Reciprocal Vectors

Abstract

Reciprocal vectors and barycentric coordinates are well-established concepts in various scientific fields, where lattices and grids are essential, e.g., in solid state physics, crystallography, in the numerical analysis of partial differential equations using finite elements, and also in computer graphics and visualisation. In preparation of the Cluster mission, Chanteur [1998] in Chapter 14 of ISSI SR-001 adopted reciprocal vectors to construct estimators for spatial derivatives from four-point measurements, to perform error analysis, and to write down the spatial aliasing condition for four-point wave analysis techniques in a very transparent form. Reciprocal vectors also entered the study on the ac- curacy of plasma moment derivatives, described in Chapter 17 of ISSI SR-001 [Vogt and Paschmann, 1998]. As will be shown below, by using the least squares approach presented in Chapter 12 of ISSI SR-001 [Harvey, 1998], reciprocal vectors are a convenient means in discontinuity analysis to express boundary parameters in terms of crossing times.

This chapter is intended to provide a conceptual introduction to reciprocal vectors, and to emphasise their importance for the analysis of data from the Cluster spacecraft mission. It is organised as follows: The crossing times approach to boundary analysis is presented in Section 4.2 as a way to motivate the use of reciprocal vectors; some of their most important properties are briefly addressed in Section 4.3; then Section 4.4 deals with various aspects of the spatial gradient reconstruction problem; magnetic curvature estimation is reviewed in Section 4.5, while Section 4.6 contains a discussion on the errors of boundary analysis and curvature estimation. Finally, in Section 4.7 we suggest a way to generalise the reciprocal vector concept to cases where the number of spacecraft, N, is not four.