gifIsolate Vertex

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Creating Isolate Vertex Sets

  1. Select a Primal or Dual Isolate tab on the Toolbar.
  2. Set the Isolate Mode by clicking the Vertex button.
  3. Click on a vertex in the manifold. The Isolate Vertex/Hub Log Options window appears:
  4. png

  5. Select the logging options and click “OK”.
  6. Unique Vertex hubs are printed in the Log along with their respective properties.
  7. The Isolate Vertex pop-up menu activates with a list of isolate sets. When you select an item from the Isolate menu, only labels for the selected set appear in the view, whilst other labels are hidden.

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Set Criteria

The Isolate Vertex tool treats vertices in the quad-edge manifold as pseudo-hubs and calculates:

  1. The angles between the edges leaving the vertex.
  2. The angles between the edges and the hub normal.
  3. The angles between the unfolded edges on the hub plane.

The above criteria are used to determine whether two hubs are considered equal; these criteria may be changed using modifier keys and the “Flip Symmetric Hubs Option”.

Modifer Keys

  1. No modifer key: Primal or Dual edges are scanned.
  2. Control key: only the spaceframe edges are scanned.
  3. Shift key: all edges including the spaceframe are scanned.

Basic Logging

Basic Logging creates simple set lists, just like Isolate Edges or Isolate Faces. For example, a 4V Icosahedron Class I, Method 1, truncated at the equator:

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UNIQUE HUBS.
  Flip Symmetric: OFF.
  Print Hub Connections: OFF.
  Scale Factor = 10.000000000
         Units = Feet and Inches

TsQEIcosahedron 4V, Class I, Method 1
  Primal Lattice
   Total  vertices = 92
   Unique hubs = 9
   Hub     Quantity
    V1      [6]      
    V2      [5]      
    V3      [30]      
    V4      [5]      
    V5      [10]      
    V6      [5]      
    V7      [25]      
    V8      [5]      
    V9      [1] 

Use “Flip Symmetric” when comparing Hubs option

This option is only available when using Basic Logging. “Flip Symmetric” modifies the criteria used to determine hub equality...

Flip Symmetric OFF

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UNIQUE HUBS.
  Flip symmetric: OFF.
  Print hub connections: OFF.
  Scale Factor = 10.000000000
         Units = Feet and Inches

TsQEIcosahedron 7V, Class I, Method 1
  Primal Lattice
   Total  vertices = 492
   Unique hubs = 9
   Hub     Quantity
    V1      [12]      
    V2      [60]      
    V3      [60]      
    V4      [60]      
    V5      [60]      
    V6      [60]      
    V7      [60]      
    V8      [60]      
    V9      [60]  

Consider the hubs V4 on the left and right of the PPT (circled blue). The hubs are identical, sharing three unique angles, a, b, c:

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The hubs labelled V4 are reflections of each other. But what is more important is that the hub on the left can be superimposed on its right counterpart by successive rotations. (Imagine picking the left hub up with its attached struts, laying it over the right hub, then rotating until it aligns with the one below). In this case, both the angles between edges, and the angles between the edges hub normal match.

Now consider Hubs V6 and V8 (circled black). These hubs are also reflections of each other and have six unique angles. However, they are not considered equal because they cannot be superimposed by successive rotations. The hubs can only be superimposed by “flipping”…

Flip Symmetric ON

When Flip Symmetric is on, hubs that can only be superimposed by flipping are considered equal. In this case, V8 has become V6. Consequently, the number of V6 hubs has doubled and the number of unique hubs has decreased from 9 to 8. When flipped, the vertex angles between V6 & V8 match; the angles between the two hub normals also match but the struts of the flipped hub are now pointing in the opposite direction - ie outwards.

