History

• An expression for computing a rotation index of a piecewise smooth, regular, and plane curve on the basis of its parametrized loops and its branch points was proposed by Aléssio. A formal proof that validates this result is also given. The expression is applied to a surface-surface intersection algorithm for determining the last point to be traced in the parametric domain of a closed intersection curve. Differently from the local differential properties commonly used in SSI algorithms, the rotation index is a global geometrical property which reflects the behavior of the traced curve. Hence, more reliable decision can be made.
• Two other marching procedures based on the differential geometric properties: marching along a helix and marching along the (locally canonical) polynomial form of a curve were presented by Aléssio in his PhD thesis. Formulas derived by Ye and Maekawa were used for obtaining the exact tangent, normal, binormal, and torsion at almost any point of contact of the intersection curve of regular surfaces. In the vicinity of singular points, however, the results of these formulas are found to be numerically unreliable. The problem was circumvented by a simple yet robust algorithm for estimating differential geometric properties. In this way, the procedure only stops when a border of the parametric domain is reached or a cusp is met, which led us to pursue stopping conditions for the tracing of a closed regular curve.
• An algorithm for estimating the curvature of intersection curves. The estimated curvature is then applied in the computation of direction and size of tracing steps. In his PhD thesis, Andrade showed that one can trace a curve with insignificant deviations, even when it contains singular points. The procedure stops only when a border of the parametric domain is reached or a singular point is (rarely) encountered. To avoid ``infinite iterations'' as they move along a closed curve, they fixed the maximal number of points traced at a predefined value.
• During his Mestrado thesis, Silveira implemented a set of robust Boolean operators for "non-manifold" objects on top of the topological model TDM. The emphasis of his work is on the topological consistency of the operators. For test purpose, geometric intersections restricted to segment-to-segment, segment-to-plane, and plane-to-plane were also included.