The special shape classes which may usefully be identified are discussed here. In addition to these, RIBALD implements a “none of the above” class, for which the figure of merit is the product of the figures of merit for the object not meeting each particular class.
9.3.1 Normalons
Normalons—objects where all face normals are aligned with one of the three co-ordinate axes—are reasonably common in engineering practice (one survey [143]
suggests that 30% of parts are normalons). Identifying these simplifies topological reconstruction (Chapter 10) and makes the process of fitting face normals trivial (Chapter 11).
For the object to be a normalon, all edges must be aligned with one of the coordinate axes, so if the bundling process (Chapter 5) has identified more than three groups of parallel lines the object cannot be classified as a normalon. Also, in a normalon, there are four more convex turns than concave turns in the outer loop
of any fully-visible face (so, for example, the object in Figure 5.2, page 91, cannot be axis-aligned). For this purpose, collinear lines at extended trihedral vertices such as those in Figure B.156 on page 315 (and also at K-type vertices) are neither right turns nor left turns; although one of the faces in the figure has five vertices, the object portrayed is a normalon.
If the drawing meets these criteria, the base figure of merit for the object being a normalon is 1.0.
In principle, every junction in a drawing of a normalon must meet the Perkins criteria (Chapter 7). This is true of accurate projections but not necessarily of freehand line drawings—Perkins also observed that, empirically, drawings appear axis-aligned to the human eye if one set of faces is drawn rectangularly and the third axis as a diagonal—for example, Figure 9.1 meets the empirical but not the mathematical criteria for a normalon, and should be classified as such. Perkins went on to report [126] that while the human eye will reject assumptions which contradict these criteria if other, valid assumptions of rectangularity or symmetry are found elsewhere, it will sometimes impose rectangularity in defiance of the mathematical criteria if doing so creates some order in a non-rectangular and asymmetric drawing.
Figure 9.1: Technically Invalid Axis-Aligned Drawing
Since the object may still be a normalon even if the drawing fails to meet the Perkins criteria, the figure of merit for axis-alignment is reduced for each line which breaks these criteria (by multiplying by the figure of merit for parallelism between the line and the nearest mathematically-correct line), but the hypothesis is not rejected entirely.
The figure of merit is also decreased if the assumption of parallelism is uncertain—
it is multiplied by the parallelism figure of merit between each line and the mean
orientation of lines in that bundle.
Identification of normalons as described here takes O(n) time. No alternative methods of identifying normalons or assessing figures of merit have been tested (alternatives involving generating face normals were considered but rejected as they involve extra processing and provide no obvious benefit).
9.3.2 Semi-Normalons
The requirements for semi-normalons relax the requirements for a normalon—most, but not all, face normals must be aligned with one of the three coordinate axes.
Semi-normalons are common in engineering practice—one survey [143] showed that although only 30% of the parts covered could be described using axially-aligned blocks and cylinders, 85% could be described using axially-aligned blocks, wedges and cylinders (it is assumed that the axially-aligned cylinders are mostly drilled holes). Adding axially-aligned wedges to a normalon will in general give a semi-normalon, as in Figure B.91.
Semi-normalons are identified as follows:
• if there are more than N bundles of parallel lines, exit—the object is not a normalon (RIBALD uses N = 6)
• list the sets of three bundles which meet at junctions of three lines
• count the number of junction bundle set occurrences (at junctions of three lines) or potential occurrences (at junctions of two lines) of each set
• if no set occurs more than any other, exit—the object is not a normalon
• the object might be a normalon—estimate the figure of merit
If the object is a semi-normalon, the three bundles which occur together more of-ten than any other three are obviously the three mutually-orthogonal special bundles V , B0 and B1 (vertical and two in the base plane) described in Chapter 5.7; if V , B0 or B1 was originally some other bundle, it is updated to reflect this new knowledge about the object.
Initially, the figure of merit is calculated as for normalons. It is multiplied by the proportion of junctions which contributed to the count of the best set of three bundles.
Identification of semi-normalons, as described here, takes O(n) time.
9.3.3 Semi-Normalons with Mirror Symmetry
Classifying an object as a semi-normalon is of only moderate utility—it can help in reconstructing the topology, but it does not of itself give any clues about the geometry of non-axially-aligned hidden faces. In practice, many semi-normalons have a single predominant mirror symmetry which reflects aligned edges to axis-aligned edges—55% of the test drawings in Appendix B which can be labelled meet this requirement. Since this combination provides enough information to deduce the face normals of many hidden faces, it is detected as a special case. The figure of merit is the product of that for semi-normalons (as above) and the highest figure of merit of any mirror chain (see Chapter 8).
If the drawing meets the requirements for this class, the figure of merit for this class, once calculated, is then subtracted from the figure of merit for the previous class (that of semi-normalon without mirror symmetry).
It would be simple to extend this idea further to a combination of semi-normalon with either a C2 or C4 rotation, which would also provide enough information to deduce the face normals of many hidden faces, but such objects appear to be less common in engineering practice. RIBALD does not identify semi-normalons with rotational symmetry as a class.
9.3.4 Artefact-Axis-Aligned Solids
Similar to the concept of a normalon is one where the mirror planes and rotation axes define three mutually-orthogonal axes—see for example Figure 9.2. Such objects seem to be relatively rare, and RIBALD does not identify such objects as a special class.