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UNIQUE HUBS.
  Flip symmetric: ON.
  Print hub connections: OFF.
  Scale Factor = 10.000000000
         Units = Feet and Inches

TsQEIcosahedron 7V, Class I, Method 1
  Primal Lattice
   Total  vertices = 492
   Unique hubs = 8
   Hub     Quantity
    V1      [12]      
    V2      [60]      
    V3      [60]      
    V4      [60]      
    V5      [60]      
    V6      [120]      
    V7      [60]      
    V8      [60] 

The Flip Symmetric option was implemented because some geodesic hub designs allow a hub to flip at the required joint in the dome. Eg hubs that are cut from sheet metal as "star" shapes with flanges that attach to wooden stuts. Such a hub need only be creased and fixed in the opposite sense:

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Connectivity Logging

Connectivity Logging lists all edges leaving a hub, groups them by their UN_ID, and gives their connections to adjoining hubs. In addition, any hub variants may be analysed and printed in the log. Connectivity logs can be very long because they contain all the required connection data to build a dome.

IMPORTANT: Connectivity Logging requires that the Isolate sets for Edges are generated first, otherwise the UN_IDs for edges will be missing in the log.

Edges are listed in counter-clockwise order around the vertex. The starting edge is arbitrary, so the following edge groups are identical:

            Edge Group 1             Edge Group 2
             E5 (to V6)                E5 (to V7)  <--
             E7 (to V9)                E4 (to V2)
             E5 (to V8)                E5 (to V6)
             E5 (to V7)  <--           E7 (to V9)
             E4 (to V2)                E5 (to V8)

To enable Connectivity Logging for spaceframes, you must first analyse Primal and Dual hubs without the Shift modifier keys. This ensures all hubs are labelled, so that when you analyse again with the Shift modifier keys, there will be no '?' in place of hub labels.

For example, to get a full connective log for a dome that uses Primal, Dual and Spaceframe edges, use the following sequence (assuming that ‘Generate Dual’ and ‘Generate Spaceframe’ are ON).

  1. Ensure that “Connectivity Logging” is ON.
  2. Do 'ISOLATE EDGES' for the Primal, Dual and Spaceframe edges respectively.
  3. Do 'ISOLATE VERTEX' for the Primal vertices.
  4. Do 'ISOLATE VERTEX' for the Dual vertices.
  5. Do 'ISOLATE VERTEX' again for the Primal vertices, but this time, hold down the SHIFT modifier key to include the Spaceframe connections.
  6. Do 'ISOLATE VERTEX' again for the Dual verticies, but this time, hold down the SHIFT modifier key to include the Spaceframe connections.

NOTE. There is no point in holding down the SHIFT modifier key at stage 3 because the Dual hubs have not been processed yet.

After performing the above sequence, Geodesica has the required information to print out the edge connections for all hubs. This might seem a little tedious, but just think of the data it provides at your disposal: with the connectivity log alone, you have all the required connectivity information to build the frame of a very complex dome.

Print Hub Variants

Whilst all the hubs in a set may be considered equal in terms of their edge angles, the identity of the connecting hubs will sometimes vary according to the dome in question; and a truncated dome will also have minor variations around the truncation plane. I call such hubs “Variants” When “Print Hub Variants” is checked, the Log lists the number of variants for each hub, and prints the connections for each variant. These lists can be very long. The following shows the Log for a 4V, Class I, Method 1, Icosahedron, truncated at the equator. The log has been edited for so that only the connectivity and variants for V7 are shown. The explanation follows after...

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UNIQUE HUBS.
  Flip Symmetric: OFF.
  Print Hub Connections: ON.
  Scale Factor = 10.000000000
         Units = Feet and Inches

TsQEIcosahedron 4V, Class I, Method 1
  Primal Lattice
   Total  vertices = 92
   Unique hubs = 9
   
   (NOTE: the log has been cut for brevity to start at V7...)

