三角不等式演示：-3/2 ≤ cosα + cosβ + cosγ ≤ 3
2017-10-14 05:15
本GeoGebra范例通过几何构造与向量分析，直观演示三角函数不等式 -3/2 ≤ cosα + cosβ + cosγ ≤ 3 的成立条件与极值情况。利用点A、B、C、D及对应向量构建角度关系，结合滑块交互控制参数变化，动态展示余弦和的取值范围与边界条件。适用于三角函数不等式证明教学、数学竞赛培训与探究性学习。
所属专题
GeoGebra曲面互动教学资源集
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.389.0" app="classic" platform="d" id="6f938031-0497-473a-99c3-37f993aca550" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="919" height="731" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.5788900979325353" 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="500" 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="2" visible="false" inframe="true" stylebar="true" location="3" size="256" window="988,39,250,809" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="300" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="true" stylebar="true" location="1" size="367" window="862,311,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" 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="3" size="908" window="100,100,600,400" />
</views>
<toolbar show="true" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="811" height="596"/>
<coordSystem xZero="209.00000000000006" yZero="399.99999999999994" scale="50.000000000000014" yscale="49.99999999999996"/>
<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="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<spreadsheetView>
</spreadsheetView>
<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>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="-0.6942774580079478" yZero="-0.9753880553997574" zZero="-0.08290282225001888" scale="71.68043416804" yscale="74.46441338963609" zscale="53.52340730579783" xAngle="19.0" zAngle="-31.923456168174738"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0471975511965976" tickExpression="π / 3"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0471975511965976" tickExpression="π / 3"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0" tickExpression="1"/>
<plate show="false"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="false" show="false" size="1"/>
<projection type="1"/>
</euclidianView3D>
<construction title="" author="" date="">
<expression label ="dw" exp="1"/>
<element type="numeric" label="dw">
<value val="1.0"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<expression label ="f" exp="f(x, y) = cos(x) + cos(y) + cos((-x) - y)"/>
<element type="functionNVar" label="f">
<show object="false" label="false" ev="4"/>
<condition showObject="showGlobal"/>
<objColor r="102" g="204" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label ="inD" exp="inD(x, y) = (((x ≥ 0) * (y ≥ 0)) * (x + y ≤ π))"/>
<element type="functionNVar" label="inD">
<show object="false" label="false" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label ="a" exp="z = 0" type="plane"/>
<element type="plane3d" label="a">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="204" b="255" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="1.0" w="0.0"/>
<fading val="0.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label ="b" exp="z = -1" type="plane"/>
<element type="plane3d" label="b">
<show object="true" label="false" ev="4"/>
<condition showObject="showGlobal"/>
<objColor r="153" g="204" b="255" alpha="0.6499999761581421"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="1.0" w="1.0"/>
<fading val="0.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label ="inD2" exp="inD2(x, y) = (((x ≤ π) * (y ≤ π)) * (x + y ≥ π))"/>
<element type="functionNVar" label="inD2">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="85" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="f2" exp="f2(x, y) = f(x, y) / inD2(x, y)" />
<element type="functionNVar" label="f2">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="51" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label ="x0" exp="π / 3"/>
<element type="numeric" label="x0">
<value val="1.0471975511965976"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<expression label="z0" exp="f((2 * x0), (2 * x0))" />
<element type="numeric" label="z0">
<value val="-1.5"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<selectionAllowed val="false"/>
</element>
<expression label="A" exp="((2 * x0), (2 * x0), z0)" type="point" />
<element type="point3d" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="2.0943951023931953" y="2.0943951023931953" z="-1.5" w="1.0"/>
<pointSize val="7"/>
</element>
<expression label ="B" exp="(0, 0, 3)" type="point"/>
<element type="point3d" label="B">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="3.0" w="1.0"/>
<pointSize val="7"/>
</element>
<expression label ="C" exp="(π, 0, -1)" type="point"/>
<element type="point3d" label="C">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="3.141592653589793" y="0.0" z="-1.0" w="1.0"/>
<pointSize val="7"/>
</element>
<expression label ="D" exp="(0, π, -1)" type="point"/>
<element type="point3d" label="D">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="3.141592653589793" z="-1.0" w="1.0"/>
<pointSize val="7"/>
</element>
<expression label="text1" exp=""\left( \frac{2π}{3}, \frac{2π}{3}, -\frac{3}{2} \right) ""/>
<element type="text" label="text1">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="A2 + (0, 0, -0.3)"/>
<labelOffset x="-67" y="0"/>
</element>
<element type="boolean" label="showGlobal">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<checkbox fixed="true"/>
</element>
<expression label="btnShow" exp=""▷Show/Hide Global Function ""/>
<element type="text" label="btnShow">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="250" b="205" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="true" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="713" y="41"/>
<ggbscript val="SetValue(showGlobal, !showGlobal)"/>
</element>
<expression label ="x1" exp="π / 2"/>
<element type="numeric" label="x1">
<value val="1.5707963267948966"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<element type="numeric" label="lineWidth">
<value val="1.0"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<ggbscript onUpdate="SetLineThickness(f, lineWidth) SetLineThickness(f1, lineWidth) SetLineThickness(f2, lineWidth)"/>
</element>
<expression label="text2" exp=""▶ Show/Hide Grid Lines ""/>
<element type="text" label="text2">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="204" g="255" b="204" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.0" size="0" style="1"/>
