Editing Appendix/Ramblings/VirtualReality (section)

==Ellipsoid==

Here, we have added another desirable element to the visual scene, specifically, one octant of the surface of an ellipsoid whose axis ratios are b/a = 0.75 and c/a = 0.50.  We generated the data needed to define the relevant vertices and polygons using a home-developed fortran algorithm.  This data was printed out from the fortran code in a format that created a new <geometry id="EllipsoidOctant"> subsection for the COLLADA-based xml file; see the Example #9 code presented immediately below.

<table border="0" align="center" width="80%">
<tr><th align="center" colspan="2">Example #9A &#8212; Building One Octant of an Ellipsoid</th></tr>
<tr><td align="left">
<div style="height: 400px; width: 700px; overflow: scroll;">
<pre>
<?xml version="1.0" encoding="utf-8"?>
<COLLADA xmlns="http://www.collada.org/2005/11/COLLADASchema" version="1.4.1">
<!--
On 23 September 2019 I copied the following cube-animation file from github:
https://github.com/chinedufn/collada-dae-parser/blob/master/test/fixture/animated-blender-cube.dae
I have made substantial modifications in order to better understand how the animation instructions
actually work.

THIS FILE NAME:  Dropbox/3Dviewers/Clock/Arrow07.PERFECT.dae
It uses "polylist" and correct implementation of "normals" for all polygons of the Clock.
It also displays **two** clocks:  [Initially, we made one spin and the other not.]
It also adds a red "minute hand" to the clock; now, this is the only segment that spins.
-->
  <asset>
    <contributor>
      <author>jet53man</author>
      <authoring_tool>Raw XML</authoring_tool>
    </contributor>
    <created>2019-09-24</created>
    <modified>2019-09-29</modified>
    <unit name="meter" meter="1"/>
    <up_axis>Z_UP</up_axis>
  </asset>
  <library_images/>
    <library_effects>
        <effect id="ID5">
            <profile_COMMON>
                <technique sid="COMMON">
                    <lambert>
                        <diffuse>
                            <color>0 0 1 1</color>
                        </diffuse>
                    </lambert>
                </technique>
            </profile_COMMON>
        </effect>
        <effect id="InnerID5">
            <profile_COMMON>
                <technique sid="COMMON">
                    <lambert>
                        <diffuse>
                            <color>0.4 0.0 0.7 0.4</color>
                        </diffuse>
                    </lambert>
                </technique>
            </profile_COMMON>
        </effect>
        <effect id="ArrowID5">
            <profile_COMMON>
                <technique sid="COMMON">
                    <lambert>
                        <diffuse>
                            <color>1.0 0.0 0.0 1.0</color>
                        </diffuse>
                    </lambert>
                </technique>
            </profile_COMMON>
        </effect>
        <effect id="EllipID5">
            <profile_COMMON>
                <technique sid="COMMON">
                    <lambert>
                        <diffuse>
                            <color>0.0 1.0 0.0 0.9</color>
                        </diffuse>
                    </lambert>
                </technique>
            </profile_COMMON>
        </effect>
    </library_effects>
    <library_materials>
        <material id="ID6" name="material">
            <instance_effect url="#ID5" />
        </material>
        <material id="InnerID6" name="Inner_material">
            <instance_effect url="#InnerID5" />
        </material>
        <material id="ArrowID6" name="Arrow_material">
            <instance_effect url="#ArrowID5" />
        </material>
        <material id="EllipID6" name="Ellip_material">
            <instance_effect url="#EllipID5" />
        </material>
    </library_materials>
    <library_nodes>
        <node id="ID3" name="cube_component">
            <instance_geometry url="#Clock_OuterEdge">
                <bind_material>
                    <technique_common>
                        <instance_material symbol="Material2" target="#ID6">
                            <bind_vertex_input semantic="UVSET0" input_semantic="TEXCOORD" input_set="0" />
                        </instance_material>
                    </technique_common>
                </bind_material>
            </instance_geometry>
            <instance_geometry url="#Clock_InnerEdge">
                <bind_material>
                    <technique_common>
                        <instance_material symbol="Material2" target="#InnerID6">
                            <bind_vertex_input semantic="UVSET0" input_semantic="TEXCOORD" input_set="0" />
                        </instance_material>
                    </technique_common>
                </bind_material>
            </instance_geometry>
        </node>
        <node id="ArrowID3" name="cube_component">
            <instance_geometry url="#Clock_ArrowEdge">
                <bind_material>
                    <technique_common>
