Abstract: We define hyperbolic barycentric coordinates (HBC) that describe the position of a point in the hyperbolic plane with respect to the vertices of a given geodesic polygon. We construct explicitly three kinds of HBC, namely hyperbolic Wachspress, mean values and discrete harmonic coordinates. [Read More] These coordinates have properties which resemble those of the planar ones, and they are invariant by the Lorentzian transformations. Furthermore, we figure out the HBC on the Poincaré disk model. The HBC associated to a point in a hyperbolic triangle are unique. We develop two expressions of these coordinates, taking into account the parameters of a point inside the triangle. [Read Less]

Abstract: This paper presents two algorithms for blending between two closed curves in the hyperbolic plane in a manner that guarantees that the intermediate curves are closed. We deal with hyperbolic discrete curves on Poincaré disc which is a famous model of the hyperbolic plane. We use the linear interpolation approach of the geometric invariants of hyperbolic polygons namely hyperbolic side lengths, exterior angles and geodesic discrete curvature. [Read More] We formulate the closing condition of a hyperbolic polygon in terms of its geodesic side lengths and exterior angles. This is to be able to generate closed intermediate curves. Finally, some experimental results are given to illustrate that the proposed methods generate aesthetic blending of closed hyperbolic curves. [Read Less]

Abstract: The study of planar and spherical geometric subdivision schemes was done in Dyn and Hormann (2012); Bellaihou and Ikemakhen (2020). In this paper we complete this study by examining the hyperbolic case. We define general interpolatory geometric subdivision schemes generating curves on the hyperbolic plane by using geodesic polygons and the hyperbolic trigonometry. [Read More] We show that a hyperbolic interpolatory geometric subdivision scheme is convergent if the sequence of maximum edge lengths is summable and the limit curve is G¹-continuous if in addition the sequence of maximum angular defects is summable. In particular, we study the case of bisector interpolatory schemes. Some examples are given to demonstrate the properties of these schemes and some fascinating images on Poincaré disk are produced from these schemes. [Read Less]

Abstract: Shape morphing is a continuous deformation in time between two shapes (curves, surfaces, ...). For planar curves, most efficient methods for blending between two closed curves are based on the construction of the morph curve involving its signed curvature function. The latter is obtained by linear interpolation of the signed curvature functions of the source and target curves (Sederberg et al. (1993), Saba et al. (2014) and Surazhsky and Elber (2002)). [Read More] When the key curves (source and target curves) are closed, the intermediate curve is not necessarily closed, but with a closing process, we look for a closed curve close enough to the open intermediate one. In this paper, we propose two algorithms for blending between two spherical closed curves such that the morph curves remain closed and spherical. Our two methods are based firstly on the approximation of smooth curves by geodesic polygons and secondly on the interpolation of the notion of discrete geodesic curvature and the spherical side lengths of polygons. We solve the problem of closing the morph geodesic polygon by imposing its closing conditions on the sphere and by minimizing the difference of discrete geodesic curvatures. [Read Less]

Abstract: This paper is devoted to the study of semi-symmetric curvature algebraic tensors on a Lorentzian vector space and to the classification of 4-dimensional simply-connected semi-symmetric homogeneous Lorentzian manifolds.