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204    C. Liang et al.
                           11.3.1 Tangent vector field

                           The tangent vector on C serves as the key to our analysis of topology of the curve. It
                           serves as an important indicator of topological feature of C. While it is computation-
                           ally prohibitive to compute the tangent vector at each point on C, we can reach some
                           useful conclusion about the topology of the curve by looking into the tangent vector
                           field defined below:


                                                                           e x  e y  e z

                                t = t x (x)e x + t y (x)e y + t z (x) e z = %f ∧%g = ∂ x f∂ y f∂ z f     (11.7)


                                                                         ∂ x g∂ y g∂ z g

                           where e x , e y and e z are the unit vectors along the principle axis x, y and z, respec-
                           tively; t x , t y and t z are functions of x =(x, y, z).
                              Singularities on the curve can be easily characterized, as t vanishes at those
                           points. In [8], the author also tried to localize the point having a tangent parallel
                           to a virtual sweeping plane. They are connected together with the singularities to
                           form the final topological graph. In order to do this, the whole curve is projected
                           onto some principle projection planes. However, the projected planar curve in many
                           cases has a very different topology as C. In our proposed algorithm, we exploit the
                           subdivision along all three principle axes simultaneously and the critical events are
                           either reduced to regular case (such as for tangents) or localized (such as for inter-
                           sections). The topology graph can be built without explicitly computing the exact
                           position of the singularities.

                           11.3.2 Regularity test

                           In this section, we are going now to describe how to detect boxes, for which the
                           topology of the curve can be determined. We will use the following notions:
                                                                                             n
                                                            n
                           Definition 2. We say that a curve C∈ R is regular in a compact domain D ⊂ R ,
                           if its topology is uniquely determined from its intersection with the boundary D.
                           The aim of the method is to give a simple criterion for the regularity of a curve in a
                           box.
                              To form the topological graph for this domain, we only need to compute the
                           intersections between the curve and the boundary of this domain, and there exists
                           a unique graph to link these intersections so that this graph complies to the true
                           topology of the original curve.


                           2D case:

                           For 2D planar algebraic curve C defined by a polynomial equation f(x, y)=0,
                           and denoting the partial derivative of f w.r.t x by ∂ x f, we have the following direct
                           property:
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