                        <instance_material symbol="Material2" target="#ArrowID6">
                            <bind_vertex_input semantic="UVSET0" input_semantic="TEXCOORD" input_set="0" />
                        </instance_material>
                    </technique_common>
                </bind_material>
            </instance_geometry>
        </node>
        <node id="EllipID3" name="ellipsoid_component">
            <instance_geometry url="#EllipsoidOctant">
                <bind_material>
                    <technique_common>
                        <instance_material symbol="Material2" target="#EllipID6">
                            <bind_vertex_input semantic="UVSET0" input_semantic="TEXCOORD" input_set="0" />
                        </instance_material>
                    </technique_common>
                </bind_material>
            </instance_geometry>
        </node>
    </library_nodes>
    <library_geometries>
        <geometry id="Clock_OuterEdge">
            <mesh>
<!--
            **********************************************
            *
            *  Insert first portion of clock's structure
            *
            **********************************************
-->
                <source id="Clock_Positions">
                  <float_array id="Coordinates01" count="123"> 0.0  1.0  0.2 1.0  1.0  0.2 1.0  0.0  0.2 1.0 -1.0  0.2 0.0 -1.0  0.2 -1.0 -1.0  0.2 -1.0  0.0  0.2 -1.0  1.0  0.2 0.0  1.0 -0.2 1.0  1.0 -0.2 1.0  0.0 -0.2 1.0 -1.0 -0.2 0.0 -1.0 -0.2 -1.0 -1.0 -0.2 -1.0  0.0 -0.2 -1.0  1.0 -0.2 0.9  0.0  0.2 0.779423 0.45 0.2 0.45 0.779423 0.2 0.0  0.9  0.2 -0.45 0.779423 0.2 -0.779423 0.45 0.2 -0.9 0.0 0.2 -0.779423 -0.45 0.2 -0.45 -0.779423 0.2 0.0 -0.9 0.2 0.45 -0.779423 0.2 0.779423 -0.45 0.2 -0.9  0.0  -0.2 -0.779423 0.45 -0.2 -0.45 0.779423 -0.2 -0.0  0.9  -0.2 0.45 0.779423 -0.2 0.779423 0.45 -0.2 0.9 0.0 -0.2 0.779423 -0.45 -0.2 0.45 -0.779423 -0.2 -0.0 -0.9 -0.2 -0.45 -0.779423 -0.2 -0.779423 -0.45 -0.2  0.0 0.0 -0.2</float_array>
                  <technique_common>
                    <accessor source="#Coordinates01" count="41" stride="3">
                            <param name="X" type="float" />
                            <param name="Y" type="float" />
                            <param name="Z" type="float" />
                    </accessor>
                  </technique_common>
                </source>
                <source id="Clock_Normals">
                  <float_array id="InnerNormals01" count="18">
 1.0  0.0  0.0
-1.0  0.0  0.0
 0.0  1.0  0.0
 0.0 -1.0  0.0
 0.0  0.0  1.0
 0.0  0.0 -1.0
                  </float_array>
                    <technique_common>
                        <accessor source="#Normals01" count="6" stride="3">
                            <param name="X" type="float" />
                            <param name="Y" type="float" />
                            <param name="Z" type="float" />
                        </accessor>
                    </technique_common>
                </source>
                <vertices id="Clock_Vertices">
                    <input semantic="POSITION" source="#Clock_Positions" />
                </vertices>
                <polylist count="56" material="Material2">
                    <input semantic="VERTEX" source="#Clock_Vertices" offset="0" />
                    <input semantic="NORMAL" source="#Clock_Normals" offset="1" />
<vcount> 3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3</vcount>
<p> 0 2 1 2 8 2  1 2 9 2 8 2  7 2 0 2 15 2  0 2 8 2 15 2 1 0 2 0 9 0  2 0 10 0 9 0  2 0 3 0 10 0  3 0 11 0 10 0 3 3 4 3 11 3  4 3 12 3 11 3  4 3 5 3 12 3  5 3 13 3 12 3 5 1 6 1 13 1  6 1 14 1 13 1  6 1 7 1 14 1  7 1 15 1 14 1 2 4 17 4 16 4  2 4 1 4 17 4  1 4 18 4 17 4  1 4 0 4 18 4  0 4 19 4 18 4 0 4 20 4 19 4  0 4 7 4 20 4  7 4 21 4 20 4  7 4 6 4 21 4  6 4 22 4 21 4 6 4 23 4 22 4  6 4 5 4 23 4  5 4 24 4 23 4  5 4 4 4 24 4  4 4 25 4 24 4 4 4 26 4 25 4  4 4 3 4 26 4  3 4 27 4 26 4  3 4 2 4 27 4  2 4 16 4 27 4 14 5 29 5 28 5  14 5 15 5 29 5  15 5 30 5 29 5  15 5 8 5 30 5  8 5 31 5 30 5 8 5 32 5 31 5  8 5 9 5 32 5  9 5 33 5 32 5  9 5 10 5 33 5  10 5 34 5 33 5 10 5 35 5 34 5  10 5 11 5 35 5  11 5 36 5 35 5  11 5 12 5 36 5  12 5 37 5 36 5 12 5 38 5 37 5  12 5 13 5 38 5  13 5 39 5 38 5  13 5 14 5 39 5  14 5 28 5 39 5 </p>
                </polylist>
            </mesh>
        </geometry>
        <geometry id="Clock_InnerEdge">
            <mesh>
<!--
            **********************************************
            *
            *  Insert second portion of clock's structure
            *
            **********************************************
-->
                <source id="Clock_InnerPositions">
                  <float_array id="InnerCoordinates01" count="123">
