Sections of Spheres
2014-11-03 17:11
This GeoGebra applet demonstrates various cross-sections of a sphere. Through interactive 3D geometric construction, users can observe different shapes formed when a plane intersects a sphere, such as circles and ellipses. Utilizing commands like Sphere, Plane, and IntersectPath, this example helps learners intuitively understand the properties and variations of spherical sections in solid geometr
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.279.0" id="423c38d8-d0b9-4820-b2dc-63939aa1ad03" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="922" height="635" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.6062906724511931" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="180,143,700,550" />
<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="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="1" visible="true" inframe="false" stylebar="false" location="1" size="355" window="100,100,600,400" />
<view id="2" visible="false" inframe="true" stylebar="true" location="3" size="250" window="335,0,249,703" />
<view id="16" visible="false" inframe="false" stylebar="false" location="2" size="116" 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="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="true" stylebar="false" location="1" size="480" 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="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 | 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="false" location="3" size="559" 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="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="355" height="635"/>
<coordSystem xZero="171.47177417389105" yZero="723.8679101240506" scale="25.657905911535316" yscale="25.657905911535348"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="1" 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"/>
<grid distX="1" distY="1" distTheta="0.5235987755982988"/>
</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"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0" yZero="0" zZero="-1.5" scale="50" xAngle="16" zAngle="-60"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" 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="true" show="false" size="2"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<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="0.1" speed="1" type="1" playing="false"/>
<coords x="2" y="18" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="B + (-2, -2)" a1="B + (2, -2)" a2="B + (2, 2)" a3="B + (-2, 2)"/>
<output a0="poly3" a1="e" a2="f" a3="g" a4="i"/>
</command>
<element type="polygon" label="poly3">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="false"/>
<coords x="0" y="4" z="-64"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-4" y="0" z="16"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="-4" z="80"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="4" y="0" z="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="PointIn">
<input a0="poly3"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="3" y="-2"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="2" y="18" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
<caption val="移動截面"/>
</element>
<command name="Translate">
<input a0="(0, 0, 0)" a1="Vector[B, C]"/>
<output a0="A"/>
</command>
<element type="point3d" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<element type="numeric" label="h">
<value val="1.45"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-4" max="6" absoluteScreenLocation="true" width="150" x="45" y="208" fixed="true" horizontal="false" showAlgebra="false"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.05" speed="1" type="0" playing="false"/>
</element>
<command name="Polygon">
<input a0="A + (-4, -4)" a1="A + (4, -4)" a2="A + (4, 4)" a3="A + (-4, 4)"/>
<output a0="poly1" a1="a" a2="b" a3="c" a4="d"/>
</command>
<element type="polygon3d" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment3d" label="a">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="b">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="c">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="d">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="numeric" label="s">
<value val="0"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-52" y="27"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="100" x="168" y="99" fixed="false" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="t">
<value val="0"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="100" x="170" y="149" fixed="false" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<command name="Rotate">
<input a0="Rotate[Translate[poly1, Vector[A, A + (0, 0, h)]], (90° * s), A + (0, 0, h), yAxis]" a1="(90° * t)" a2="A" a3="zAxis"/>
<output a0="poly2"/>
</command>
<element type="polygon3d" label="poly2">
<lineStyle thickness="1" type="0" typeHidden="2"/>
<show object="true" label="false" ev="4"/>
<objColor r="99" g="219" b="219" alpha="0.15" dynamicr="0.38824" dynamicg="0.85882" dynamicb="0.85882" dynamica="o2" colorSpace="0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<expression label="text1" exp=""→""/>
<element type="text" label="text1">
<show object="false" label="false" ev="4"/>
<condition showObject="x(C) < x(B) + 2"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<startPoint exp="C"/>
<labelOffset x="8" y="7"/>
</element>
<expression label="text2" exp=""←""/>
<element type="text" label="text2">
<show object="false" label="false" ev="4"/>
<condition showObject="x(C) > x(B) - 2"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<startPoint exp="C"/>
<labelOffset x="-27" y="7"/>
</element>
<expression label="text3" exp=""↑""/>
<element type="text" label="text3">
<show object="false" label="false" ev="4"/>
<condition showObject="y(C) < y(B) + 2"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<startPoint exp="C"/>
<labelOffset x="-9" y="-11"/>
</element>
<expression label="text4" exp=""↓""/>
<element type="text" label="text4">
<show object="false" label="false" ev="4"/>
<condition showObject="y(C) > y(B) - 2"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<startPoint exp="C"/>
<labelOffset x="-9" y="25"/>
</element>
<expression label="text5" exp=""截面高度""/>
<element type="text" label="text5">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="6" y="39"/>
</element>
<expression label="text6" exp=""水平""/>