Figure 9.2: Tapered Wedge Figure 9.3: A Frustum
9.3.5 Extrusions
Right extrusions are a particularly common class of engineering objects (one cata-logue [12] consists entirely of extrusions), and the topology and geometry are easily deduced—a right extrusion has two identical, parallel end cap faces joined by rect-angles. More general extrusions are possible, but much less common.
The requirements for an extrusion are:
• no more than one fully-visible face (the front end cap) is other than a convex quadrilateral
• all vertices can be labelled as trihedral
• each vertex has at least one interpretation with no more than one concave edge
• all edges leaving the front end cap (the side edges) are in the same bundle
• all vertices are either on the front end cap or on side edges (occluding T -junctions need not be—Figures B.44 and B.413 show extrusions)—this may seem obvious, but cannot be omitted (for example, Figure B.108 meets all of the other require-ments)
• each partial face may, when reconstructed, be a convex quadrilateral (i.e. no more than four visible vertices and no concave corners)
Note that, exhaustive as they appear, there is still a problem with these require-ments, in that Figure B.223 meets them and is identified as an extrusion.
If these requirements are met, the figure of merit is the product of
• the figures of merit for each side line being parallel in 2D to the average orientation of this bundle
• the figures of merit for each line on the front end cap being parallel in 2D to the corresponding line on the back end cap (where this line is visible)
For drawings containing only quadrilaterals, RIBALD assumes that each region in turn is the front end cap of an extrusion, and picks the interpretation which produces the best figure of merit.
Identification of extrusions, as described here, takes O(n) time—although iden-tification of quadrilateral extrusions appears to require O(n2) time, these are all topologically equivalent to a cube and are not the limiting case for large n.
9.3.6 Extrusions with Side Holes
The class of extrusions, as described above, includes objects with through holes in the direction of extrusion, such as Figure B.413, but not objects with side-to-side holes, such as Figure B.498 or Figure B.513. Although there are several figures in the test set with side-to-side holes, this is an artefact of the way the test data was generated—most of the through holes were cylindrical in the originals.
RIBALD does not identify extrusions with side holes as an additional class, partly because of the low frequency of such objects, and partly because of the difficulty of distinguishing drawings such as Figure B.415 from those such as Figure B.438—in the latter case, it is clear to a human that a pocket, not a through hole, is intended.
This recommendation may need to be reviewed in future work if cylindrical holes are to be permitted. A more general class, that of extrusion with a single additional feature (boss, pocket, side-to-side hole, or slot), may be more useful—there has not been time to investigate this idea.
9.3.7 Frusta
Right frusta such as Figure 9.3 can be identified by similar criteria to those used for extrusions, although in this case, the lines joining the visible front face and the back face should, if extended, meet at a single point rather than being parallel. The figure of merit for the hypothesis that the object is a frustum depends on how close
to a single point these lines come, and is reduced if the object is likely to be an extrusion (this is in order to prevent the system misclassifying an extrusion as a frustum where the nearly-parallel lines meet a long way away).
For drawings containing only quadrilaterals, RIBALD assumes that each region in turn is the front end cap of a frustum, and picks the interpretation which produces the best figure of merit.
The figure of merit for an object being a frustum is the product of
• 1 − Fx, where Fx is the figure of merit for the object being an extrusion
• the figures of merit for each side line being parallel in 2D to a line joining its start junction to the apex of the extended frustum
• the figures of merit for each line on the front end cap being parallel in 2D to the corresponding line on the back end cap
Identification of frusta, as described here, takes O(n) time—although identifica-tion of quadrilateral frusta appears to require O(n2) time, these are all topologically equivalent to a cube and are not the limiting case for large n.
9.3.8 Platonic and Archimedean Solids
Although the Platonic and Archimedean solids such as Figures B.116 and B.119 are useful test cases for the ideas in Chapters 10 and 11 for handling general-case rotations, it is preferable in practice to treat them as a special class of object, particularly since neither bundling (Chapter 5) nor inflation (Chapter 7) handles them well. This class is identified as follows:
• If any face is only partially visible, exit—the object is not regular
• If any edge is concave, exit—the object is not regular
• If any vertex has no all-convex interpretation, exit—the object is not regular
• Initial set of possible regular objects = { all Platonic and Archimedean solids }
• For each vertex
– determine the number of corners of each face touching this vertex
– eliminate from the set of possible regular objects any for which the required numbers of faces is not a superset of the numbers at this vertex
• If the set of possible regular objects is not empty, the object might be regular—
assess the merit
RIBALD does not check the requirement for alternating faces such as would be required for a drawing to be interpreted as (for example) Figure B.127; errors resulting from this omission are noted in Section 9.5.
The base figure of merit for any drawing which meets the requirements is 1.0.
This is multiplied by the mean merit for each face being a regular polygon: for odd-sided faces, this is the product of the figure of merit for parallelism of an edge and the undrawn line linking the two vertices either side of the opposite vertex; for even-sided faces, it is the product of the figures of merit for parallelism of opposite edges.
Identification of Platonic and Archimedean solids, as described here, takes O(n) time.
9.3.9 Rotationally Symmetric Solids
If the entire drawing is consistent with Cn symmetry, n > 4, the object could be classified as rotationally symmetric. However, the set of test drawings includes no examples in this class which could not equally well be classed as extrusions (Figure B.50 etc) or Platonic or Archimedean solids. Thus, RIBALD does not identify rotationally-symmetric objects as a special class.