0.0  1.0  0.2 1.0  1.0  0.2 1.0  0.0  0.2 1.0 -1.0  0.2 0.0 -1.0  0.2 -1.0 -1.0  0.2 -1.0  0.0  0.2 -1.0  1.0  0.2 0.0  1.0 -0.2 1.0  1.0 -0.2 1.0  0.0 -0.2 1.0 -1.0 -0.2 0.0 -1.0 -0.2 -1.0 -1.0 -0.2 -1.0  0.0 -0.2 -1.0  1.0 -0.2 0.9  0.0  0.2 0.779423 0.45 0.2 0.45 0.779423 0.2 0.0  0.9  0.2 -0.45 0.779423 0.2 -0.779423 0.45 0.2 -0.9 0.0 0.2 -0.779423 -0.45 0.2 -0.45 -0.779423 0.2 0.0 -0.9 0.2 0.45 -0.779423 0.2 0.779423 -0.45 0.2 -0.9  0.0  -0.2 -0.779423 0.45 -0.2 -0.45 0.779423 -0.2 -0.0  0.9  -0.2 0.45 0.779423 -0.2 0.779423 0.45 -0.2 0.9 0.0 -0.2 0.779423 -0.45 -0.2 0.45 -0.779423 -0.2 -0.0 -0.9 -0.2 -0.45 -0.779423 -0.2 -0.779423 -0.45 -0.2  0.0 0.0 -0.2</float_array>
                  <technique_common>
                    <accessor source="#InnerCoordinates01" count="41" stride="3">
                            <param name="X" type="float" />
                            <param name="Y" type="float" />
                            <param name="Z" type="float" />
                    </accessor>
                  </technique_common>
                </source>
                <source id="Clock_InnerNormals">
                  <float_array id="InnerNormals01" count="54">
 1.0  0.0  0.0
-1.0  0.0  0.0
 0.0  1.0  0.0
 0.0 -1.0  0.0
 0.0  0.0  1.0
 0.0  0.0 -1.0
 -0.965926 -0.258819 0.0
 -0.707107 -0.707107 0.0
 -0.258819 -0.965926 0.0
  0.258819 -0.965926 0.0
  0.707107 -0.707107 0.0
  0.965926 -0.258819 0.0
  0.965926  0.258819 0.0
  0.707107  0.707107 0.0
  0.258819  0.965926 0.0
 -0.258819  0.965926 0.0
 -0.707107  0.707107 0.0
 -0.965926  0.258819 0.0
                  </float_array>
                    <technique_common>
                        <accessor source="#InnerNormals01" count="18" stride="3">
                            <param name="X" type="float" />
                            <param name="Y" type="float" />
                            <param name="Z" type="float" />
                        </accessor>
                    </technique_common>
                </source>
                <vertices id="Clock_InnerVertices">
                    <input semantic="POSITION" source="#Clock_InnerPositions" />
                </vertices>
                <polylist count="24" material="Material2">
                    <input semantic="VERTEX" source="#Clock_InnerVertices" offset="0" />
                    <input semantic="NORMAL" source="#Clock_InnerNormals" offset="1"/>
<vcount>4 4 4 4  4 4 4 4  4 4 4 4  3 3 3 3  3 3 3 3  3 3 3 3</vcount>
<p> 16 6 17 6 33 6 34 6 17 7 18 7 32 7 33 7 18 8 19 8 31 8 32 8 19 9 20 9 30 9 31 9 20 10 21 10 29 10 30 10 21 11 22 11 28 11 29 11 22 12 23 12 39 12 28 12 23 13 24 13 38 13 39 13 24 14 25 14 37 14 38 14 25 15 26 15 36 15 37 15 26 16 27 16 35 16 36 16 27 17 16 17 34 17 35 17 28 5 29 5 40 5 29 5 30 5 40 5 30 5 31 5 40 5 31 5 32 5 40 5 32 5 33 5 40 5 33 5 34 5 40 5 34 5 35 5 40 5 35 5 36 5 40 5 36 5 37 5 40 5 37 5 38 5 40 5 38 5 39 5 40 5 39 5 28 5 40 5 </p>
              </polylist>
            </mesh>
        </geometry>
        <geometry id="Clock_ArrowEdge">
            <mesh>
<!--
            **********************************************
            *
            *  Insert arrow portion of clock's structure
            *
            **********************************************
-->
                <source id="Clock_ArrowPositions">
                  <float_array id="ArrowCoordinates01" count="30"> 0.0  0.85 0.25 0.1  0.0 0.25 0.0 -0.2 0.25 -0.1  0.0 0.25 0.0  0.85 0.05 0.1  0.0 0.05 0.0 -0.2 0.05 -0.1  0.0 0.05 0.0  0.0 0.25 0.0  0.0 0.05 </float_array>
                  <technique_common>
                    <accessor source="#ArrowCoordinates01" count="10" stride="3">
                            <param name="X" type="float" />
                            <param name="Y" type="float" />
                            <param name="Z" type="float" />
                    </accessor>
                  </technique_common>
                </source>
                <source id="Clock_ArrowNormals">
                  <float_array id="ArrowNormals01" count="18">
 0  0  1
 0  0 -1
 0.993151  0.116841 0.0
 0.447214 -0.894427 0.0
-0.447214 -0.894427 0.0
-0.993151  0.116841 0.0
                  </float_array>
                    <technique_common>
                        <accessor source="#ArrowNormals01" count="6" stride="3">
                            <param name="X" type="float" />
                            <param name="Y" type="float" />
                            <param name="Z" type="float" />
                        </accessor>
                    </technique_common>
                </source>
                <vertices id="Clock_ArrowVertices">
                    <input semantic="POSITION" source="#Clock_ArrowPositions" />
                </vertices>
                <polylist count="12" material="Material2">
                    <input semantic="VERTEX" source="#Clock_ArrowVertices" offset="0" />
                    <input semantic="NORMAL" source="#Clock_ArrowNormals" offset="1"/>
<vcount>3 3 3 3 3 3 3 3 4 4 4 4</vcount>
<p> 0 0 8 0 1 0 1 0 8 0 2 0 2 0 8 0 3 0 3 0 8 0 0 0 4 1 5 1 9 1 5 1 6 1 9 1 6 1 7 1 9 1 7 1 4 1 9 1 0 2 1 2 5 2 4 2 1 3 2 3 6 3 5 3 2 4 3 4 7 4 6 4 3 5 0 5 4 5 7 5 </p>
              </polylist>
            </mesh>
        </geometry>
<geometry id="EllipsoidOctant">
  <mesh>
<source id="Ellipsoid_Positions">
<float_array id="Ellipsoid_Coordinates" count="135">
