The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3–manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson’s Q– theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be suff...

A quadrilateral cylinder of length m and breadth n is the Cartesian product of a m-cycle(with m vertices) and a n-path(with n vertices). Write the vertices of the two cycles on the boundary of the quadrilateral cylinder as x1, x2, · · · , xm and y1, y2, · · · , ym, respectively, where xi corresponds to yi(i = 1, 2, . . . , m). We denote by Qm,n,r, the graph obtained from quadrilateral cylinder ...

Because of their improved numerical properties, quadrilateral meshes have become a popular representation for finite elements computations and computer animation. In this paper we address the problem of optimally representing quadrilateral meshes as generalized triangle strips (with one swap bit per triangle). This is important because 3D rendering hardware is optimized for rendering triangle m...

We present a detailed study of the basic reductions of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net). To make this study, it is necessary to introduce new important ingredients in the by now well established theory of quadrilateral lattices. In particular, we introduce the notions of forward and backwa...

A recent note induced me to resume the classic problem (Po) and its natural generalization: (P0) To construct a square, the sides of which go through four given points. (P) To construct a quadrilateral Q(XYZT) similar to a given quadrilateral q with its sides XY, YZ, • • • , respectively passing through four given points K, L, M, N. The solution of (P0) is classic: we immediately get a locus fo...

An approach based on the use of genetic algorithms to detect quadrilateral shapes in images is presented in this paper. The proposed approach finds the best sets of four edge points that are the vertices of quadrilateral shapes in the image. The proposed method uses the evidence provided by the image resulting of the application of an edge detection operator to the input image. Individuals havi...

Generating quadrilateral meshes is very important in many industrial applications such as finite element analysis and B-spline surface fitting. However, it is still a challenging task to design appropriate vertex connectivity in the quadrilateral meshes by respecting the shapes of the target object and its boundary. This paper presents an approach for interactively editing such mesh topology in...

Acetabular fractures involving predominantly the anterior column associated with a disruption of the quadrilateral surface can be treated with instrumentation implementing the stabilization of the quadrilateral surface. The recently introduced suprapectineal quadrilateral buttress plate is specifically designed to prevent secondary medial subluxation of the femoral head, especially in elderly p...

Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of Catmull-Clark is an example of the former, while the Doo-Sabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Beginning ...

Fig. 1 Let ABCD be a circumscribed quadrilateral, that is, a quadrilateral which has an incircle. Let this incircle have the centerO and the radius ρ and touch its sides AB, BC, CD, DA at the points X, Y, Z, W, respectively. Then, for some very obvious reasons, OX ⊥ AB, OY ⊥ BC, OZ ⊥ CD, OW ⊥ DA and OX = OY = OZ = OW = ρ. Moreover, AW = AX, BX = BY, CY = CZ, DZ = DW, since the two tangents from...