Ordering of Angle Sizes

The geometric description of shape involves both magnitude and direction information. In the representation described above, all distances in a scene are related by their magnitudes, but angles are simply divided into four broad categories by quadrant, with the boundaries between quadrants providing three distinguished points in the angle quantity space.

This is sufficient for expressing qualitative direction of neighbouring boundary segments, and for determining overall convexity or concavity around a simple boundary. The addition of a further four qualitative regions by including very-acute and very-obtuse angles (both concave and convex) allows several trigonometric operations to be included in the repertoire of geometric techniques for the reasoning system. Table 5.1 lists a set of rules in qualitative trigonometry which become considerably more powerful with the addition of these further qualitative angles.

Further information about angles could be represented by creating a partial ordering of angle sizes similar to the distance ordering. The requirements for angle information are rather different to those for size, however. Absolute distance values need never be expressed, as long as any required relationship between distances in the scene can be represented. The one exception to this is the contact distance (zero), which is a distinguished absolute point. There are a number of distinguished absolute points in the angle quantity space, however, because there are particular angle values which always have the same properties (there are no geometric properties that are always associated with a given absolute size).

The current system has distinguished points for qualitative description of angles at $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ (straight, convex-right and concave-right angles). The very-acute and very-obtuse ranges referred to in the next chapter might also start at a distinguished point; some of the trigonometric properties assigned to very-acute are true of any angle less than $45^{\circ}$, for instance. The addition of a number of other distinguished points could also be justified by the fact that they correspond to useful trigonometric properties. These might include $30^{\circ}$, $60^{\circ}$, $110^{\circ}$, $150^{\circ}$, $210^{\circ}$, $240^{\circ}$, $300^{\circ}$ and $330^{\circ}$, to name a few.

The need for a greater number of distinguished points in a partial ordering can be easily accomodated by using special absolute value nodes in the network. The contact list in the distance ordering is one such special node. The approach taken by Simmons in his common-sense arithmetic system [Sim86] allows any node in the network to be assigned an absolute value, and his method could be used to create a partial angle ordering, in which the nodes have an absolute value slot that can be used for normal trigonometric operations where they are necessary. This would maintain the robust nature of the qualitative representation, while providing an adjustable number of absolute distinguished points.^4.4