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Fig.9. A query of an arrangement of arcs in Shape Machine under isometry transformations. the two isometric matches T(x) of the identity rule is given in Figure 9(c). ‘he previous example showed the query of a compound arc whose structural features, including centers of a nd intersections of arcs, could be retrieved from the database of CAD system. The next example in Figur howcases the search of a shape consisting of a rotational arrangement of three arcs such that each arc has ¢ ndpoint registered in the database of a shape to be queried. Clearly, this query can be extended for a variety hapes consisting of three or more arcs or combinations of arcs and straight lines without caring about whet! neir boundaries, endpoints and intersections are registered in the database of the CAD system to be searched. T uery here yields two results because the three-fold dihedral symmetry of the three-lens polygon of order € ivided by the order of symmetry of the rotational symmetry of order 3 as long as both shapes share three symme lements, namely the rotations of order 0°, 120°, and 240° around their centers. The query x of the rotatio: rrangement of the three arcs is shown in Figure 9(a), the trefoil design that is queried in Figure 9(b) and one he two isometric matches 7(x) of the identity rule is given in Figure 9(c). 2.4. Find shapes consisting of arcs under similarity transformations — Figure 9 A query of an arrangement of arcs in Shape Machine under isometry transformations. the two isometric matches T(x) of the identity rule is given in Figure 9(c). ‘he previous example showed the query of a compound arc whose structural features, including centers of a nd intersections of arcs, could be retrieved from the database of CAD system. The next example in Figur howcases the search of a shape consisting of a rotational arrangement of three arcs such that each arc has ¢ ndpoint registered in the database of a shape to be queried. Clearly, this query can be extended for a variety hapes consisting of three or more arcs or combinations of arcs and straight lines without caring about whet! neir boundaries, endpoints and intersections are registered in the database of the CAD system to be searched. T uery here yields two results because the three-fold dihedral symmetry of the three-lens polygon of order € ivided by the order of symmetry of the rotational symmetry of order 3 as long as both shapes share three symme lements, namely the rotations of order 0°, 120°, and 240° around their centers. The query x of the rotatio: rrangement of the three arcs is shown in Figure 9(a), the trefoil design that is queried in Figure 9(b) and one he two isometric matches 7(x) of the identity rule is given in Figure 9(c). 2.4. Find shapes consisting of arcs under similarity transformations

Related Figures (29)

Fig.1. A query of a closed polyline in Shape Machine under isometry transformations. Figure 1(b), identifies two embedded isometric hexagons, one of them shown highlighted in Figure 1(c). The search for closed polygons can be extended for arrangements of polygons even though these arrangements are he initial model identifies no more isometric instances of the square in the model. A pictorial query of a pentagon elation between two isosceles right triangles reflected along the leading diagonal of an underlying square upon

Fig.2 A query of an arrangement of two polygons in Shape Machine under isometry transformations.

Fig.3. A query of an arrangement of open polylines in Shape Machine under isometry transformations. queried in Figure 3(b) and one of the eight isometric matches of the identity rule is given in Figure 3(c). 2.2. Find shapes under similarity transformations number of non-equivalent instances. The query x of the lower-case k is shown in Figure 3(a), the design that is

Fig.4. A query of a polygon in Shape Machine under similarity transformations n a spatial relation is shown in Figure 5. The specificity of this spatial relation is quite involved: All three polygons

Fig.5. A query of an arrangement of polygons in Shape Machine under similarity transformations. division of the orders of their symmetry groups equal to 5. share one symmetry element - a mirror refection and the complete non-equivalent matches T(x) is equal to the

Fig.6. A query of an arrangement of lines in Shape Machine under similarity transformations. 2.3. Find shapes consisting of arcs under isometric transformations the shape x in Figure 6(a) in the design in Figure 6(b) and give the possible ten matches T(x), one of which is The second family of examples of visual queries in Shape Machine is constructed around shapes that consist of

Fig.7. A query of a closed arc shape in Shape Machine under isometry transformations. and one of the three isometric matches T(x) of the identity rule is given in Figure 7(c).