    V7      [25]      Edge Group 1 for V7 at Vertex UN_ID: 4947
                       E6 (to V7)
                       E6 (to V8)
                       E5 (to V4)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 2 for V7 at Vertex UN_ID: 4953
                       E2 (to V3)
                       E5 (to V2)
                       E6 (to V8)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

                      Edge Group 3 for V7 at Vertex UN_ID: 5691
                       E6 (to V7)
                       E6 (to V8)
                       E5 (to V4)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 4 for V7 at Vertex UN_ID: 5697
                       E2 (to V3)
                       E5 (to V2)
                       E6 (to V8)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

                      Edge Group 5 for V7 at Vertex UN_ID: 5877
                       E6 (to V7)
                       E6 (to V8)
                       E5 (to V4)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 6 for V7 at Vertex UN_ID: 5883
                       E2 (to V3)
                       E5 (to V2)
                       E6 (to V8)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

                      Edge Group 7 for V7 at Vertex UN_ID: 6063
                       E6 (to V7)
                       E6 (to V8)
                       E5 (to V4)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 8 for V7 at Vertex UN_ID: 6069
                       E2 (to V3)
                       E5 (to V2)
                       E6 (to V8)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

                      Edge Group 9 for V7 at Vertex UN_ID: 6156
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 10 for V7 at Vertex UN_ID: 6162
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

                      Edge Group 11 for V7 at Vertex UN_ID: 6173
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)

                      Edge Group 12 for V7 at Vertex UN_ID: 6249
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 13 for V7 at Vertex UN_ID: 6255
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

                      Edge Group 14 for V7 at Vertex UN_ID: 6266
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)

                      Edge Group 15 for V7 at Vertex UN_ID: 6342
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 16 for V7 at Vertex UN_ID: 6348
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

                      Edge Group 17 for V7 at Vertex UN_ID: 6359
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)

                      Edge Group 18 for V7 at Vertex UN_ID: 6435
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 19 for V7 at Vertex UN_ID: 6441
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

                      Edge Group 20 for V7 at Vertex UN_ID: 6452
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)

                      Edge Group 21 for V7 at Vertex UN_ID: 6528
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 22 for V7 at Vertex UN_ID: 6534
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

                      Edge Group 23 for V7 at Vertex UN_ID: 6545
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)

                      Edge Group 24 for V7 at Vertex UN_ID: 6621
                       E6 (to V7)
                       E6 (to V8)
                       E5 (to V4)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)

                      Edge Group 25 for V7 at Vertex UN_ID: 6627
                       E2 (to V3)
                       E5 (to V2)
                       E6 (to V8)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)

         TOTAL VARIANTS for V7 = 3
           ----------------------
           Variant ID 1 for V7
           Quantity: 5
           Vertex UN_IDs: 4947
                          5691
                          5877
                          6063
                          6621
           Edge Group
                       E6 (to V7)
                       E6 (to V8)
                       E5 (to V4)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)
           ----------------------
           Variant ID 2 for V7
           Quantity: 5
           Vertex UN_IDs: 4953
                          5697
                          5883
                          6069
                          6627
           Edge Group
                       E2 (to V3)
                       E5 (to V2)
                       E6 (to V8)
                       E6 (to V7)
                       E5 (to V5)
                       E2 (to V3)
           ----------------------
           Variant ID 3 for V7
           Quantity: 15
           Vertex UN_IDs: 6156
                          6162
                          6173
                          6249
                          6255
                          6266
                          6342
                          6348
                          6359
                          6435
                          6441
                          6452
                          6528
                          6534
                          6545
           Edge Group
                       E5 (to V5)
                       E2 (to V3)
                       E2 (to V3)
                       E5 (to V5)
                       E6 (to V7)
                       E6 (to V7)
           ----------------------

   (NOTE: the remainder of the log has been omitted for brevity).

There are 25 identical hubs for group V7. The total number of variants in this group is 3. There are 5 hubs of variant 1, 5 hubs of variant 2, and 15 hubs of variant 3. Note that each variant is listed with its vertex UN_ID’s, followed by the edge group. Every hub of a variant not only has the same edge group, but also the same connecting hubs. The total quantity of hubs for the variants (25) matches the total number of identical hubs in the group.