13。 Then tabs along the edges of faces 4 and 16 are glued to faces 1 and 9。 This means that the first stage of erection is to open out the

Fig。 8。   Closing the side faces at different speeds。

carton so that faces 1, 2, 3 and 4 form a parallelogram and finally a square。 It is this initial gluing that makes the carton a skillet [6]。

Fig。 11 shows two possible spanning trees for the face graph。 Below each is the corresponding net with the appropriate faces separated。 Also shown separated are the flaps at the bottom of the carton, faces 5-8。 Note that the nets do not fall into two pieces since there are joins between faces at the left and right sides as indicated by the arrows; for example faces 1 and 4 join。 In both trees, the edge joining nodes 2 and 3 is removed。 This is on the assumption

that during the opening process, face 1 is held fixed and face 4 is driven through 90◦  as suggested in Fig。 9。 To assemble the    main

walls of the carton, a constraint is required to bring faces 2 and 3 together。

A little more care is required with the gussets involved in the gable-end itself。 This is because there are now cases of five faces with a common vertex。 Consider the tree labelled (a)。 Assuming that face 9 is driven, faces 10 and 11 can be treated as a dyad。 Once the position of face 11 is known, faces 12 and 13 can be dealt with as a second dyad; then faces 14 and 15。 This leaves faces 16 as 9 as a final dyad。 But the position of face 9 is already established。 So the final dyad is handled by only adjusting face  16。

In practice, simply driving face 9 is unlikely to be successful in driving all the other faces because of the tendency of the panels to distort as they transmit the motion。 Instead, both faces 9 and 13 are driven。 This means that the second spanning tree, labelled (b) in Fig。 11, is more appropriate。 It means that every face is at most distance 3 away (in terms of path-length within the tree) from a driven face。 A general principle is that one wishes to choose the faces that are to be driven and the spanning tree so as to ensure that every face is close to a driven face and thus problems with distortion are reduced。 Using spanning tree (b), faces 10 and 11 are solved as dyad。 The dyad between faces 11 and 12 is then handled by adjusting face 12 alone。 A similar approach is applied for the other faces around the gable-end。

From the point of view of the simulation, the pre-processor can set up the commands in the same way as before, with two angles declared as variable for each dyad。 However, when only one angle is allowed to change, a ‘‘fix’’ command is given to ensure that the other  is  unaltered  during  constraint resolution。

5。Extension to origami

The previous sections have looked at the simulation of the erection of cartons。 Although there is interest in more complicated packs [5], practical cartons tend to be fairly simple structures。 This is partly because of the need to be able to manipulate them by machine or manually。 In order to see if the previous simulation

Fig。 10。    Net and face tree for gable-end skillet。

techniques work with added complexity, this section looks at their application to an example from origami。 This is the ‘‘flapping bird’’ (e。g。 [41]) which is a standard origami construction。

As with most origami models, the bird starts with a square of paper。 The net used for the simulation is shown in Fig。 12。 The upper right corner of the net contains the fold lines that form the head of the bird。 Stages in the simulation are shown in Fig。 13。

As with the carton tray, there is a need to introduce a number of ‘‘dummy’’ faces to allow turning of parts once dyads have been fully folded。 There are 17 dummy faces shown as shaded regions in Fig。 12。 The 13 central dummy faces all have zero size and are co- incident with the central point; showing them larger in the figure distorts the shape of neighbouring faces。 The central face allows the simulation to show the parts folding symmetrical downwards and upwards about the mid-point of the square (as for example in passing between stages 1 and 3 in Fig。 13)。 Its 12 neighbouring dummy faces allow the rotation of the main triangular parts about a common vertical axis (as in going between stages 3 and 4 of the