    0.0000    0.0000    0.5000
    0.0000    0.7500    0.0000
    0.1000    0.7462    0.0000
    0.2000    0.7348    0.0000
    0.3000    0.7155    0.0000
    0.4000    0.6874    0.0000
    0.5000    0.6495    0.0000
    0.6000    0.6000    0.0000
    0.7000    0.5356    0.0000
    0.8000    0.4500    0.0000
    0.9000    0.3269    0.0000
    1.0000    0.0000    0.0000
    0.0000    0.7262    0.1250
    0.0968    0.7225    0.1250
    0.1936    0.7115    0.1250
    0.2905    0.6927    0.1250
    0.3873    0.6656    0.1250
    0.4841    0.6289    0.1250
    0.5809    0.5809    0.1250
    0.6778    0.5186    0.1250
    0.7746    0.4357    0.1250
    0.8714    0.3165    0.1250
    0.9682    0.0000    0.1250
    0.0000    0.6495    0.2500
    0.0866    0.6463    0.2500
    0.1732    0.6364    0.2500
    0.2598    0.6196    0.2500
    0.3464    0.5953    0.2500
    0.4330    0.5625    0.2500
    0.5196    0.5196    0.2500
    0.6062    0.4638    0.2500
    0.6928    0.3897    0.2500
    0.7794    0.2831    0.2500
    0.8660    0.0000    0.2500
    0.0000    0.4961    0.3750
    0.0661    0.4936    0.3750
    0.1323    0.4861    0.3750
    0.1984    0.4732    0.3750
    0.2646    0.4547    0.3750
    0.3307    0.4296    0.3750
    0.3969    0.3969    0.3750
    0.4630    0.3543    0.3750
    0.5292    0.2976    0.3750
    0.5953    0.2162    0.3750
    0.6614    0.0000    0.3750
</float_array>
<technique_common>
  <accessor source="#Ellipsoid_Coordinates" count="   45" stride="3">
        <param name="X" type="float" />
        <param name="Y" type="float" />
        <param name="Z" type="float" />
  </accessor>
</technique_common>
</source>
<source id="Ellipsoid_Normals">
<float_array id="Ellipsoid_Normals01" count="210">
    0.0092    0.2443    0.9696
    0.0276    0.2419    0.9699
    0.0460    0.2371    0.9704
    0.0644    0.2295    0.9712
    0.0829    0.2190    0.9722
    0.1015    0.2051    0.9735
    0.1203    0.1868    0.9750
    0.1393    0.1627    0.9768
    0.1587    0.1289    0.9789
    0.1854    0.0567    0.9810
    0.0237    0.6314    0.7751
    0.0237    0.6314    0.7751
    0.0713    0.6261    0.7764
    0.0713    0.6261    0.7764
    0.1193    0.6154    0.7792
    0.1193    0.6154    0.7792
    0.1680    0.5985    0.7833
    0.1680    0.5985    0.7833
    0.2176    0.5746    0.7890
    0.2176    0.5746    0.7890
    0.2685    0.5422    0.7962
    0.2685    0.5422    0.7962
    0.3211    0.4987    0.8051
    0.3211    0.4987    0.8051
    0.3760    0.4393    0.8159
    0.3760    0.4393    0.8159
    0.4344    0.3529    0.8287
    0.4344    0.3529    0.8287
    0.5149    0.1575    0.8427
    0.5149    0.1575    0.8427
    0.0320    0.8520    0.5226
    0.0320    0.8520    0.5226
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    0.0964    0.8461    0.5242
    0.1617    0.8340    0.5276
    0.1617    0.8340    0.5276
    0.2287    0.8148    0.5328
    0.2287    0.8148    0.5328
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    0.2981    0.7871    0.5400
    0.3708    0.7488    0.5493
    0.3708    0.7488    0.5493
    0.4481    0.6958    0.5613
    0.4481    0.6958    0.5613
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    0.5315    0.6209    0.5762
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    0.6239    0.5069    0.5947
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    0.7533    0.2304    0.6160
    0.0369    0.9817    0.1870
    0.0369    0.9817    0.1870
    0.1112    0.9759    0.1878
    0.1112    0.9759    0.1878
    0.1869    0.9639    0.1894
    0.1869    0.9639    0.1894
    0.2652    0.9449    0.1919
    0.2652    0.9449    0.1919
    0.3473    0.9172    0.1954
    0.3473    0.9172    0.1954
    0.4348    0.8780    0.2001
    0.4348    0.8780    0.2001
    0.5298    0.8227    0.2061
    0.5298    0.8227    0.2061
    0.6353    0.7421    0.2139
    0.6353    0.7421    0.2139
    0.7564    0.6146    0.2240
    0.7564    0.6146    0.2240
    0.9292    0.2842    0.2361
    0.9292    0.2842    0.2361
</float_array>
<technique_common>
  <accessor source="#Ellipsoid_Normals01" count="   70" stride="3">
        <param name="X" type="float" />
        <param name="Y" type="float" />
        <param name="Z" type="float" />
  </accessor>
</technique_common>
</source>
<vertices id="Ellipsoid_Vertices">
  <input semantic="POSITION" source="#Ellipsoid_Positions" />
</vertices>
<polylist count="   70" material="Material2">
  <input semantic="VERTEX" source="#Ellipsoid_Vertices" offset="0" />
  <input semantic="NORMAL" source="#Ellipsoid_Normals" offset="1" />
<vcount> 3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  3 3 3 3 3  </vcount>
<p>
    0    0   35    0   34    0
    0    1   36    1   35    1
    0    2   37    2   36    2
    0    3   38    3   37    3
    0    4   39    4   38    4
    0    5   40    5   39    5
    0    6   41    6   40    6
    0    7   42    7   41    7
    0    8   43    8   42    8
    0    9   44    9   43    9
   34   10   24   10   23   10
   34   11   35   11   24   11
   35   12   25   12   24   12