Fig.8 A query of an arrangement of arcs in Shape Machine under isometry transformations in Figure 7(b) and one of the six isometric matches T(x) of the identity rule is given in Figure 7(c). compound arc equal to 1. The query x of the compound arc is shown in Figure 7(a), the trefoil design that is queried

Fig.10. A query of an arrangement of arcs in Shape Machine under similarity transformations. A second family of examples of visual queries in Shape Machine of shapes that consist of arcs that can be second family of examples OF visual queries 1n Shape Machine of shapes that consist oF arcs that can be arched under similarity transformations can be constructed by taking any arrangements of arcs irrelevan hether they specify closed shapes or connected arcs in continuous circuits. Even more, this inquiry can b ‘tended by asking that the boundaries of the arcs (endpoints) are not necessarily registered in the databas e CAD system that underlies the specification of a shape that will be searched. An example of a query 0 shape is given in Figure 11. Here a shape consisting of six arcs is queried with three points defined as th tersections of three pairs of arcs while the other six endpoints are indefinitely defined upon the circumfe ‘the arc. The lengths of the arcs are all the same and they are identically disposed around a center giving ape a dihedral symmetry of order 6 equal to the order of symmetry of the trefoil shape that is queried. T 1ery X of the six arcs is shown in Figure 11(a), the trefoil design that is queried in Figure 11(b) and the si ES De! ay Be SE Ca ee oot, ee: eT i a ee, Po, Pn TE PEG Ae ee Be ee Mr ee oe. ne. Me me Ara searched under similarity transformations can be constructed by taking any arrangements of arcs irrelevant of

Fig.11. A query of an arrangement of arcs in Shape Machine under similarity transformations. similarity match T(x) of the identity rule is given in a highlighted form in Figure 1 1(c). A last example of a visual query in Shape Machine of shapes that consist of arcs and that can be searched under parts of a trefoil shape yielding a highly expressive shape that defies a discursive description. Clearly the

Fig.12. A query of an arrangement of arcs in Shape Machine under similarity transformations. 2.5. Exceptions noted: Indeterminate applications of rules shape. The query x of the fragmented shape (arrangements of arcs) is shown in Figure 12(a), the trefoil design

Fig. 13. A two-step query of an arrangement of lines in Shape Machine under similarity transformations given in a highlighted form in Figure 13(c).

Fig.14. A two-step query of an arrangement of lines in Shape Machine under similarity transformations. Finally, these two-step visual queries in Shape Machine can be combined with the rest of the armature of the query x of the y-shape is shown in Figure 14(a), the nested square design that is queried in Figure 14(b) and one

Fig. 15. A two-step query of an arrangement of lines in Shape Machine under similarity transformations. scalar transformation of 1.5 of the identity rule is given in a highlighted form in Figure 15(c). 3. Replacement rules

Fig.16. A construction in Shape Machine using a single shape rule in the schema x > YT (x + t(x))

Fig.17. A construction in Shape Machine using a single shape rule in the schema x > >} T(y) ina determinate way.

Fig. 18. A construction in Shape Machine using a single shape rule in the schema x > )} T(y) in an indeterminate (two-step) 3.2. Substitute shapes consisting of lines and arcs under isometries and similarities

Fig.19. A construction in Shape Machine using a single shape rule in the schema x > }} T(y) ina determinate way. shape 7(y) is given in Figure 19(b); a final design after the execution of a series of rules is shown in Figure 19(c). A simple reworking of the same schema rule with a substitution of a shape made of straight lines in the LHS with pattern. The shapes x in the LHS and y in the RHS of the rule are shown in Figure 19(a); one of the sixteen

Fig.20. A construction in Shape Machine using a single shape rule in the schema x > )} T(y) ina determinate way. after the execution of a series of rules is shown in Figure 20(c). transformation T and its corresponding replacement of the shape T(y) is given in Figure 20(b); a final design These replacement rules need not confine to well-structured or conventionalized geometries as in the 4 x 4 RHS of the rule are shown in Figure 20(a); one of the 16 possible matches of the shape 7(x) under a similarity

Fig.21. A construction in Shape Machine using a single shape rule in the schema x > XTC) ina determinate way. a final design after the execution of all possible rule applications is shown in Figure 21(c).

Fig.22. The Celtic knot grammar: Configuration stage. applications is shown in Figure 22(c).

Fig.23. The Celtic knot grammar: Fabric stage.

Fig.24. A Celtic knot grammar: Modifying stage. and a design after the double execution of the untying rule is shown in Figure 24(c).

Fig.25, An automated specification of a mechanical gear in the DrawScript mode of the Shape Machine. automated generation of the gear in Shape Machine is given in Figure 25(c). the circle with the pentagonal polyline that outlines the tooth of the gear is shown in Figure 25(b). The complete

Fig.26. Editing of the profile of the teeth of a gear in the DrawScript mode of the Shape Machine. gear grammar is shown in Figure 26(b). The new generated variation of the gear is given in Figure 26(c).

Fig.27. Editing of the number of teeth of a gear in the DrawScript mode of the Shape Machine.

Fig.28. Portm-Ino: Framework stage

The third stage of the grammar, Architectonics, completes the generation of the plan. This stage, consist of rules

Fig.30. Portm-Ino: Architectonics stage gridiron plan in the language is shown in Figure 30(c)