3D Spherical Coordinate System Demonstration
2020-12-05 14:50
This GeoGebra applet dynamically demonstrates the spherical coordinate system in 3D space. By constructing a sphere, equatorial circle, and orthogonal projection lines, it visually illustrates the spherical coordinate parameters of a point, including radial distance, polar angle (φ), and azimuthal angle (λ). Combined with trigonometric functions and geometric projections, it clearly explains the c
In collections
Spherical Coordinates | GeoGebra
<?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="900" height="700" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.34" 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="1" visible="false" inframe="false" stylebar="false" location="1,3" size="214" 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="false" location="1,1" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="372" window="806,334,700,550" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="306" 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="900" 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="false" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="900" height="700"/>
<coordSystem xZero="82.99999999999994" yZero="286.00000000000017" scale="49.99999999999998" yscale="50.00000000000003"/>
<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="29" zAngle="-347.4248962261055"/>
<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="O">
<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="O" 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="0.15000000596046448"/>
<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>
<element type="point3d" label="I">
<show object="true" 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="O" 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>
<command name="Point">
<input a0="equatorial"/>
<output a0="A"/>
</command>
<element type="point3d" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.4339786937911469" y="0.9009231339772167" z="0" w="1"/>
<pointSize val="4"/>
</element>
<element type="point3d" label="K">
<show object="true" 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>
<command name="CircumcircleArc">
<input a0="(-K)" a1="A" a2="K"/>
<output a0="arcAB"/>
</command>
<element type="conicpart" label="arcAB">
<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="-2.220446049250313e-16" A3="0" A4="-0.7071067811865476" A5="-0.7071067811865476"/>
<eqnStyle style="parametric"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="arcAB"/>
<output a0="P"/>
</command>
<element type="point3d" label="P">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.32520716654677817" y="0.6751176126129694" z="0.6621604850535714" w="1"/>
<pointSize val="5"/>
</element>
<element type="vector3d" label="i">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<coords x="1" y="0" z="0" w="0"/>
<lineStyle thickness="5" type="0" typeHidden="2"/>
</element>
<element type="vector3d" label="j">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="153" b="0" 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="1" z="0" w="0"/>
<lineStyle thickness="5" type="0" typeHidden="2"/>
</element>
<element type="vector3d" label="k">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="255" 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="1" w="0"/>
<lineStyle thickness="5" type="0" typeHidden="2"/>
</element>
<command name="Angle">
<input a0="I" a1="O" a2="A"/>
<output a0="λ"/>
</command>
<element type="angle3d" label="λ">
<angleStyle val="1"/>
<emphasizeRightAngle val="false"/>
<value val="1.1218919818706372"/>
<show object="true" label="true" ev="4"/>
<objColor r="51" g="153" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="6"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
<selectionAllowed val="false"/>
</element>
<command name="Angle">
<input a0="A" a1="O" a2="P"/>
<output a0="φ"/>
</command>
<element type="angle3d" label="φ">
<angleStyle val="1"/>
<emphasizeRightAngle val="false"/>
<value val="0.7236982007535776"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="153" b="51" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<decoration type="6"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="50"/>
<selectionAllowed val="false"/>
</element>
<command name="Segment">
<input a0="O" a1="P"/>
<output a0="sOB"/>
</command>
<element type="segment3d" label="sOB">
<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="20" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="cosφ" exp="cos(φ)" />
<element type="numeric" label="cosφ">
<value val="0.7493620567079836"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<expression label="sinφ" exp="z(P)" />
<element type="numeric" label="sinφ">
<value val="0.6621604850535714"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<element type="point3d" label="J">
<show object="true" 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="text4" exp=""x軸""/>
<element type="text" label="text4">
<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="2" size="12" style="0"/>
<startPoint exp="I"/>
<labelOffset x="-58" y="55"/>
</element>
<expression label="text5" exp=""y軸""/>
<element type="text" label="text5">
<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="2" size="12" style="0"/>
<startPoint exp="J"/>
<labelOffset x="68" y="42"/>
</element>
<expression label="text6" exp=""z軸""/>
<element type="text" label="text6">
<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="2" size="12" style="0"/>
<startPoint exp="K"/>
<labelOffset x="11" y="-79"/>
</element>
<expression label="text1" exp=""(1,0,0)""/>
<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="-6" y="13"/>
</element>
<expression label="text2" exp=""(0,1,0)""/>
<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="29" y="-31"/>
</element>
<expression label="text3" exp=""(0,0,1)""/>
<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="-71" y="-36"/>
</element>
<command name="Segment">
<input a0="O" a1="A"/>
<output a0="sOA"/>
</command>
<element type="segment3d" label="sOA">
<show object="true" 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"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="P" a1="sOA" a2="space"/>
<output a0="L"/>
</command>
<element type="line3d" label="L">
<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="0.32520716654677817" oy="0.6751176126129694" oz="0.6621604850535714" ow="1" vx="7.552743098812658e-17" vy="-3.638190053010345e-17" vz="1" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<command name="Intersect">
<input a0="L" a1="sOA"/>
<output a0="P'"/>
</command>
<element type="point3d" label="P'">
<show object="true" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.3252071665467781" y="0.6751176126129694" z="0" w="1"/>
