## Spatial rings and domes

The conventional scissor-like elements can also be used to form spatial rings. The process is as follows. First, we build a closed chain consisting of

na straight scissor-like elements that are made of rods of equal length 2a with a pivot in the middle. A similar closed chain can be formed by nb elements with rod length of 2b with again a middle pivot. Both chains are symmetric about the middle plane on which all middle pivots of scissorlike elements lie. Obviously, both of the chains could become very flexible while na and nb are large because of the tolerance in each revolute. One way to stiffen the structure is to put one chain inside of the other and then connect them with a number of intermediate ties which are also conventional scissor-like elements to form an integrated larger assembly. The question that remains to be answered is what the conditions are for the assembly to retain mobility one. Name the first chain as the inner loop and the second the outer loop. For the inner loop, each of the scissor-like elements occupies a sector with a corresponding central angle

. (4.1)

Denote by θ the pivoting angle for elements in the inner loop and let the elements in the outer loop have the same deployment angle, see Figure 4.4(a) and (b). The projection lengths of the inner and outer elements, La and Lb, are

and , (4.2a, b)

respectively, whereas the heights of the elements, Ha and Hb, are

and . (4.3a, b)

To effectively connect the two loops with evenly distributed sets of elements, nb should be either na or 2na. Three different ways of arranging these elements have been found. Figure 4.5 shows the top projection view

of three rings, referred to as concepts A, B and C hereafter. The first two have nb = na and their top projections consist of na trapezia and 2na isosceles triangles, respectively, but for the third one, nb = 2na, and the projection has na trapezia and na isosceles triangles. To ensure the mobility of the rings, further geometrical conditions have to be met, which are discussed in next section.