Geometric Construction and Optimization of a Torus
2016-10-26 14:29
This applet demonstrates the geometric construction of a torus in 3D space. Using unit vectors, segments, and surface commands, it builds the parametric equations and 3D model. It also integrates calculus, Riemann sums, and optimization to explore the geometric center and related extrema. Ideal for visualizing solid geometry and calculus concepts.
In collections
微積分
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.278.0" id="b5be86e2-2933-4302-87f8-d40d35e4e681" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="833" height="672" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.2785114045618247" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 || 2020 , 2021 , 2022 || 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="3,0" size="378" window="100,100,250,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 | 1009 || 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="375" window="869,281,700,550" />
<view id="1" visible="false" inframe="false" stylebar="true" location="3" size="232" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="true" stylebar="true" location="1" size="291" window="940,24,500,821" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" 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="true" location="1" size="833" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 | 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="true" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="232" height="520"/>
<coordSystem xZero="209.0" yZero="400.00000000000006" scale="49.99999999999999" yscale="49.999999999999964"/>
<evSettings axes="false" grid="false" gridIsBold="false" 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="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="3"/>
</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"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="-0.05673874156961328" yZero="0.12441381319889058" zZero="-0.355462963998873" scale="88.10227653349054" xAngle="23.0" zAngle="726.7594434052323"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="2" showNumbers="false"/>
<plate show="false"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="false" show="false" size="1"/>
<projection type="1"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="point3d" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="0.0" z="0.0" w="1.0"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="B">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.49859506416906996" y="1.1368805258415087" z="1.7416906514039834" w="1.0"/>
<pointSize val="5"/>
</element>
<command name="UnitVector">
<input a0="Vector[A, B]"/>
<output a0="u"/>
</command>
<element type="vector3d" label="u">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.233116132064324" y="0.5315439518943155" z="0.8143205119459673" w="0.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
<startPoint exp="A"/>
</element>
<command name="UnitVector">
<input a0="Vector[If[x(u)^(2) + y(u)^(2) ≟ 0, (1, 0, 0), ((-y(u)), x(u), 0)]]"/>
<output a0="v"/>
</command>
<element type="vector3d" label="v">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.9157990963477575" y="0.4016366705476863" z="0.0" w="0.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
</element>
<expression label="w" exp="u ⊗ v" type="vector" />
<element type="vector3d" label="w">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.3270609791766657" y="-0.7457539889775601" z="0.5804154579471998" w="0.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
</element>
<element type="numeric" label="r1">
<value val="1.1"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="30.0" y="50.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="r2">
<value val="2.0"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="30.0" y="90.0" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Min">
<input a0="abs(r1)" a1="abs(r2)"/>
<output a0="r"/>
</command>
<element type="numeric" label="r">
<value val="1.1"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Max">
<input a0="abs(r1)" a1="abs(r2)"/>
<output a0="R"/>
</command>
<element type="numeric" label="R">
<value val="2.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="x0" exp="x(A)" />
<element type="numeric" label="x0">
<value val="0.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="y0" exp="y(A)" />
<element type="numeric" label="y0">
<value val="0.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="z0" exp="z(A)" />
<element type="numeric" label="z0">
<value val="0.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Surface">
<input a0="x0 + ((s * cos(t)) * x(v)) + ((s * sin(t)) * x(w))" a1="y0 + ((s * cos(t)) * y(v)) + ((s * sin(t)) * y(w))" a2="z0 + ((s * cos(t)) * z(v)) + ((s * sin(t)) * z(w))" a3="s" a4="r" a5="R" a6="t" a7="0" a8="(2 * π)"/>
<output a0="ring"/>
</command>
<element type="surfacecartesian3d" label="ring">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.53"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="a" exp="(2 * u)" type="vector" />
<element type="vector3d" label="a">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.466232264128648" y="1.063087903788631" z="1.6286410238919347" w="0.0"/>
<lineStyle thickness="6" type="0" typeHidden="1"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="f"/>
</command>
<element type="segment3d" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="b" exp="(x(u) * x) + (y(u) * y) + (z(u) * z) = (x(u) * x0) + (y(u) * y0) + (z(u) * z0)" type="plane" />
<element type="plane3d" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="192" g="192" b="192" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.233116132064324" y="0.5315439518943155" z="0.8143205119459673" w="0.0"/>
<fading val="0.0"/>
<lineStyle thickness="0" type="0" typeHidden="2"/>
</element>
</construction>
</geogebra>