   35   13   36   13   25   13
   36   14   26   14   25   14
   36   15   37   15   26   15
   37   16   27   16   26   16
   37   17   38   17   27   17
   38   18   28   18   27   18
   38   19   39   19   28   19
   39   20   29   20   28   20
   39   21   40   21   29   21
   40   22   30   22   29   22
   40   23   41   23   30   23
   41   24   31   24   30   24
   41   25   42   25   31   25
   42   26   32   26   31   26
   42   27   43   27   32   27
   43   28   33   28   32   28
   43   29   44   29   33   29
   23   30   13   30   12   30
   23   31   24   31   13   31
   24   32   14   32   13   32
   24   33   25   33   14   33
   25   34   15   34   14   34
   25   35   26   35   15   35
   26   36   16   36   15   36
   26   37   27   37   16   37
   27   38   17   38   16   38
   27   39   28   39   17   39
   28   40   18   40   17   40
   28   41   29   41   18   41
   29   42   19   42   18   42
   29   43   30   43   19   43
   30   44   20   44   19   44
   30   45   31   45   20   45
   31   46   21   46   20   46
   31   47   32   47   21   47
   32   48   22   48   21   48
   32   49   33   49   22   49
   12   50    2   50    1   50
   12   51   13   51    2   51
   13   52    3   52    2   52
   13   53   14   53    3   53
   14   54    4   54    3   54
   14   55   15   55    4   55
   15   56    5   56    4   56
   15   57   16   57    5   57
   16   58    6   58    5   58
   16   59   17   59    6   59
   17   60    7   60    6   60
   17   61   18   61    7   61
   18   62    8   62    7   62
   18   63   19   63    8   63
   19   64    9   64    8   64
   19   65   20   65    9   65
   20   66   10   66    9   66
   20   67   21   67   10   67
   21   68   11   68   10   68
   21   69   22   69   11   69
</p>
</polylist>
  </mesh>
</geometry>
    </library_geometries>
  <library_animations>
    <animation id="Cube_rotation_euler_Z">
      <source id="Cube_rotation_euler_Z-input">
<!--        <float_array id="Cube_rotation_euler_Z-input-array" count="5">0 0.2083333 0.4166666 0.625 0.8333333</float_array> -->
<!--        <float_array id="Cube_rotation_euler_Z-input-array" count="5">0 0.4167 0.8333 1.25 1.666</float_array> -->
        <float_array id="Cube_rotation_euler_Z-input-array" count="17">0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.25 2.5 2.75 3.0 3.25 3.5 3.75 4.0</float_array>
        <technique_common>
          <accessor source="#Cube_rotation_euler_Z-input-array" count="5" stride="1">
            <param name="TIME" type="float"/>
          </accessor>
        </technique_common>
      </source>
      <source id="Cube_rotation_euler_Z-output">
<!--        <float_array id="Cube_rotation_euler_Z-output-array" count="5">0 -22.5 -45 -67.5 -90</float_array> -->
        <float_array id="Cube_rotation_euler_Z-output-array" count="17">0 -22.5 -45 -67.5 -90 -112.5 -135.0 -157.5 -180.0 -202.5 -225.0 -247.5 -270.0 -292.5 -315.0 -337.5 -360.0</float_array>
        <technique_common>
          <accessor source="#Cube_rotation_euler_Z-output-array" count="5" stride="1">
            <param name="ANGLE" type="float"/>
          </accessor>
        </technique_common>
      </source>
      <sampler id="Cube_rotation_euler_Z-sampler">
        <input semantic="INPUT" source="#Cube_rotation_euler_Z-input"/>
        <input semantic="OUTPUT" source="#Cube_rotation_euler_Z-output"/>
      </sampler>
      <channel source="#Cube_rotation_euler_Z-sampler" target="jetman/rotationZ.ANGLE"/>
    </animation>
    <animation id="Cube_scale_Z">
      <source id="Cube_scale_Z-input">
<!--        <float_array id="Cube_scale_Z-input-array" count="5">0 0.2083333 0.4166666 0.625 0.8333333</float_array>  -->
        <float_array id="Cube_scale_Z-input-array" count="5">0 0.0694 0.1389 0.2083 0.2778</float_array>
        <technique_common>
          <accessor source="#Cube_scale_Z-input-array" count="5" stride="1">
            <param name="TIME" type="float"/>
          </accessor>
        </technique_common>
      </source>
      <source id="Cube_scale_Z-output">
<!--    <float_array id="Cube_scale_Z-output-array" count="5">0.5 0.4 0.3 0.2 0.1</float_array>  -->
        <float_array id="Cube_scale_Z-output-array" count="5">1.0 1.0 1.0 1.0 1.0</float_array>
        <technique_common>
          <accessor source="#Cube_scale_Z-output-array" count="5" stride="1">
            <param name="Z" type="float"/>
          </accessor>
        </technique_common>
      </source>
      <sampler id="Cube_scale_Z-sampler">
        <input semantic="INPUT" source="#Cube_scale_Z-input"/>
        <input semantic="OUTPUT" source="#Cube_scale_Z-output"/>
      </sampler>
      <channel source="#Cube_scale_Z-sampler" target="jetman/scale.Z"/>
    </animation>
  </library_animations>
    <library_controllers/>
    <library_visual_scenes>
        <visual_scene id="EntireLayout">
<!--
***********
*
* Evidently, it is important for the "id" of this node to be shared
* with the "directory" name that is used in the animation "channel"
* instructions.  For example, "jetman" is used here.
