Circle on a Sphere
2014-11-18 05:35
This GeoGebra applet demonstrates the geometric construction and properties of a circle on a sphere. Using 3D graphing tools, it visualizes the intersection of a sphere and a plane to form a circle, along with related geometric elements such as the center, radius, and chord length. It is suitable for teaching and exploring solid geometry and spatial coordinate geometry, helping users understand th
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.24.0" id="f3b5093a-1079-4c80-b7db-de685dbd839f" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="944" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.3305084745762712" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="107,180,700,550" />
<view id="1" visible="false" inframe="false" stylebar="true" location="1,3" size="278" 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="1,1" size="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3,2" size="289" window="252,303,250,400" />
<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="16" visible="true" inframe="false" stylebar="true" location="3" size="312" window="92,243,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" 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 , 22 , 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="612" window="677,272,600,400" />
</views>
<toolbar show="true" 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="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="278" height="444"/>
<coordSystem xZero="92.66666666666667" yZero="277.5" scale="50.00000000000006" yscale="50.00000000000006"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="2" 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>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="312" height="444"/>
<coordSystem xZero="156.41649999999976" yZero="342.4354999999999" scale="56.37500000000006" yscale="56.37500000000003"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="2" 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="3"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="-0.08191347940713095" yZero="0.3497383425562194" zZero="-0.21730616594020424" scale="122.40650327501348" xAngle="28.0" zAngle="232.88823529411764"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" gridType="0"/>
<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="Circle on a Sphere" author="" date="">
<element type="numeric" label="k">
<value val="0.0"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200.0" x="46.0" y="41.0" fixed="true" horizontal="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.001" speed="1" type="0" playing="false"/>
</element>
<expression label="b" exp="z = k" type="plane" />
<element type="plane3d" label="b">
<show object="false" label="true"/>
<objColor r="99" g="219" b="219" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="1.0" w="0.0"/>
<fading val="0.1"/>
</element>
<element type="point3d" label="P">
<show object="true" label="true"/>
<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="1.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Sphere">
<input a0="(0, 0, 0)" a1="1"/>
<output a0="a"/>
</command>
<element type="quadric" label="a">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.17"/>
<layer val="0"/>
<labelMode val="0"/>
<matrix A0="1.0" A1="1.0" A2="1.0" A3="-1.0" A4="0.0" A5="0.0" A6="0.0" A7="-0.0" A8="-0.0" A9="-0.0"/>
</element>
<command name="IntersectPath">
<input a0="b" a1="a"/>
<output a0="c"/>
</command>
<element type="conic3d" label="c">
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.04"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" 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="-1.0" A3="0.0" A4="0.0" A5="0.0"/>
<eqnStyle style="implicit"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Circumference">
<input a0="c"/>
<output a0="d"/>
</command>
<element type="numeric" label="d">
<value val="6.283185307179586"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="A"/>
</command>
<element type="point3d" label="A">
<show object="true" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-0.9967518404706838" y="-0.08053426921692705" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Midpoint">
<input a0="A" a1="P"/>
<output a0="B"/>
</command>
<element type="point3d" label="B">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.4983759202353419" y="-0.040267134608463526" z="0.5" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Line">
<input a0="B" a1="(0, 0, 0)"/>
<output a0="h"/>
</command>
<element type="line3d" label="h">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coords ox="-0.4983759202353419" oy="-0.040267134608463526" oz="0.5" ow="1.0" vx="0.4983759202353419" vy="0.040267134608463526" vz="-0.5" vw="0.0"/>
</element>
<command name="Plane">
<input a0="P" a1="A" a2="(0, 0, 0)"/>
<output a0="e"/>
</command>
<element type="plane3d" label="e">
<show object="false" label="true"/>
<objColor r="99" g="219" b="219" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.08053426921692705" y="-0.9967518404706838" z="0.0" w="-0.0"/>
<fading val="0.1"/>
</element>
<command name="IntersectPath">
<input a0="e" a1="a"/>
<output a0="f"/>
</command>
<element type="conic3d" label="f">
<show object="false" label="true"/>
<objColor r="255" g="127" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" 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="1.9999999999999996" A1="1.9999999999999993" A2="0.0" A3="-1.1102230246251565E-16" A4="-1.414213562373095" A5="-1.4142135623730947"/>
<eqnStyle style="implicit"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Intersect">
<input a0="h" a1="f"/>
<output a0="C" a1="D"/>
</command>
<element type="point3d" label="C">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.7048099855569923" y="0.05694632788119214" z="-0.7071067811865477" w="1.0"/>
<pointSize val="3"/>
</element>
<element type="point3d" label="D">
<show object="false" label="false"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.7048099855569924" y="-0.05694632788119215" z="0.7071067811865475" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="CircumcircleArc">
<input a0="A" a1="D" a2="P"/>
<output a0="p"/>
</command>
<element type="conicpart" label="p">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" 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="0.0" A3="0.0" A4="-0.3826834323650897" A5="-0.9238795325112867"/>
<eqnStyle style="implicit"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Length">
<input a0="p"/>
<output a0="i"/>
</command>
<element type="numeric" label="i">
<value val="1.5707963267948974"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="text2" exp=""圓周 ＝" + d + """ />
<element type="text" label="text2">
<show object="true" label="false" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="100" y="139"/>
</element>
<command name="Segment">
<input a0="P" a1="A"/>
<output a0="g"/>
</command>
<element type="segment3d" label="g">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""半徑 AP = " + g + """ />
<element type="text" label="text1">
<show object="true" label="false" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="88" y="102"/>
</element>
<expression label="pp" exp="d / ((2 * i))" />
<element type="numeric" label="pp">
<value val="1.999999999999999"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="text3" exp=""圓周率 = " + pp + """ />
<element type="text" label="text3">
<show object="true" label="false" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<absoluteScreenLocation x="101" y="175"/>
</element>
</construction>
</geogebra>