<absoluteScreenLocation x="714" y="78"/>
<ggbscript val="SetValue(lineWidth, 1-lineWidth)"/>
</element>
<expression label="inPad" exp="inPad(x, y) = ((abs(x) ≤ dw) * (abs(y) ≤ dw))" />
<element type="functionNVar" label="inPad">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="fmax" exp="fmax(x, y) = 3 / inPad(x, y)" />
<element type="functionNVar" label="fmax">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="inPad2" exp="inPad2(x, y) = ((abs(x - (2 * x0)) ≤ dw) * (abs(y - (2 * x0)) ≤ dw))" />
<element type="functionNVar" label="inPad2">
<show object="false" label="false" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="fmin" exp="fmin(x, y) = -3 / 2 / inPad2(x, y)" />
<element type="functionNVar" label="fmin">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="153" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="PolyLine">
<input a0="((2 * x1), 0, -1)" a1="((2 * x1), (2 * x1), -1)" a2="(0, (2 * x1), -1)" a3="((2 * x1), 0, -1)"/>
<output a0="g"/>
</command>
<element type="polyline3d" label="g">
<lineStyle thickness="4" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="PolyLine">
<input a0="(π, 0, -1)" a1="(π, (-π), -1)" a2="(0, (-π), -1)" a3="((-π), 0, -1)" a4="((-π), π, -1)" a5="(0, π, -1)"/>
<output a0="h"/>
</command>
<element type="polyline3d" label="h">
<lineStyle thickness="4" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="showGlobal"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<expression label="f1" exp="f1(x, y) = (f(x, y) + 0.01) / inD(x, y)" />
<element type="functionNVar" label="f1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="text3" exp=""(0,0,3)""/>
<element type="text" label="text3">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="B2 + (0, 0, 0.7)"/>
<labelOffset x="10" y="-7"/>
</element>
<command name="Point">
<input a0="Ray[A, A + (0, 0, -1)]"/>
<output a0="A2"/>
</command>
<element type="point3d" label="A2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="2.0943951023931953" y="2.0943951023931953" z="-3.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Vector">
<input a0="A2" a1="A"/>
<output a0="vecA"/>
</command>
<element type="vector3d" label="vecA">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="1.5" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint exp="A2"/>
</element>
<command name="Point">
<input a0="Ray[D, D + (0, 0, 1)]"/>
<output a0="D2"/>
</command>
<element type="point3d" label="D2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="3.141592653589793" z="1.1429095629697348" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Vector">
<input a0="D2" a1="D"/>
<output a0="vecD"/>
</command>
<element type="vector3d" label="vecD">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="-2.142909562969735" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint exp="D2"/>
</element>
<expression label="text4" exp=""(0,π,-1)""/>
<element type="text" label="text4">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="D2"/>
<labelOffset x="-38" y="-43"/>
</element>
<command name="Point">
<input a0="Ray[C, C + (0, 0, 1)]"/>
<output a0="C2"/>
</command>
<element type="point3d" label="C2">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="3.141592653589793" y="0.0" z="1.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Vector">
<input a0="C2" a1="C"/>
<output a0="vecC"/>
</command>
<element type="vector3d" label="vecC">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="-2.0" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint exp="C2"/>
</element>
<expression label="text5" exp=""(π,0,-1)""/>
<element type="text" label="text5">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<startPoint exp="C2"/>
<labelOffset x="-39" y="-43"/>
</element>
<command name="Point">
<input a0="Ray[B, B + (1, 1, 2)]"/>
<output a0="B2"/>
</command>
<element type="point3d" label="B2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.5689779000186377" y="0.5689779000186377" z="4.137955800037275" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Vector">
<input a0="B2" a1="B"/>
<output a0="vecB"/>
</command>
<element type="vector3d" label="vecB">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="-0.5689779000186377" y="-0.5689779000186377" z="-1.1379558000372754" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint exp="B2"/>
</element>
<command name="PointIn">
<input a0="b"/>
<output a0="P"/>
</command>
<element type="point3d" label="P">
<show object="true" label="false" ev="4"/>
<condition showObject="showGlobal"/>
<objColor r="0" g="0" b="204" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="-2.3368637071388263" y="-2.458031289646729" z="-1.0" w="1.0"/>
<pointSize val="5"/>
</element>
<command name="Point">
<input a0="Ray[P, P + (0, 0, 1)]"/>
<output a0="Q"/>
</command>
<element type="point3d" label="Q">
<show object="true" label="false" ev="4"/>
<condition showObject="showGlobal"/>
<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"/>
<isShape val="false"/>
<coords x="-2.3368637071388263" y="-2.458031289646729" z="0.0" w="1.0"/>
<pointSize val="5"/>
</element>
<expression label="level" exp="z = z(Q)" type="plane" />
<element type="plane3d" label="level">
<show object="true" label="false" ev="4"/>
<condition showObject="showGlobal"/>
<objColor r="153" g="204" b="255" alpha="0.6000000238418579"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="1.0" w="-0.0"/>
<fading val="0.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="IntersectPath">
<input a0="level" a1="f"/>
<output a0="leveCurve"/>
</command>
<element type="implicitpoly" label="leveCurve">
<show object="true" label="true" ev="4"/>
<condition showObject="showGlobal"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="4"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="5" type="0" typeHidden="2"/>
<userinput show="false"/></element>
<command name="Segment">
<input a0="P" a1="Q"/>
<output a0="segLevel"/>
</command>
<element type="segment3d" label="segLevel">
<show object="true" label="false" ev="4"/>
<condition showObject="showGlobal"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text6" exp=""z=" + (LaTeX[z(Q)]) + """ />
<element type="text" label="text6">
<show object="true" label="false" ev="20"/>
<condition showObject="showGlobal"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="Q"/>
<labelOffset x="-28" y="-45"/>
</element>
</construction>
</geogebra>