球面坐标系与经纬度角度演示
2020-12-05 10:41
本GeoGebra范例演示了三维空间中的球面几何与坐标系构造。通过构建球体、赤道平面及正交线，展示了球面上点的经度与纬度参数，以及它们与原点、赤道面和坐标轴之间的空间角度与线段关系。适用于立体几何与球面坐标系的直观教学与探索。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.688.0" app="classic" platform="w" id="3ce7f76e-7072-4743-9fd5-5fff4fe81bc6" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1282" height="700" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.21450858034321374" 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="1,3,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1,1" size="353" window="806,334,700,550" />
<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="1" visible="true" inframe="false" stylebar="false" location="3" size="275" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="204" tab="ALGEBRA" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="336" 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="true" 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="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="999" window="100,100,600,400" />
</views>
<toolbar show="false" items="0 39 | 1 501 67 , 5 19 , 72 | 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="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="275" height="700"/>
<coordSystem xZero="127.99999999999994" yZero="199.0000000000002" scale="49.999999999999986" yscale="50.000000000000064"/>
<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="10"/>
<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="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.002902716953633755" yZero="0.00222001118167392" zZero="-0.1367654444121021" scale="290.05895092056574" xAngle="21" zAngle="-731.9342966338529"/>
<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="2"/>
<projection type="1"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="point3d" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="64" g="64" b="64" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="0" w="1"/>
<pointSize val="5"/>
</element>
<command name="Sphere">
<input a0="A" a1="1"/>
<output a0="s"/>
</command>
<element type="quadric" label="s">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="204" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="0" x1="0" y1="1" z1="0" x2="0" y2="0" z2="1"/>
<matrix A0="1" A1="1" A2="1" A3="-1" A4="0" A5="0" A6="0" A7="0" A8="0" A9="0"/>
<eqnStyle style="specific"/>
</element>
<expression label="I" exp="(1, 0, 0)" type="point"/>
<element type="point3d" label="I">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-53" y="-23"/>
<labelMode val="3"/>
<fixed val="true"/>
<coords x="1" y="0" z="0" w="1"/>
<pointSize val="1"/>
</element>
<command name="Circle">
<input a0="A" a1="I" a2="zAxis"/>
<output a0="equatorial"/>
</command>
<element type="conic3d" label="equatorial">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-1" A3="0" A4="0" A5="0"/>
<eqnStyle style="parametric"/>
</element>
<element type="point3d" label="K">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="-30"/>
<labelMode val="3"/>
<fixed val="true"/>
<coords x="0" y="0" z="1" w="1"/>
<pointSize val="1"/>
</element>
<element type="point3d" label="J">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="15" y="-8"/>
<labelMode val="3"/>
<fixed val="true"/>
<coords x="0" y="1" z="0" w="1"/>
<pointSize val="1"/>
<caption val="(0,1,0)"/>
</element>
<expression label="P" exp="((cos(48°) * cos(16°)), (cos(48°) * sin(16°)), sin(48°))" type="point"/>
<element type="point3d" label="P">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.6432096214727517" y="0.184437391032981" z="0.7431448254773942" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="A" a1="P"/>
<output a0="sOP"/>
</command>
<element type="segment3d" label="sOP">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""x軸""/>
<element type="text" label="text1">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="I"/>
<labelOffset x="-75" y="28"/>
</element>
<expression label="text2" exp=""y軸	""/>
<element type="text" label="text2">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="J"/>
<labelOffset x="77" y="22"/>
</element>
<expression label="text3" exp=""z軸""/>
<element type="text" label="text3">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="K"/>
<labelOffset x="18" y="-49"/>
</element>
<command name="Plane">
<input a0="A" a1="I" a2="J"/>
<output a0="E"/>
</command>
<element type="plane3d" label="E">
<show object="false" label="false" ev="4"/>
<objColor r="99" g="219" b="219" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="1" w="0"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="OrthogonalLine">
<input a0="P" a1="E"/>
<output a0="L_{PE}"/>
</command>
<element type="line3d" label="L_{PE}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.6432096214727517" oy="0.184437391032981" oz="0.7431448254773942" ow="1" vx="0" vy="0" vz="1" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L_{PE}" a1="E"/>
<output a0="P'"/>
</command>
<element type="point3d" label="P'">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.6432096214727517" y="0.184437391032981" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="P" a1="P'"/>
<output a0="sPP'"/>
</command>
<element type="segment3d" label="sPP'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="P'"/>
<output a0="sOP'"/>
</command>
<element type="segment3d" label="sOP'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="P'" a1="xAxis" a2="space"/>
<output a0="L_{P'x}"/>
</command>
<element type="line3d" label="L_{P'x}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.6432096214727517" oy="0.184437391032981" oz="0" ow="1" vx="0" vy="1" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L_{P'x}" a1="xAxis"/>
<output a0="P''"/>
</command>
<element type="point3d" label="P''">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.6432096214727517" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="P'" a1="P''"/>
