Orthographic Projection of a Cube
2020-05-20 01:21
This example demonstrates the orthographic projection of a cube onto a specified plane. By constructing the eight vertices, twelve edges, and six faces of a cube using vector, plane, and rotation commands, it illustrates the geometric principles of orthogonally projecting a 3D solid along a specific direction onto a projection plane. Users can observe how the projected shapes of faces and edges ch
In collections
立体几何
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.583.0" app="classic" platform="w" id="1cdb3c17-0c8e-4aee-b795-d77f69991169" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="897" height="553" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.27201783723522854" orientation="1" />
</panes>
<views>
<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="false" stylebar="true" location="1,1" size="370" window="668,766,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="3" size="244" 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" size="295" window="376,617,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="232" window="533,393,600,400" />
<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="128" 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="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" 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="false" location="1" size="645" window="652,324,600,400" />
</views>
<toolbar show="false" 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="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="14"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="244" height="553"/>
<coordSystem xZero="222.99999999999991" yZero="714.5000000000002" scale="49.99999999999997" yscale="50.00000000000002"/>
<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>
<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" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="2.8" yZero="-9" zZero="0" scale="50" xAngle="20" zAngle="-60"/>
<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"/>
<clipping use="false" show="false" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<expression label="A" exp="(0, 0, 0)" type="point"/>
<element type="point3d" label="A">
<show object="false" label="true" ev="5"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0" y="0" z="0" w="1"/>
<pointSize val="5"/>
</element>
<expression label="B" exp="(2, 0, 0)" type="point"/>
<element type="point3d" label="B">
<show object="false" label="true" ev="5"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="2" y="0" z="0" w="1"/>
<pointSize val="5"/>
</element>
<command name="Point">
<input a0="Circle[B, Distance[A, B], Segment[A, B]]"/>
<output a0="C"/>
</command>
<element type="point3d" label="C">
<show object="false" label="true" ev="5"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="2" y="2" z="0" w="1"/>
<pointSize val="5"/>
</element>
<command name="Cube">
<input a0="A" a1="B" a2="C"/>
<output a0="a" a1="D" a2="E" a3="F" a4="G" a5="H" a6="faceABCD" a7="faceADHE" a8="faceABFE" a9="faceBCGF" a10="faceCDHG" a11="faceEFGH" a12="edgeAD" a13="edgeAB" a14="edgeBC" a15="edgeCD" a16="edgeAE" a17="edgeDH" a18="edgeEH" a19="edgeEF" a20="edgeBF" a21="edgeFG" a22="edgeCG" a23="edgeGH"/>
</command>
<element type="polyhedron" label="a">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0.44999998807907104"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="point3d" label="D">
<show object="false" label="true" ev="5"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="2" z="0" w="1"/>
<pointSize val="4"/>
</element>
<element type="point3d" label="E">
<show object="false" label="true" ev="5"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="0" z="2" w="1"/>
<pointSize val="4"/>
</element>
<element type="point3d" label="F">
<show object="false" label="true" ev="5"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2" y="0" z="2" w="1"/>
<pointSize val="4"/>
</element>
<element type="point3d" label="G">
<show object="false" label="true" ev="5"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2" y="2" z="2" w="1"/>
<pointSize val="4"/>
</element>
<element type="point3d" label="H">
<show object="false" label="true" ev="5"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="2" z="2" w="1"/>
<pointSize val="4"/>
</element>
<element type="polygon3d" label="faceABCD">
<lineStyle thickness="2" type="0" typeHidden="1" opacity="178"/>
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0.44999998807907104"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="polygon3d" label="faceADHE">
<lineStyle thickness="2" type="0" typeHidden="1" opacity="178"/>
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0.44999998807907104"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="polygon3d" label="faceABFE">
<lineStyle thickness="2" type="0" typeHidden="1" opacity="178"/>
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0.44999998807907104"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="polygon3d" label="faceBCGF">
<lineStyle thickness="2" type="0" typeHidden="1" opacity="178"/>
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0.44999998807907104"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="polygon3d" label="faceCDHG">
<lineStyle thickness="2" type="0" typeHidden="1" opacity="178"/>