*
***********
-->
          <node id="TwoClocks">
            <node id="firstclock" name="instance_1">
              <translate>0.0 1.0 0.0</translate>
              <rotate sid="rotationX">1 0 0 90</rotate>
              <rotate sid="rotationZ">0 0 1 0</rotate>
<!--
              <rotate sid="rotationY">0 1 0 90</rotate>
-->
              <scale sid="scale">1 1 1</scale>
              <instance_node url="#ID3" />
            </node>
            <node id="jetman" name="instance_Arrow">
              <translate>0.0 1.0 0.0</translate>
              <rotate sid="rotationX">1 0 0 90</rotate>
              <rotate sid="rotationZ">0 0 1 0</rotate>
<!--
              <rotate sid="rotationY">0 1 0 90</rotate>
-->
              <scale sid="scale">1 1 1</scale>
              <instance_node url="#ArrowID3" />
            </node>
            <extra>
              <technique profile="FCOLLADA">
                <start_time>0</start_time>
                <end_time>4.0</end_time>
              </technique>
            </extra>
            <node id="secondClock" name="instance_2">
              <translate>10.0 0.0 0.0</translate>
<!--
              <rotate sid="rotationZ">0 0 1 90</rotate>
-->
              <rotate sid="rotationX">1 0 0 90</rotate>
              <rotate sid="rotationY">0 1 0 -90</rotate>
              <scale sid="scale">1 1 1</scale>
              <instance_node url="#ID3" />
            </node>
            <node id="EllipScene" name="instance_Ellip">
              <translate>10.0 0.0 0.0</translate>
<!--
              <rotate sid="rotationZ">0 0 1 90</rotate>
-->
              <rotate sid="rotationX">1 0 0 90</rotate>
              <rotate sid="rotationY">0 1 0 -90</rotate>
              <scale sid="scale">5.0 5.0 5.0</scale>
              <instance_node url="#EllipID3" />
            </node>
          </node>
        </visual_scene>
    </library_visual_scenes>
    <scene>
        <instance_visual_scene url="#EntireLayout" />
    </scene>
</COLLADA>
</pre>
</div>
</td>
<td align="center" bgcolor="lightgrey">[[File:EllipsoidOctantMerged54.png|250px|EllipsoidOctantMerged54]]</td>
</tr></table>

Here we document the various lines of the COLLADA-based <xml> code that require a unique new "id" tag when a new geometry node is added:
[[File:FlowChartEllipsoidOctant2.png|right|Flow Chart]]
<ul>
  <li><b><library_effects></b></li>
  <ul>
    <li><effect id="<font color="orange">EllipID5</font>"></li>
  </ul>
  <li><b><library_materials></b></li>
  <ul>
    <li><material id="<font color="lightgreen">EllipID6</font>" name="Ellip_material"></li>
    <li><instance_effect url="#<font color="orange">EllipID5</font>" /></li>
  </ul>
  <li><b><library_nodes></b></li>
  <ul>
    <li><node id="<font color="lightblue">EllipID3</font>" name="ellipsoid_component"></li>
    <li><instance_geometry url="#<font color="pink">EllipsoidOctant</font>"></li>
    <li><instance_material symbol="Material2" target="#<font color="lightgreen">EllipID6</font>"></li>
  </ul>
  <li><b><library_geometries></b> &#8212; this group provided by fortran output</li>
  <ul>
    <li><geometry id="<font color="pink">EllipsoidOctant</font>"></li>
    <li><source id="Ellipsoid_Positions"></li>
    <li><float_array id="Ellipsoid_Coordinates" count="135"></li>
    <li><accessor source="#Ellipsoid_Coordinates" count="   45" stride="3"></li>
    <li><source id="Ellipsoid_Normals"></li>
    <li><float_array id="Ellipsoid_Normals01" count="210"></li>
    <li> <accessor source="#Ellipsoid_Normals01" count="   70" stride="3"></li>
    <li><vertices id="Ellipsoid_Vertices"></li>
    <li><input semantic="POSITION" source="#Ellipsoid_Positions" /></li>
    <li><polylist count="   70" material="Material2"></li>
    <li><input semantic="VERTEX" source="#Ellipsoid_Vertices" offset="0" /></li>
    <li><input semantic="NORMAL" source="#Ellipsoid_Normals" offset="1" /></li>
  </ul>
  <li><b><library_visual_scenes></b></li>
  <ul>
    <li><node id="EllipScene" name="instance_Ellip"></li>
    <li><instance_node url="#<font color="lightblue">EllipID3</font>" /></li>
  </ul>
</ul>


<table border="0" align="center" width="90%">
<tr><th align="center" colspan="2">Example #9B &#8212; FORTRAN code that Generated Vertices &amp; Polygons</th></tr>
<tr><td align="left">
<div style="height: 400px; width: 100%; overflow: scroll;">
<pre>
      Program ThreeDEllipsoids
!!!!!!!!
!
!  Surface of one Octant of the Ellipsoid
!
!!!!!!!!
