In the Bézier formalism, an arc of a conic is a rational curve of degree 2 with control polygon {P, Q, R} for which the weights can be normalized to {1, w, 1}. The parametrization of the conic arc is C(t) = (1 − t) 2 P + 2wt(1 − t)Q + t 2 R (1 − t) 2 + 2wt(1 − t) + t 2 , t ∈ [0, 1]. Abstract Synthetic derivation of closed for-mulae of the geometric characteristic of a conic given in Bézier form...

This paper presents a practical representation containing a parameter of rational cubic conic sections and research’s deeply the inner properties. Firstly, the parameter how to affect the control points, inner weights and shoulder point is discussed. Secondly, the inner relation between the parameter and the weights of the nonstandard-form quadratic rational conic sections is analyzed in detail...

Conic map projections are appropriate for mapping regions at medium and large scales with east–west extents at intermediate latitudes. Conic projections are appropriate for these cases because they show the mapped area with less distortion than other projections. In order to minimize the distortion of the mapped area, the two standard parallels of conic projections need to be selected carefully...

In our present investigation, with the help of basic (or q-) calculus, we first define a new domain which involves Janowski function. We also subclass class q-starlike functions, maps open unit disk U, given by U= z:z?C and z <1, onto this generalized conic type domain. study here some such potentially useful results as, for example, sufficient conditions, closure results, Fekete-Szegö inequ...

Conic sections embedded in a torus must be circles of special types: (i) proole circles, (ii) cross-sectional circles , and (iii) Yvone-Villarceau circles. Based on this classiication, we present eecient and robust geometric algorithms that detect and compute all degenerate conic sections (circles) in torus/plane and torus/natural-quadric intersections. 2 Introduction Simple surfaces (such as p...

We investigate two different textures of smectic A liquid crystals. These textures are particularly symmetric when they are observed at crossed polars optical microscopy. For both textures, a model has been made in order to examine the link between the defective macroscopic texture and the microscopic disposition of the layers. We present in particular in the case of some hexagonal tiling of ci...

The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a, b, c, d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of non-intersecting lines, the Lindström-GesselViennot theorem leads to a formula as a sum of...

Keywords: Conic sections Bézier curves Eccentricity Asymptotes Axes Foci In this paper, we address the calculation of geometric characteristics of conic sections (axes, asymptotes, centres, eccentricity, foci) given in Bézier form in terms of their control polygons and weights, making use of real and complex projective and affine geometry and avoiding the use of coordinates.