Fig。 11。   Possible spanning trees。

simulation)。 The remaining four dummy faces are used to simulate the raising of the wings (as in stages 7 to 9)。

The thick edges shown in Fig。 12 indicate which edges of the net are regarded as being cut so as to leave a spanning tree。 In each case, the faces on either side form a dyad, and these can be set in the correct orientation by setting up and resolving constraints。 As with the skillet example, there is a need with some dyads to allow only one of the two faces to move。 An example is the triangular face along the top edge of the net at the right hand corner。 In the simulation, when this is aligned with the edge to its left, that face has already been positioned (with reference to the neighbour on its left), so only the triangular face can be changed。

Fig。 14。    Two folding panels with a dyad。

Fig。 12。  Starting net for origami bird。

rotated about OR through an angle φ, and face ORCQ is rotated about OR through the same angle but in the opposite sense。 Clearly the two points A can be kept together by taking θ to be zero。 To find the other possibility, consider the transform matrix for face OAP 。 Since point O is fixed, the transform matrix is that of a rotation。 It

can be obtained by combining a rotation of α about the z-axis, a rotation of θ about the new y-axis, a reverse rotation of −α about the z-axis, and a rotation of φ about the x-axis。 This is  represented

by the following product。

。1

cos φ − sin φ

sin φ cos φ

。 。 cos α sin α 。

− sin α cos α

1

。cos θ − sin θ 。 。cos α − sin α 。

× 1

sin θ cos θ

sin α cos α 。

1

This product can be formed and used to transform, by pre- multiplication, the point A = [a 0 0]T regarded as part of face OAP 。

The y-component of the transformed point is found to be

a cos α[− sin α cos φ cos θ + sin α cos φ − sin φ sin θ ]。

By symmetry, when the two points A coincide, the above component is zero and so, assuming that cos α is non-zero,

0 = sin α cos φ(1 − cos θ) − sin φ sin θ

Fig。 13。    Folding of origami bird。

。 2  1 。 1 1

As noted previously, there are typically two solutions for   any

= sin α cos φ

1

2 sin θ

2

− 2 sin φ sin 2 θ cos 2 θ 。

dyad assembly and so there is a need to ensure that the appropriate

So either sin  2 θ is zero, which is the trivial solution,  or

one is found。 In the constraint modelling environment, constraint resolution is carried out by a search process。 The result achieved can be controlled by ensuring the search starts out in a configura-

tan

1

2 θ =

tan φ

。

sin α