!          
      real*8 bovera,covera
      real*8 zlayer(5),xvalue(4,11),yvalue
      real*8 pt3xx(200),pt3yy(200),pt3zz(200)
      real*8 xmax,dx
      real*8 x1,y1,z1,x2,y2,z2,x3,y3,z3,xnorm,ynorm,znorm
      integer n,nzmax,m,mxmax,ncount,npt
      integer ell,nverts,numtri,nt1,nt2,nfloats
      integer ellmin,ellmax,nzslice
      integer triangle(200,4)
  501 format(8x,'pt num',6x,'x',10x,'y',10x,'z',/)
  502 format(I10,3F15.7)
  503 format(//,5x,'nverts = ',I5,/)
  504 format(/,6x,'Triangle No.',3x,'Vertex1',3x,'Vertex2',3x,'Vertex3')
  505 format(I10,5x,3I10,5x,3f13.5)
  507 format(//,5x,'numtri = ',I5,/)
  510 format(/,'nzslice, numtri, ellmin, ellmax = ',4I8)
  599 format('<geometry id="EllipsoidOctant">')
  600 format(2x,'<mesh>')
  601 format('<source id="Ellipsoid_Positions">',/,'<float_array id="Ellipsoid_Coordinates" count="',I3,'">'/)
  602 format(3F10.4)
  603 format('</float_array>')
  604 format('<technique_common>')
  605 format(2x,'<accessor source="#Ellipsoid_Coordinates" count="',I5,&
     &'" stride="3">')
  606 format(8x,'<param name="X" type="float" />')
  607 format(8x,'<param name="Y" type="float" />')
  608 format(8x,'<param name="Z" type="float" />')
  609 format(2x,'</accessor>',/,'</technique_common>',/,'</source>')
  610 format('<source id="Ellipsoid_Normals">',/,'<float_array id="Ellipsoid_Normals01" count="',I3,'">'/)
  611 format(2x,'<accessor source="#Ellipsoid_Normals01" count="',I5,&
     &'" stride="3">')
  612 format('<vertices id="Ellipsoid_Vertices">')
  613 format(2x,'<input semantic="POSITION"',&
     &' source="#Ellipsoid_Positions" />')
  614 format('</vertices>')
  615 format('<polylist count="',I5,'" material="Material2">')
  616 format(2x,'<input semantic="VERTEX"',&
     &' source="#Ellipsoid_Vertices" offset="0" />')
  617 format(2x,'<input semantic="NORMAL"',&
     &' source="#Ellipsoid_Normals" offset="1" />',/,'<vcount></vcount>',/,'<p>')
  618 format('</p>',/,'</polylist>')
  619 format(2x,'</mesh>')
  620 format('</geometry>')
  621 format(6I5)
! zlayer is z/a; its vector length is (nzmax+1). 
! xvalue is x/a; its array lengths are (nzmax,mxmax+1).
! yvalue is y/a.
      bovera = 0.75d0
      covera = 0.50d0
      nzmax  = 4
      mxmax  = 10
      zlayer(nzmax+1) = covera
      pt3xx(1) = 0.0d0
      pt3yy(1) = 0.0d0
      pt3zz(1) = zlayer(nzmax+1)
      ncount  = 1
      do n = 1,nzmax
        zlayer(n) = covera*dfloat((n-1))/dfloat(nzmax)
        xmax = dsqrt(1.0d0 - (zlayer(n)/covera)**2)
        dx   = xmax/dfloat(mxmax)
        do m = 1,mxmax+1
          xvalue(n,m) = dfloat(m-1)*dx
            ncount = ncount+1            
            if(m.eq.(mxmax+1))then
            yvalue = 0.0d0
            else
            yvalue = bovera*dsqrt(1.0d0 - (zlayer(n)/covera)**2 - xvalue(n,m)**2)
            endif
            pt3xx(ncount) = xvalue(n,m)
            pt3yy(ncount) = yvalue
            pt3zz(ncount) = zlayer(n)
        enddo
      enddo
      nverts = ncount
      nfloats = 3*nverts
!     write(*,503)nverts
      write(*,599)
      write(*,600)
      write(*,601)nfloats
!     write(*,501)
!     do n=1,ncount
!     npt = n - 1
!     write(*,502)npt,pt3xx(n),pt3yy(n),pt3zz(n)
!     enddo
      do n=1,ncount
      write(*,602)pt3xx(n),pt3yy(n),pt3zz(n)
      enddo
      write(*,603)
      write(*,604)
      write(*,605)nverts
!!!
!!!   Finished determining coordinates of all [nzmax x (mxmax+1) + 1] vertices.
!!!
!!!   Now identify triangles
!     write(*,504)
      nt1 = 0
      nt2 = nverts - mxmax
      numtri = 0
      do ell=1,mxmax
      triangle(ell,1) = nt1
      triangle(ell,2) = nt2-1
      triangle(ell,3) = nt2
      triangle(ell,4) = ell-1
      nt2 = nt2 + 1
!     write(*,505)triangle(numtri+1,4),triangle(numtri+1,1),&
!    &triangle(numtri+1,2),triangle(numtri+1,3)
      numtri = numtri+1
      enddo
        numtri = numtri+1
      do nzslice=1,nzmax-1
!!!!!
!
!   Phase 2
!
!!!!!
!     nzslice=1
        ellmin = nverts - nzslice*(mxmax+1)
        ellmax = ellmin + mxmax - 1
!       numtri = numtri+1
!       write(*,510)nzslice,numtri,ellmin,ellmax
        do ell=ellmin,ellmax
        triangle(numtri,1) = ell
        triangle(numtri,2) = ell-(mxmax+1)
        triangle(numtri,3) = ell - mxmax
        triangle(numtri,4) = numtri-1
          numtri = numtri + 1
        triangle(numtri,1) = ell 
        triangle(numtri,2) = ell - mxmax
        triangle(numtri,3) = ell + 1
        triangle(numtri,4) = numtri-1
          numtri = numtri + 1
        enddo
      enddo
      numtri = numtri-1
!!!!!
!
!     write(*,507)numtri
!