<output a0="sP'P''"/>
</command>
<element type="segment3d" label="sP'P''">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="P" a1="A" a2="P'"/>
<output a0="φ_{1}"/>
</command>
<element type="angle3d" label="φ_{1}">
<angleStyle val="1"/>
<value val="0.837758040957278"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="255" g="153" b="51" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Angle">
<input a0="P'" a1="A" a2="P''"/>
<output a0="λ_{1}"/>
</command>
<element type="angle3d" label="λ_{1}">
<angleStyle val="1"/>
<value val="0.27925268031909223"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 1"/>
<objColor r="0" g="153" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<element type="numeric" label="n">
<value val="1"/>
<show object="true" label="true"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<bgColor r="0" g="153" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="4" absoluteScreenLocation="true" width="150" x="37" y="84" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="text4" exp=""1.\ 維也納\\2.\ 峇里島\\3.\ 紐約\\ 4.\ 巴西利亞""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="12" style="0"/>
<startPoint x="-1.6199999999999994" y="-2.339999999999997" z="1"/>
</element>
<expression label="Q" exp="((cos(-8°) * cos(115°)), (cos(-8°) * sin(115°)), sin(-8°))" type="point"/>
<element type="point3d" label="Q">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-0.4185053698688818" y="0.8974876619542299" z="-0.13917310096006544" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="A" a1="Q"/>
<output a0="sOQ"/>
</command>
<element type="segment3d" label="sOQ">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="Q" a1="E"/>
<output a0="L_{QE}"/>
</command>
<element type="line3d" label="L_{QE}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="-0.4185053698688818" oy="0.8974876619542299" oz="-0.13917310096006544" ow="1" vx="0" vy="0" vz="1" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L_{QE}" a1="E"/>
<output a0="Q'"/>
</command>
<element type="point3d" label="Q'">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.4185053698688818" y="0.8974876619542299" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="A" a1="Q'"/>
<output a0="sOQ'"/>
</command>
<element type="segment3d" label="sOQ'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="Q'" a1="xAxis" a2="space"/>
<output a0="L_{Q'x}"/>
</command>
<element type="line3d" label="L_{Q'x}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="-0.4185053698688818" oy="0.8974876619542299" oz="0" ow="1" vx="0" vy="1" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="L_{Q'x}"/>
<output a0="Q''"/>
</command>
<element type="point3d" label="Q''">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.4185053698688818" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Angle">
<input a0="Q" a1="A" a2="Q'"/>
<output a0="φ_{2}"/>
</command>
<element type="angle3d" label="φ_{2}">
<angleStyle val="1"/>
<value val="0.13962634015954603"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="255" g="153" b="51" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Segment">
<input a0="Q" a1="Q'"/>
<output a0="sQQ'"/>
</command>
<element type="segment3d" label="sQQ'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="Q'" a1="Q''"/>
<output a0="sO'O''"/>
</command>
<element type="segment3d" label="sO'O''">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="R" exp="((cos(40°) * cos(-74°)), (cos(40°) * sin(-74°)), sin(40°))" type="point"/>
<element type="point3d" label="R">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 3"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.2111504647396207" y="-0.7363691805566739" z="0.6427876096865393" w="1"/>
<pointSize val="3"/>
</element>
<command name="Angle">
<input a0="I" a1="A" a2="Q'"/>
<output a0="λ_{2}"/>
</command>
<element type="angle3d" label="λ_{2}">
<angleStyle val="1"/>
<value val="2.007128639793479"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 2"/>
<objColor r="0" g="153" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="OrthogonalLine">
<input a0="R" a1="E"/>
<output a0="L_{RE}"/>
</command>
<element type="line3d" label="L_{RE}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.2111504647396207" oy="-0.7363691805566739" oz="0.6427876096865393" ow="1" vx="0" vy="0" vz="1" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L_{RE}" a1="E"/>
<output a0="R'"/>
</command>
<element type="point3d" label="R'">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 3"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.2111504647396207" y="-0.7363691805566739" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="OrthogonalLine">
<input a0="R'" a1="xAxis" a2="space"/>
<output a0="L_{R'x}"/>
</command>
<element type="line3d" label="L_{R'x}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.2111504647396207" oy="-0.7363691805566739" oz="0" ow="1" vx="0" vy="-1" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="L_{R'x}"/>
<output a0="R''"/>
</command>
<element type="point3d" label="R''">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 3"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.2111504647396207" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="(0, 0)" a1="R"/>
<output a0="sOR"/>
</command>
<element type="segment3d" label="sOR">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="R" a1="R'"/>
<output a0="sRR'"/>
</command>
<element type="segment3d" label="sRR'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="R'" a1="R''"/>
<output a0="sR'R''"/>
</command>
<element type="segment3d" label="sR'R''">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="(0, 0)" a1="R'"/>
<output a0="sOR'"/>
</command>
<element type="segment3d" label="sOR'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="R" a1="(0, 0)" a2="R'"/>
<output a0="φ_{3}"/>
