Planar double chain linkages 39
D of a angulated scissor- like element, does not change when the angulated
beams AEC and BED are rotated relative to one another if two beams are
identical, i.e.
,
1
(3.2)
and
.
(3.3)
Let us draw two perpendiculars EG and EH on OD and OC, respectively.
Considering deltoid OGEH, we have
.
(3.4)
Because
substituting the above into Eq. (3.4) yields
.
(3.5)
As ∠AEC is predetermined, α is a constant regardless of the rotation
between angulated beams AEC and BED.
This particular geometrical feature of the angulated scissor- like elements
enables the construction of mobile plane loops. For a total of n angulated
scissor- like elements, each of which has a subtended central angle α i (i = 1,
2, . . ., n), to form a mobile double chain, there must be
.
(3.6)
The Hoberman sphere has a number of predominant planar double chains
composed of angulated scissor- like elements placed along the great circles
of the sphere. For each double chain Eq. (3.6) holds.
Eq. (3.6) concerns angles only. It alone is not enough for the loop to be
mobile. Figure 3.5(a) shows that a double chain of six angulated scissorlike elements, each consisting of a pair of identical beams that subtend a
central angle of π/3. Eq. (3.6) is therefore satisfied and the lines linking two
end connectors always remain parallel. However, there is a gap between
the end connectors of the first and last elements. This gap may be bridged