Bezier Surface
2017-02-27 15:25
This GeoGebra example demonstrates how to create three-dimensional graphics using Bezier surfaces. By using a set of control points, Bezier surfaces can generate smooth and continuous surfaces. In this example, commands such as Sum, Surface, BezierSurface are used to construct the surface, and commands like Sequence, Zip, Net3D are used to generate a three-dimensional grid. This example is suitabl
In collections
Interactive 3D Surfaces in GeoGebra
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.332.0" id="aaa33da9-0062-49dd-9ea0-0cc547fc1171" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="950" height="750" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.2505263157894737" orientation="1" />
</panes>
<views>
<view id="1" visible="false" inframe="false" stylebar="true" location="3,1" size="515" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="true" stylebar="true" location="1,1" size="291" window="934,562,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 | 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1" size="375" window="457,176,700,550" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="238" window="100,100,250,400" />
<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="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="false" stylebar="true" location="1" size="900" 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 , 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="948" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 | 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 | 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="948" height="748"/>
<coordSystem xZero="209.00000000000003" yZero="400" scale="50.00000000000001" yscale="49.999999999999964"/>
<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>
<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="-0.2291222213624815" yZero="0.07922545646593868" zZero="-3.735531444456539" scale="30.811614116376763" xAngle="23" zAngle="1049"/>
<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="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="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="-5.985525214657774" y="-10.205031584643983" z="3.227640786954958" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="B">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="-6.926895465307741" y="-6.777747697372908" z="-4.440892098500626e-16" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="C">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="-5.627859836681675" y="-2.293403007563397" z="5.161200455680718" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="D">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="-1.3104345066691472" y="-10.68742964994433" z="0" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="E">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-51" y="56"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="-0.7640462058165729" y="-6.982186810562658" z="-3.1415150041213713" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="F">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="-0.7488092613961973" y="-2.4746346303235542" z="0" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="G">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="2.704129325043379" y="-10.416571798768759" z="5.2769470056001" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="H">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="4.129584613280219" y="-6.838860018314685" z="0" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="I">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="3.1422614934117377" y="-3.3151785431233964" z="5.402426723296202" w="1"/>
<pointSize val="5"/>
</element>
<expression label ="mat" exp="{{(0, 0, 0), (0, 1, 1), (0, 2, 0), (0, 3, 1)}, {(1, 0, 0), (1, 1, 1), (1, 2, 0), (1, 3, 1)}, {(2, 0, 0), (2, 1, 1), (2, 2, 0), (2, 3, 1)}}"/>
<element type="list" label="mat">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<expression label="m" exp="Dimension[mat] - 1" />
<element type="numeric" label="m">
<value val="2"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="n" exp="Dimension[Element[mat, 1]] - 1" />
<element type="numeric" label="n">
<value val="3"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="mat2" exp="{{A, B, C}, {D, E, F}, {G, H, I}}" />
<element type="list" label="mat2">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sum">
<input a0="Sum[Sequence[Sequence[((((((x(Element[mat, j + 1, k + 1]) * Binomial[m, j]) * Binomial[n, k]) * (1 - x)^(m - j)) * x^(j)) * (1 - y)^(n - k)) * y^(k)), k, 0, n], j, 0, m]]"/>
<output a0="bx"/>
</command>
<element type="functionNVar" label="bx">
<show object="false" label="false" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Sum">
<input a0="Sum[Sequence[Sequence[((((((y(Element[mat, j + 1, k + 1]) * Binomial[m, j]) * Binomial[n, k]) * (1 - x)^(m - j)) * x^(j)) * (1 - y)^(n - k)) * y^(k)), k, 0, n], j, 0, m]]"/>
<output a0="by"/>
</command>
<element type="functionNVar" label="by">
<show object="false" label="false" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Sum">
<input a0="Sum[Sequence[Sequence[((((((z(Element[mat, j + 1, k + 1]) * Binomial[m, j]) * Binomial[n, k]) * (1 - x)^(m - j)) * x^(j)) * (1 - y)^(n - k)) * y^(k)), k, 0, n], j, 0, m]]"/>
<output a0="bz"/>
</command>
<element type="functionNVar" label="bz">
<show object="false" label="false" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="bx(s, t)" a1="by(s, t)" a2="bz(s, t)" a3="s" a4="0" a5="1" a6="t" a7="0" a8="1"/>
<output a0="bf"/>
</command>