<element type="text" label="text6">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="111" y="105"/>
</element>
<expression label="text7" exp=""鉛垂""/>
<element type="text" label="text7">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="277" y="104"/>
</element>
<expression label="text8" exp=""旋轉截面 1""/>
<element type="text" label="text8">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="178" y="73"/>
</element>
<expression label="text9" exp=""旋轉截面 2""/>
<element type="text" label="text9">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="178" y="180"/>
</element>
<element type="point3d" 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="0.1" speed="1" type="0" playing="false"/>
<coords x="-2" y="0" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="Rotate">
<input a0="D" a1="120°" a2="(0, 0, 0)"/>
<output a0="E"/>
</command>
<element type="point3d" label="E">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.9999999999999996" y="-1.7320508075688774" z="0" w="1"/>
<pointSize val="3"/>
</element>
<command name="PointIn">
<input a0="poly2"/>
<output a0="G"/>
</command>
<element type="point3d" label="G">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-2.12936353016407" y="-1.1412336435859558" z="1.45" w="1"/>
<pointSize val="3"/>
</element>
<command name="PointIn">
<input a0="poly2"/>
<output a0="H"/>
</command>
<element type="point3d" label="H">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="-2.0681958328316843" y="1.9691673239635037" z="1.45" w="1"/>
<pointSize val="3"/>
</element>
<command name="PointIn">
<input a0="poly2"/>
<output a0="I"/>
</command>
<element type="point3d" label="I">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-1.0961314359440615" y="-1.8956721123706046" z="1.45" w="1"/>
<pointSize val="3"/>
</element>
<command name="Plane">
<input a0="G" a1="I" a2="H"/>
<output a0="n"/>
</command>
<element type="plane3d" label="n">
<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="3.259913369479597" w="-4.726874385745416"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<element type="point3d" label="K">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="-2" y="8.326672684688674e-17" z="1.65" w="1"/>
<pointSize val="3"/>
</element>
<element type="point3d" label="L_1">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="0.9999999999999996" y="-1.7320508075688774" z="1.65" w="1"/>
<pointSize val="3"/>
</element>
<element type="point3d" label="O">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="0.9999999999999996" y="1.7320508075688767" z="1.65" w="1"/>
<pointSize val="3"/>
</element>
<element type="point3d" label="P">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="NaN" y="NaN" z="NaN" w="NaN"/>
<pointSize val="3"/>
</element>
<element type="point3d" label="J">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="NaN" y="NaN" z="NaN" w="NaN"/>
<pointSize val="3"/>
</element>
<element type="point3d" label="L">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="NaN" y="NaN" z="NaN" w="NaN"/>
<pointSize val="3"/>
</element>
<element type="segment3d" label="p">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="q">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="r">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="a_1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="b_1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment3d" label="c_1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="numeric" label="o1">
<value val="0.5"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="150" x="83" y="401" fixed="true" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<expression label="text10" exp=""立體圖形""/>
<element type="text" label="text10">
<show object="true" label="false" ev="8"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="125" y="432"/>
</element>
<expression label="text11" exp=""透明""/>
<element type="text" label="text11">
<show object="true" label="false" ev="8"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="34" y="409"/>
</element>
<expression label="text12" exp=""不透明""/>
<element type="text" label="text12">
<show object="true" label="false" ev="8"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="249" y="409"/>
</element>
<element type="numeric" label="o2">
<value val="0.15"/>
<show object="true" label="false"/>
<objColor r="0" g="153" b="204" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-80" y="2"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="150" x="83" y="360" fixed="true" horizontal="true" showAlgebra="false"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<expression label="text13" exp=""平面""/>
<element type="text" label="text13">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="144" y="345"/>
</element>
<expression label="text14" exp=""透明""/>
<element type="text" label="text14">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="34" y="363"/>
</element>
<expression label="text15" exp=""不透明""/>
<element type="text" label="text15">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="153" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="248" y="365"/>
</element>
<command name="OrthogonalLine">
<input a0="A" a1="n"/>
<output a0="d_1"/>
</command>
<element type="line3d" label="d_1">
<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" oy="0" oz="0" ow="1" vx="0" vy="0" vz="3.259913369479597" vw="0"/>
<eqnStyle style="parametric" parameter="λ"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="105" y="452"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="SetViewDirection[d_1]"/>
<font serif="false" sizeM="2" size="20" style="0"/>
<caption val="從截面觀察"/>
</element>
<command name="Sphere">
<input a0="(0, 0, 2)" a1="2"/>
<output a0="j"/>
</command>
<element type="quadric" label="j">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="215" b="0" alpha="0.5" dynamicr="1" dynamicg="0.84314" dynamicb="0" dynamica="o1" colorSpace="0"/>
<layer val="0"/>
<labelMode val="0"/>
<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="0" A4="0" A5="0" A6="0" A7="0" A8="0" A9="-2"/>
</element>
<command name="IntersectPath">
<input a0="n" a1="j"/>
<output a0="k"/>
</command>
<element type="conic3d" label="k">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1.9999999999999996" A1="1.9999999999999996" A2="4.278206545690533" A3="0" A4="-2.0934572616789415" A5="-4.354750255170183"/>
<eqnStyle style="implicit"/>
</element>
</construction>
</geogebra>