]> A Practical Guide to Tensegrity Design: 6.2.3 Dome Step 3: Adjust the base points

6.2.3 Dome Step 3: Adjust the base points

Table 6.20 shows the unadjusted coordinate values for the base points. The mathematical programming problem for the dome treats these three points as fixed. As mentioned in Section 6.1, according to some definitions of tensegrity, once these points are fixed the structure is no longer a tensegrity since it is no longer self-supporting.5

Practically speaking, this seems a useful approach to developing a dome. So, the self-support requirement some definitions make for a true tensegrity will be ignored and the constraints to fix the base points will be included with the point constraints discussed in Chapter 3. Certainly it would be possible to develop a dome which met the self-support requirement. Another truncation technique would need to be developed, but the truncation would probably be a little more ragged looking and the structural support from the fixed base points would probably be missed. The resulting dome would be more mobile though.

To facilitate construction and perhaps make the structure more aesthetically pleasing, the base points for the not-quite-a-tensegrity dome being designed here are adjusted to lie evenly spaced on a circle about the symmetry axis of the dome. This section gives the details of how that adjustment is made.

 Point
Coordinates
 $x$ $y$ $z$
 Sphere Point Transform Number
 $P22$ $P23$ $P24$
 -3.78438 0.063992 0.205336 -3.12849 -1.75383 0.751127 -1.01487 -3.51733 0.871082
 $P2$ 9 $P6$ 9 $P3$ 11

Table 6.20: 6ν T-Octahedron Dome: Base Point Initial Raw Coordinate Values

The symmetry axis is the line through the origin and the point $1.01.01.0$. It is convenient to normalize the corresponding vector so it has length 1.0, so the vector $13 13 13$ is used whenever the symmetry axis is needed for computations and is called $A$.

The adjusted points lie on a circle chosen so that the points are moved as little as possible because of the adjustment. The radius of the circle is the average distance of the unadjusted points from the symmetry axis.6 This value is called $ravg$. In addition, the adjusted points are selected so that they all have the same value when projected onto the symmetry axis, and it will be the average of the values for the three unadjusted points. This common value is called $havg$. The projection is computed by taking the dot product of the point with $A$. The component of the point orthogonal to the axis is called $Pi⊥$ and is computed using the formula $Pi⊥ =Pi- Pi⋅A ⁢A$. This data is summarized in Table 6.21.

 Point
$P⊥$ Coordinates
 $x$ $y$ $z$
$r$   $h$
 $P22$ $P23$ $P24$ Average
 -2.612699 1.235678 1.377022 -1.751428 -0.376766 0.751127 0.205503 -2.296956 2.091453 N/A N/A N/A
 3.20145 2.78184 3.11326 3.03219
 -2.02942 -2.38515 -2.11375 -2.1761

Table 6.21: 6ν T-Octahedron Dome: Raw Base Point Characteristics

The adjusted value of $P22$, call it $P22*$, is generated using the formula $P22* =havg ⁢A+ ravg P22⊥ ⁢ P22⊥$. The adjusted values for the other two points are generated by by rotating $P22*$ about the symmetry axis by $2⁢π 9$ and $4⁢π 9$. Nine is chosen as the divisor for the two rotation angles since there are nine base points when all symmetry transformations are taken into account. The general matrix for rotating a point about a normalized (so it has length one) vector $xyz$ by an angle $θ$ is:7

$x2+ 1-x2 ⁢cosθ x⁢y⁢ 1-cosθ -z⁢sinθ x⁢z⁢ 1-cosθ +y⁢sinθ x⁢y⁢ 1-cosθ +z⁢sinθ y2+ 1-y2 ⁢cosθ y⁢z⁢ 1-cosθ -x⁢sinθ x⁢z⁢ 1-cosθ -y⁢sinθ y⁢z⁢ 1-cosθ +x⁢sinθ z2+ 1-z2 ⁢cosθ$

In the present situation, the normalized vector in question is just $A$ and the value of $θ$ is $2⁢π 9$. Substituting these values yields the matrix:

$13+ 23⁢cos 2⁢π9 1-cos 2⁢π9 3- sin 2⁢π9 3 1-cos 2⁢π9 3+ sin 2⁢π9 3 1-cos 2⁢π9 3+ sin 2⁢π9 3 13+ 23⁢cos 2⁢π9 1-cos 2⁢π9 3- sin 2⁢π9 3 1-cos 2⁢π9 3- sin 2⁢π9 3 1-cos 2⁢π9 3+ sin 2⁢π9 3 13+ 23⁢cos 2⁢π9$

Applying this matrix once to $P22*$ yields $P23*$. Applying this matrix twice to $P22*$ yields $P24*$. The adjusted values are what appear in Table 6.17.

5 See Wang98. Wang goes beyond the definition by Pugh quoted in Chapter 1 to identify the following characteristics of a tensegrity structure:

1. It is composed of compression and tension elements.
2. The struts (compression elements) are discontinuous while the cables (tension elements) are continuous.
3. The structure is rigidified by self-stressing.
4. The structure is self-supporting.
Sometimes the second item is modified to allow struts which are attached to each other by pin joints. None of the examples discussed in this book are of that sort. Certainly all the techniques described here would apply to such tensegrities, but simpler procedures might apply in these cases and thus obviate the need for solving a mathematical programming problem. Kenner76, p. 6, uses the term "self-sufficient" to describe the quality of tensegrity structures characterized by the fourth item.

6 See Section 8.2.1 for the formula for calculating the distance of a point from a line.

7 From Rogers76, Chapter 3.