Solids of
Revolution

Determining an exhaustive catalogue of visual events for curved objects proves more complex than for polyhedra. All visual events for polyhedra are viewpoint-dependent and can be determined from an object model. Curved objects introduce a new class of viewpoint-dependent visual events, created by the object limb or rim (the locus of points where the line of sight is tangential to the surface). Due to symmetry the viewpoint space of solids of revolution is partitioned in bands identified by only one parameter.

Kriegman and Ponce (See ref. 3) report an implemented technique for computing the exact orthographic aspect graph of solids of revolution. Singularity theory is employed to create a catalogue of visual events. Views are projections of limbs (occluding contours) and creases (intersections of surface patches). The authors assume algebraic curves as solid generators and orthographic projections. Each step of the three-step algorithm presented corresponds to solving a system of polynomial equations, using both symbolic methods (elimination theory) and numerical methods (continuation).

Local singularities: top - swallowtail, middle - beak-to beak, bottom - lip
Figure 6: Local singularities: top - swallowtail, middle - beak-to beak, bottom - lip

Multilocal singularities: top - triple point, middle - tangent crossing, bottom - cusp crossing
Figure 7: Multilocal singularities: top - triple point, middle - tangent crossing, bottom - cusp crossing

Visual events are subdivided into local singularities where the viewing direction and the tangent to the edge are aligned. and multilocal singularities where two or more distinct points on either occluding contours or surface intersections project to the same point. In the upper figure, the local singularities are denoted as a swallowtail, where a smooth image contour breaks into two cusps and a t-junction, beak-to-beak, where two distinct contours meet at a point in the image, and lip where, out of nowhere, a closed curve is formed with the intersection of two cusps. In the lower figure, the multilocal singularities are, from top to bottom a triple point where three contour segments intersect at a point, a tangent crossing where two contours meet at a point and share a common tangent, and a cusp crossing where a cusp on one contour meets another contour. For a triple point on an opaque object, only two branches can be seen on one side of the transition but three are seen on the other side.

Singularities depend on the shape of the contour and the viewing direction; for example a series of singular views of an opaque vase are shown in Figure 8, below. The circular-symmetric aspect graph is shown in Figure 9 (See ref. 3 Kriegman and Ponce, 1990).

<The aspects of an opaque vase
Figure 8:   The aspects of an opaque vase

The aspect graph of an opaque vase
Figure 9: The aspect graph of an opaque vase


[ Polyhedral objects | Piecewise smooth objects ]

Comments to: Sarah Price at ICBL.