<pointSize val="4"/>
</element>
<command name="OrthogonalLine">
<input a0="P'" a1="xAxis" a2="space"/>
<output a0="M"/>
</command>
<element type="line3d" label="M">
<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="0.3252071665467781" oy="0.6751176126129694" 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="M"/>
<output a0="P''"/>
</command>
<element type="point3d" label="P''">
<show object="true" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.3252071665467781" y="0" z="0" w="1"/>
<pointSize val="4"/>
</element>
<command name="Segment">
<input a0="P" a1="P'"/>
<output a0="sBD"/>
</command>
<element type="segment3d" label="sBD">
<show object="true" 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"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="P'" a1="P''"/>
<output a0="sDE"/>
</command>
<element type="segment3d" label="sDE">
<show object="true" 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"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="cosλ" exp="cos(λ)" />
<element type="numeric" label="cosλ">
<value val="0.4339786937911469"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<expression label="sinλ" exp="sin(λ)" />
<element type="numeric" label="sinλ">
<value val="0.9009231339772167"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<expression label="text7" exp=""\cosφ""/>
<element type="text" label="text7">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) > 0"/>
<objColor r="255" g="102" 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="m_{OP'}"/>
<labelOffset x="-25" y="-20"/>
</element>
<expression label="text8" exp=""\sinφ""/>
<element type="text" label="text8">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) > 0"/>
<objColor r="255" g="102" 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="m_{PP''}"/>
<labelOffset x="5" y="-3"/>
</element>
<expression label="text9" exp=""\cosφ\sinλ""/>
<element type="text" label="text9">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) > 0° ∧ y(P) > 0°"/>
<objColor r="0" g="100" 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="m_{P'P''}"/>
<labelOffset x="-57" y="3"/>
</element>
<expression label="text10" exp=""\cosφ\cosλ""/>
<element type="text" label="text10">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) > 0° ∧ y(P) > 0°"/>
<objColor r="0" g="100" 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="m_{OP''}"/>
<labelOffset x="-78" y="-19"/>
</element>
<expression label="text7_{1}" exp=""\cos(-φ)""/>
<element type="text" label="text7_{1}">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) < 0"/>
<objColor r="255" g="102" 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="m_{OP'}"/>
<labelOffset x="-56" y="-26"/>
</element>
<expression label="text8_{1}" exp=""\sin(-φ)""/>
<element type="text" label="text8_{1}">
<show object="false" label="false" ev="37"/>
<condition showObject="z(P) < 0°"/>
<objColor r="255" g="102" 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="m_{PP''}"/>
<labelOffset x="8" y="-18"/>
</element>
<expression label="text9_{1}" exp=""\cos(-φ)\sinλ""/>
<element type="text" label="text9_{1}">
<show object="false" label="false" ev="37"/>
<condition showObject="z(P) < 0° ∧ y(P) > 0°"/>
<objColor r="0" g="100" 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="m_{P'P''}"/>
<labelOffset x="-63" y="5"/>
</element>
<expression label="text9_{2}" exp=""\cosφ\sin(-λ)""/>
<element type="text" label="text9_{2}">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) > 0° ∧ y(P) < 0°"/>
<objColor r="0" g="100" 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="m_{P'P''}"/>
<labelOffset x="-80" y="-2"/>
</element>
<expression label="text9_{3}" exp=""\cos(-φ)\sin(-λ)""/>
<element type="text" label="text9_{3}">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) < 0° ∧ y(P) < 0°"/>
<objColor r="0" g="100" 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="m_{P'P''}"/>
<labelOffset x="-55" y="7"/>
</element>
<expression label="text10_{1}" exp=""\cos(-φ)\cosλ""/>
<element type="text" label="text10_{1}">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) < 0° ∧ y(P) > 0°"/>
<objColor r="0" g="100" 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="m_{OP''}"/>
<labelOffset x="-77" y="-20"/>
</element>
<expression label="text10_{2}" exp=""\cosφ\cos(-λ)""/>
<element type="text" label="text10_{2}">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) > 0° ∧ y(P) < 0°"/>
<objColor r="0" g="100" 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="m_{OP''}"/>
<labelOffset x="-6" y="-3"/>
</element>
<expression label="text10_{3}" exp=""\cos(-φ)\cos(-λ)""/>
<element type="text" label="text10_{3}">
<show object="true" label="false" ev="37"/>
<condition showObject="z(P) < 0° ∧ y(P) < 0°"/>
<objColor r="0" g="100" 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="m_{OP''}"/>
<labelOffset x="-2" y="-3"/>
</element>
<command name="Midpoint">
<input a0="O" a1="P''"/>
<output a0="m_{OP''}"/>
</command>
<element type="point3d" label="m_{OP''}">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.16260358327338906" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<expression label="text11" exp=""P""/>
<element type="text" label="text11">
<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="2" size="12" style="0"/>
<startPoint exp="P"/>
<labelOffset x="12" y="-41"/>
</element>
<expression label="text11_{1}" exp=""P'""/>
<element type="text" label="text11_{1}">
<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="2" size="12" style="0"/>
<startPoint exp="P'"/>
<labelOffset x="-48" y="6"/>
</element>
<expression label="text11_{2}" exp=""P''""/>
<element type="text" label="text11_{2}">
<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="2" size="12" style="0"/>
<startPoint exp="P''"/>
<labelOffset x="-60" y="-4"/>
</element>
<command name="Midpoint">
<input a0="P" a1="P'"/>
<output a0="m_{PP''}"/>
</command>
<element type="point3d" label="m_{PP''}">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.32520716654677817" y="0.6751176126129694" z="0.3310802425267857" w="1"/>
<pointSize val="3"/>
</element>
<command name="Midpoint">
<input a0="O" a1="P'"/>
<output a0="m_{OP'}"/>
</command>
<element type="point3d" label="m_{OP'}">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.16260358327338906" y="0.3375588063064847" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Midpoint">
<input a0="P'" a1="P''"/>
<output a0="m_{P'P''}"/>
</command>
<element type="point3d" label="m_{P'P''}">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.3252071665467781" y="0.3375588063064847" z="0" w="1"/>
<pointSize val="3"/>
</element>
</construction>
</geogebra>