Some New Addition Formulae for Weierstrass Elliptic Functions

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialisation of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalisation of Weierstrass' functions to curves of higher genus.Comment: 20 page

Clifford’s Identity and Generalized Cayley-Menger Determinants

Trabajo presentado en el International Symposium on Advances in Robot Kinematics, celebrado en Ljubljana (Eslovenia) del 6 al 10 de diciembre de 2020Distance geometry is usually defined as the characterization and study of point sets in Rk , the k-dimensional Euclidean space, based on the pairwise distances between their points. In this paper, we use Clifford’s identity to extend this kind of characterization to sets of n hyperspheres embedded in Sn−3 or Rn−3 where the role of the Euclidean distance between two points is replaced by the so-called power between two hyperspheres. By properly choosing the value of n and the radii of these hyperspheres, Clifford’s identity reduces to conditions in terms of generalized Cayley-Menger determinants which has been previously obtained on the basis of a case-by-case analysis.This work was partially supported by the Spanish Ministry of Economy and Competitiveness through the projects DPI2017-88282-P and MDM-2016-0656

Clifford’s identity and generalized Cayley-Menger determinants

The final publication is available at link.springer.comDistance geometry is usually defined as the characterization and study of point sets in Rk, the k-dimensional Euclidean space, based on the pairwise distances between their points. In this paper, we use Clifford’s identity to extend this kind of characterization to sets of n hyperspheres embedded in ¿n-3 or Rn-3 where the role of the Euclidean distance between two points is replaced by the so-called power between two hyperspheres. By properly choosing the value of n and the radii of these hyperspheres, Clifford’s identity reduces to conditions in terms of generalized Cayley-Menger determinants which has been previously obtained on the basis of a case-by-case analysis.This work was partially supported by the Spanish Ministry of Economy and Competitiveness through the projects DPI2017-88282-P and MDM-2016-0656.Peer ReviewedPostprint (author's final draft