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0.44999998807907104"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="polygon3d" label="faceEFGH">
<lineStyle thickness="2" type="0" typeHidden="1" opacity="178"/>
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0.44999998807907104"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment3d" label="edgeAD">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeAB">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeBC">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeCD">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeAE">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeDH">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeEH">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeEF">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeBF">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeFG">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeCG">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="edgeGH">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="angle" label="α">
<angleStyle val="3"/>
<value val="0.8028514559173916"/>
<slider min="0" max="90°" absoluteScreenLocation="true" width="180" x="27" y="101" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="204" 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="Rotate">
<input a0="a" a1="α" a2="zAxis"/>
<output a0="a'"/>
</command>
<element type="polyhedron" label="a'">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="5"/>
<objColor r="0" g="153" b="204" alpha="0.6899999976158142"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<expression label="l1" exp="{A, B, C, D, E, F, G, H}" />
<element type="list" label="l1">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Rotate">
<input a0="l1" a1="α" a2="zAxis"/>
<output a0="l1'"/>
</command>
<element type="list" label="l1'">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Max">
<input a0="y(l1')"/>
<output a0="b"/>
</command>
<element type="numeric" label="b">
<value val="2.8279963415952967"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="KeepIf">
<input a0="y(Q) < b" a1="Q" a2="l1'"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Vector">
<input a0="B" a1="C"/>
<output a0="u"/>
</command>
<element type="vector3d" label="u">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="2" z="0" w="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<startPoint exp="B"/>
</element>
<command name="Plane">
<input a0="y = 10"/>
<output a0="p"/>
</command>
<element type="plane3d" label="p">
<show object="true" label="false" ev="5"/>
<objColor r="255" g="215" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="1" z="0" w="-10"/>
<fading val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="Sequence">
<input a0="Line[Element[l2, k], u]" a1="k" a2="1" a3="Length[l2]"/>
<output a0="l3"/>
</command>
<element type="list" label="l3">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Intersect[Element[l3, k], p]" a1="k" a2="1" a3="Length[l3]"/>
<output a0="l4"/>
</command>
<element type="list" label="l4">
<show object="false" label="false" ev="5"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Zip">
<input a0="Vector[P, Q]" a1="P" a2="l2" a3="Q" a4="l4"/>
<output a0="l5"/>
</command>
<element type="list" label="l5">
<show object="true" label="false" ev="5"/>
<objColor r="255" g="102" b="153" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="If">
<input a0="0° < α < 90°" a1="Polygon[Element[l4, 3], Element[l4, 2], Element[l4, 5], Element[l4, 6]]"/>
<output a0="q2"/>
</command>
<element type="polygon3d" label="q2">
<lineStyle thickness="2" type="0" typeHidden="1" opacity="191"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="If">
<input a0="α ≟ 0° ∨ α ≟ 90°" a1="Polygon[Element[l4, 1], Element[l4, 2], Element[l4, 4], Element[l4, 3]]"/>
<output a0="q1"/>
</command>
<element type="polygon3d" label="q1">
<lineStyle thickness="2" type="0" typeHidden="1" opacity="191"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="If">
<input a0="0° < α < 90°" a1="Segment[Element[l4, 1], Element[l4, 4]]"/>
<output a0="f"/>
</command>
<element type="segment3d" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="I">
<show object="false" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-4.14" y="3.426" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="J">
<show object="false" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-0.612" y="3.426" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic1">
<file name="a94857e5933a1326ec9a95938feb52d3/公众号 二维码.png"/>
<inBackground val="false"/>
<startPoint number="0" exp="I"/>
<startPoint number="1" exp="J"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="point" label="K">
<show object="false" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-3.9" y="7.286" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="L">
<show object="false" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-0.672" y="7.286" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic2">
<file name="42771fb0d6a81a4a291e3b580263b6e0/1589889590.png"/>
<inBackground val="false"/>
<startPoint number="0" exp="K"/>
<startPoint number="1" exp="L"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
</construction>
</geogebra>