This paper provides new results about efficient arithmetic on (extended) Jacobiquartic form elliptic curves y = dx + 2ax + 1. Recent works have shown thatarithmetic on an elliptic curve in Jacobi quartic form can be performed solidly fasterthan the corresponding operations in Weierstrass form. These proposals use up to 7coordinates to represent a single point. However, fast scal...

A compact Kähler surface X is a K3 surface if it is simply connected and it carries a global homolorphic symplectic form (i.e. the canonical bundle KX ∼= OX). An example is given by the Fermat quartic: consider the degree four polynomial P (X0, ..., X3) = X 4 0 + X 4 1 + X 4 2 + X 4 3 ∈ C[X0, ..., X3]. The vanishing locus S = V (P ) is an irreducible quartic hypersurface in PC, which is simply ...

Route choice through mountainous terrain requires a knowledge of how pace (the reciprocal of speed) varies with gradient of ascent or descent. To model this variation for runners, we analyse record times for 82 uphill and 14 downhill races or race stages. The pace is modelled as a polynomial function of gradient and a linear function of race duration, using ordinary least squares to obtain a be...

In this paper, we propose an elaborate geometric approach to explain the group law on Jacobi quartic curves which are seen as the intersection of two quadratic surfaces in space. Using the geometry interpretation we construct the Miller function. Then we present explicit formulae for the addition and doubling steps in Miller’s algorithm to compute Tate pairing on Jacobi quartic curves. Both the...

The moduli space M0 of semi-stable rank 2 vector bundles with fixed trivial determinant over a non-hyperelliptic curve C of genus 3 is isomorphic to a quartic hypersurface in P7 (Coble’s quartic). We show that M0 is self-dual and that its polar map associates to a stable bundle E ∈ M0 a bundle F which is characterized by dimH 0(C,E ⊗F ) = 4. The projective space PH0(C,E ⊗ F ) is equipped with a...

In this paper a new kind of quasi-quartic trigonometric polynomial base functions with two shape parameters λ and μ over the space Ω = span {1, sin t, cos t, sin2t, cos2t, sin3t, cos3t} is presented and the corresponding quasi-quartic trigonometric Bézier curves and surfaces are defined by the introduced base functions. Each curve segment is generated by five consecutive control points. The sha...

Leapfrog transformations starting with the genus 3 Klein and Dyck tessellations consisting of 24 heptagons and 12 octagons, respectively, can generate possible highly symmetrical structures for allotropes of carbon and the isosteric boron nitride, (BN)x. The Klein tessellation, alternatively described as a platonic 3,7 tessellation, corresponds to the Riemann surface for the multi-valued functi...

In this note we prove the existence of smooth Kummer surfaces in projective three-space containing sixteen mutually disjoint smooth rational curves of any given degree. Introduction Let X be a smooth quartic surface in projective three-space P. As a consequence of Nikulin’s theorem [6] X is a Kummer surface if and only if it contains sixteen mutually disjoint smooth rational curves. The classic...