<element type="surfacecartesian3d" label="bf">
<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>
<command name="BezierSurface">
<input a0="mat2"/>
<output a0="b"/>
</command>
<element type="surfacecartesian3d" label="b">
<show object="true" label="false" ev="5"/>
<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 ="pts" exp="{(1, 2, 3), (1, 1, 1), (0, 0, 1)}"/>
<element type="list" label="pts">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Segment[Element[pts, k], Element[pts, k + 1]]" a1="k" a2="1" a3="Length[pts] - 1"/>
<output a0="list2"/>
</command>
<element type="list" label="list2">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="1" type="20" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Zip">
<input a0="Polyline3D[pts]" a1="pts" a2="mat"/>
<output a0="xlines"/>
</command>
<element type="list" label="xlines">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Zip">
<input a0="Polyline3D[pts]" a1="pts" a2="Transpose[mat]"/>
<output a0="ylines"/>
</command>
<element type="list" label="ylines">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="Net3D">
<input a0="mat2"/>
<output a0="list1" a1="list3"/>
</command>
<element type="list" label="list1">
<show object="true" label="false" ev="5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="20" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="list" label="list3">
<show object="true" label="false" ev="5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="20" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label ="xy" exp="z = 0" type="plane"/>
<element type="plane3d" label="xy">
<show object="true" label="false" ev="5"/>
<objColor r="204" g="255" b="102" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0" y="0" z="1" w="0"/>
<fading val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<element type="point3d" label="A1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-4.670562961042096" y="-0.7858084778374979" z="2.775610887548994" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="A2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-1.1899330733823048" y="0.31512463840343424" z="0" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="A3">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="1.2995689717482215" y="-0.02253922896845638" z="0" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="A4">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="4.2899018254023344" y="-0.05662284588143446" z="5.480463231201134" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="B1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-4.755131915063512" y="2.6188951854486637" z="0.6631115510068171" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="B2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-1.1633072596036862" y="2.1699403040915715" z="6.044623364159645" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="B3">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="1.4865281294911166" y="2.117442885224826" z="-0.416277874825274" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="B4">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="5.041813456524569" y="1.9188719774504244" z="0" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="C1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-4.2465166889311305" y="4.506865494563587" z="4.376582160156259" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="C2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-0.7232610981136616" y="5.161877413970279" z="1.0535277756511743" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="C3">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="1.4618908422041703" y="6.511008272069638" z="0" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="C4">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="7.617445917314504" y="5.522548372276772" z="4.796636873538649" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="D1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="79" y="-14"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-3.535229666961601" y="6.705315805524151" z="12.404463780151008" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="D2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="-2.5817194531098595" y="10.66207926922481" z="2.9912338808638386" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="D3">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="2.536837103698606" y="10.971020846741087" z="1.5105819820318853" w="1"/>
<pointSize val="5"/>
</element>
<element type="point3d" label="D4">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<auxiliary val="true"/>
<coords x="6.083137873380479" y="11.82443505447446" z="7.046519048242325" w="1"/>
<pointSize val="5"/>
</element>
<expression label="mat3" exp="{{A1, A2, A3, A4}, {B1, B2, B3, B4}, {C1, C2, C3, C4}, {D1, D2, D3, D4}}" />
<element type="list" label="mat3">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="-1"/>
</element>
<command name="BezierSurface">
<input a0="mat3"/>
<output a0="a"/>
</command>
<element type="surfacecartesian3d" label="a">
<show object="true" 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>
<command name="Net3D">
<input a0="mat3"/>
<output a0="list4" a1="list5"/>
</command>
<element type="list" label="list4">
<show object="true" label="false" ev="5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="20" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="list" label="list5">
<show object="true" label="false" ev="5"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="20" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
</construction>
</geogebra>