!!!!!
      nfloats=3*numtri
      write(*,606)
      write(*,607)
      write(*,608)
      write(*,609)
      write(*,610)nfloats
      do nt1=1,numtri
      n1st = triangle(nt1,1)
      n2nd = triangle(nt1,2)
      n3rd = triangle(nt1,3)
      x1 = pt3xx(n1st+1)
      y1 = pt3yy(n1st+1)
      z1 = pt3zz(n1st+1)
      x2 = pt3xx(n2nd+1)
      y2 = pt3yy(n2nd+1)
      z2 = pt3zz(n2nd+1)
      x3 = pt3xx(n3rd+1)
      y3 = pt3yy(n3rd+1)
      z3 = pt3zz(n3rd+1)
      call Normals(x1,y1,z1,x2,y2,z2,x3,y3,z3,xnorm,ynorm,znorm)
!     write(*,505)triangle(nt1,4),triangle(nt1,1),&
!    &triangle(nt1,2),triangle(nt1,3),xnorm,ynorm,znorm
      write(*,602)xnorm,ynorm,znorm
      enddo
      write(*,603)
      write(*,604)
      write(*,611)numtri
      write(*,606)
      write(*,607)
      write(*,608)
      write(*,609)
      write(*,612)
      write(*,613)
      write(*,614)
      write(*,615)numtri
      write(*,616)
      write(*,617)
!
      do nt1=1,numtri
        write(*,621)triangle(nt1,1),triangle(nt1,4),&
     &              triangle(nt1,3),triangle(nt1,4),&
     &              triangle(nt1,2),triangle(nt1,4)
      enddo
!
      write(*,618)
      write(*,619)
      write(*,620)
      stop
      end program ThreeDEllipsoids
      Subroutine Normals(x1,y1,z1,x2,y2,z2,x3,y3,z3,xnorm,ynorm,znorm)
      real*8 x1,y1,z1,x2,y2,z2,x3,y3,z3,xnorm,ynorm,znorm
      real*8 xlength,ylength,zlength,length
      xlength = (y2-y1)*(z3-z1) - (y3-y1)*(z2-z1)
      ylength = (z2-z1)*(x3-x1) - (z3-z1)*(x2-x1)
      zlength = (x2-x1)*(y3-y1) - (x3-x1)*(y2-y1)
      length  = dsqrt(xlength**2 + ylength**2 + zlength**2)
      xnorm   = -xlength/length
      ynorm   = -ylength/length
      znorm   = -zlength/length
      return
      end
</pre>
</div>
</td>
</tr></table>
In the fortran code, "Subroutine Normals" is called upon to determine the normal (unit vector) to the plane that is uniquely defined by each individual triangle.  It accomplishes this as follows.  The vectors that extend from the coordinate origin to, respectively, the 1<sup>st</sup>, 2<sup>nd</sup> and 3<sup>rd</sup> vertex of the relevant triangle are defined by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\vec{x}}_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat\imath x_1 + \hat\jmath y_1 + \hat{k} z_1 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~{\vec{x}}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat\imath x_2 + \hat\jmath y_2 + \hat{k} z_2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~{\vec{x}}_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat\imath x_3 + \hat\jmath y_3 + \hat{k} z_3 \, .</math>
  </td>
</tr>
</table>
The vector that points from the first vertex &#8212; associated with coordinate point <math>~(x_1, y_1, z_1)</math> &#8212; to the second vertex  &#8212; associated with coordinate point <math>~(x_2, y_2, z_2)</math> &#8212; is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\vec{x}}_{12}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~{\vec{x}}_2 - {\vec{x}}_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="center">
<math>~\hat\imath(x_2 - x_1) + \hat\jmath(y_2 - y_1) + \hat{k}(z_2 - z_1) \, ;</math>
  </td>
</tr>
</table>
this vector aligns with one of the three edges of the triangle.  Likewise, the vector that points from the first vertex to the third vertex and therefore aligns with one of the other two edges of the triangle is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\vec{x}}_{13}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~{\vec{x}}_3 - {\vec{x}}_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="center">
<math>~\hat\imath(x_3 - x_1) + \hat\jmath(y_3 - y_1) + \hat{k}(z_3 - z_1) \, .</math>
  </td>
</tr>
</table>
A vector, <math>~\vec{h}_\perp</math>, that is normal to the face of this specific triangle will be given by the cross product of these two vectors, that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec{h}_\perp</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~{\vec{x}}_{12} \times {\vec{x}}_{13}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath[(y_2-y_1)(z_3-z_1) - (y_3 - y_1)(z_2-z_1)]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \hat\jmath[(z_2-z_1)(x_3-x_1) - (z_3 - z_1)(x_2-x_1)]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \hat{k} [(x_2-x_1)(y_3-y_1) - (x_3 - x_1)(y_2-y_1)] \, .
</math>
  </td>
</tr>
</table>
Hence, the unit vector that is normal to the face of this triangle is,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{n}_\perp</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm ~\frac{\vec{h}_\perp}{| \vec{h}_\perp |} \, .</math>
  </td>
</tr>
</table>
(Whether the inferior or superior sign is chosen will depend on the circumstances.  Usually the sign is chosen such that the normal vector points ''away from'' the face of the triangle that is part of the outward-facing surface and therefore will be illuminated.)  For completeness, we note that the vector magnitude that serves to normalize the right-hand side of this equation is defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~| \vec{h}_\perp |</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{
[(y_2-y_1)(z_3-z_1) - (y_3 - y_1)(z_2-z_1)]^2 + [(z_2-z_1)(x_3-x_1) - (z_3 - z_1)(x_2-x_1)]^2 + [(x_2-x_1)(y_3-y_1) - (x_3 - x_1)(y_2-y_1)]^2 } \, .
</math>
  </td>
</tr>
</table>