />
<element type="point3d" label="C">
<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="2.288227672377815" z="4.0" w="1.0"/>
<pointSize val="4"/>
</element>
<command name="Segment">
<input a0="O" a1="C"/>
<output a0="g"/>
</command>
<element type="segment3d" label="g">
<show object="false" label="true" 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>
<command name="If">
<input a0="α < 360°" a1="Surface[f, (2 * π) - α, g]" a2="Surface[t, -3 / π, u, t, -3, 3, u, 0, 4]"/>
<output a0="b"/>
</command>
<element type="surfacecartesian3d" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="If">
<input a0="α < 360°" a1="CurveCartesian[((((2 * π)) / ((2 * π) - α) * 3 / π) * sin(u)), ((((2 * π)) / ((2 * π) - α) * 3 / π) * cos(u)) + y(O), 0, u, (-2 * π) + α / 2, ((-α)) / 2]" a2="CurveCartesian[u, -3 / π, 0, u, -3, 3]"/>
<output a0="c"/>
</command>
<element type="curvecartesian3d" label="c">
<show object="true" label="true" ev="5"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<command name="If">
<input a0="α < 360°" a1="CurveCartesian[((((2 * π)) / ((2 * π) - α) * 3 / π) * sin(u)), ((((2 * π)) / ((2 * π) - α) * 3 / π) * cos(u)) + y(O), 4, u, (-2 * π) + α / 2, ((-α)) / 2]" a2="CurveCartesian[u, -3 / π, 4, u, -3, 3]"/>
<output a0="d"/>
</command>
<element type="curvecartesian3d" label="d">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<command name="Segment">
<input a0="Point[c, 1]" a1="Point[d, 1]"/>
<output a0="h"/>
</command>
<element type="segment3d" label="h">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<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="Point[c, 0]" a1="Point[d, 0]"/>
<output a0="i"/>
</command>
<element type="segment3d" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="d" a1="If[α < 360°, 0, 1]"/>
<output a0="D"/>
</command>
<element type="point3d" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="2.590100609150248" y="0.33644686817893144" z="4.0" w="1.0"/>
<pointSize val="5"/>
<caption val="B"/>
</element>
<command name="If">
<input a0="α < 360°" a1="CurveCartesian[((((2 * π)) / ((2 * π) - α) * 3 / π) * sin(u)), ((((2 * π)) / ((2 * π) - α) * 3 / π) * cos(u)) + y(O), 4 - (4 * (u + (2 * π) - α / 2) / (π - α / 2)), u, (-2 * π) + α / 2, (-π)]" a2="CurveCartesian[t, -3 / π, (4 * t / 3), t, 0, 3]"/>
<output a0="e"/>
</command>
<element type="curvecartesian3d" label="e">
<show object="false" label="true" ev="5"/>
<condition showObject="j"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<element type="boolean" label="j">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="117" y="239"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<caption val="显示最短路径"/>
</element>
<element type="point" label="E">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-7.734733437698338" y="2.615953019127293" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="F">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-3.638001669151291" y="2.615953019127293" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic1">
<file name="6650f066cc3c5e615eb4cad0974fcf0f/qrcode_for_gh_671be4471e45_258.jpg"/>
<inBackground val="true"/>
<absoluteScreenLocation x="92" y="588"/> <show object="true" label="false"/>
<condition showObject="k"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
</element>
<element type="point" label="G">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-6.2580045443848675" y="-0.7344749001108043" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="H">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-4.7653968457669516" y="-0.7344749001108043" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic2">
<file name="ce336dab3c15ac357e597d2ea854df43/b83d44e3752373ed049c91fd270a8fddthumb.png"/>
<inBackground val="true"/>
<absoluteScreenLocation x="185" y="799"/> <show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
</element>
<element type="boolean" label="k">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="324" y="239"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<caption val="帮助"/>
</element>
<expression label="text1" exp=""欢迎关注公众号\\
Geogebra 学习与应用""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<condition showObject="k"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="111" y="599"/>
</element>
</construction>
</geogebra>