</command>
<element type="angle3d" label="φ_{3}">
<angleStyle val="1"/>
<value val="0.6981317007977317"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 3"/>
<objColor r="255" g="153" b="51" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Angle">
<input a0="R'" a1="(0, 0)" a2="R''"/>
<output a0="λ_{3}"/>
</command>
<element type="angle3d" label="λ_{3}">
<angleStyle val="1"/>
<value val="1.2915436464758039"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 3"/>
<objColor r="0" g="153" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<expression label="S" exp="((cos(-16°) * cos(-48°)), (cos(-16°) * sin(-48°)), sin(-16°))" type="point"/>
<element type="point3d" label="S">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.6432096214727517" y="-0.714356655266186" z="-0.27563735581699916" w="1"/>
<pointSize val="3"/>
</element>
<command name="OrthogonalLine">
<input a0="S" a1="E"/>
<output a0="L_{SE}"/>
</command>
<element type="line3d" label="L_{SE}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.6432096214727517" oy="-0.714356655266186" oz="-0.27563735581699916" ow="1" vx="0" vy="0" vz="1" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L_{SE}" a1="E"/>
<output a0="S'"/>
</command>
<element type="point3d" label="S'">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.6432096214727517" y="-0.714356655266186" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="OrthogonalLine">
<input a0="S'" a1="xAxis" a2="space"/>
<output a0="L_{S'x}"/>
</command>
<element type="line3d" label="L_{S'x}">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="0.6432096214727517" oy="-0.714356655266186" oz="0" ow="1" vx="0" vy="-1" vz="0" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L_{S'x}" a1="xAxis"/>
<output a0="S''"/>
</command>
<element type="point3d" label="S''">
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.6432096214727517" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Segment">
<input a0="(0, 0)" a1="S"/>
<output a0="sOS"/>
</command>
<element type="segment3d" label="sOS">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="S" a1="S'"/>
<output a0="sSS'"/>
</command>
<element type="segment3d" label="sSS'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="S'" a1="S''"/>
<output a0="sS'S''"/>
</command>
<element type="segment3d" label="sS'S''">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="(0, 0)" a1="S'"/>
<output a0="sOS'"/>
</command>
<element type="segment3d" label="sOS'">
<show object="true" label="false" ev="4"/>
<condition showObject="n ≟ 4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="S" a1="(0, 0)" a2="S'"/>
<output a0="φ_{4}"/>
</command>
<element type="angle3d" label="φ_{4}">
<angleStyle val="1"/>
<value val="0.2792526803190926"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 4"/>
<objColor r="255" g="153" b="51" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Angle">
<input a0="S'" a1="(0, 0)" a2="S''"/>
<output a0="λ_{4}"/>
</command>
<element type="angle3d" label="λ_{4}">
<angleStyle val="1"/>
<value val="0.8377580409572781"/>
<show object="true" label="true" ev="4"/>
<condition showObject="n ≟ 4"/>
<objColor r="0" g="153" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<expression label="B" exp="(-2, -9)" type="point"/>
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-2" y="-9" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="C" exp="(2, -9)" type="point"/>
<element type="point" label="C">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2" y="-9" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic1">
<file name="25068968fe6fc35e13ac14deda304b36/photo-1587031547865-02677a5dedac.jpg"/>
<inBackground val="false"/>
<startPoint number="0" exp="B"/>
<startPoint number="1" exp="C"/>
<show object="true" label="false"/>
<condition showObject="n ≟ 1"/>
<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>
<expression label="D" exp="(-2, -9)" type="point"/>
<element type="point" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-2" y="-9" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="F" exp="(2, -9)" type="point"/>
<element type="point" label="F">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2" y="-9" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic2">
<file name="dcac8718b028f8bae93ce81cc05f9613/2.jpg"/>
<inBackground val="false"/>
<startPoint number="0" exp="D"/>
<startPoint number="1" exp="F"/>
<show object="true" label="false"/>
<condition showObject="n ≟ 2"/>
<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>
<expression label="G" exp="(-2, -9)" type="point"/>
<element type="point" label="G">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-2" y="-9" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="H" exp="(2, -9)" type="point"/>
<element type="point" label="H">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2" y="-9" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic3">
<file name="276b277212746f957588a2152fa3fa80/3.jpg"/>
<inBackground val="false"/>
<startPoint number="0" exp="G"/>
<startPoint number="1" exp="H"/>
<show object="true" label="false"/>
<condition showObject="n ≟ 3"/>
<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>
<expression label="L" exp="(-2, -9)" type="point"/>
<element type="point" label="L">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-2" y="-9" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="M" exp="(2, -9)" type="point"/>
<element type="point" label="M">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="2" y="-9" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="image" label="pic4">
<file name="9577a316df1108df2055c39d9af9cafb/5.jpg"/>
<inBackground val="false"/>
<startPoint number="0" exp="L"/>
<startPoint number="1" exp="M"/>
<show object="true" label="false"/>
<condition showObject="n ≟ 4"/>
<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>