1 The grand tour via geodesic interpolation of 2 - frames

Grand tours are a class of methods for visualizing multivariate data, or any finite set of points in n-space. The idea is to create an animation of data projections by moving a 2-dimensional projection plane through n-space. The path of planes used in the animation is chosen so that it becomes dense, that is, it comes arbitrarily close to any plane. One inspiration for the grand tour was the experience of trying to comprehend an abstract sculpture in a museum. One tends to walk around the sculpture, viewing it from many different angles. A useful class of grand tours is based on the idea of continuously interpolating an infinite sequence of randomly chosen planes. Visiting randomly (more precisely: uniformly) distributed planes guarantees denseness of the interpolating path. In computer implementations, 2-dimensional orthogonal projections are specified by two 1-dimensional projections which map to the horizontal and vertical screen dimensions, respectively. Hence, a grand tour is specified by a path of pairs of orthonormal projection vectors. This paper describes an interpolation scheme for smoothly connecting two pairs of orthonormal vectors, and thus for constructing interpolating grand tours. The scheme is optimal in the sense that connecting paths are geodesics in a natural Riemannian geometry. 1.0 Some terminology We define and discuss a number of key concepts that will be used in this paper. Definition: A 2-plane in Rn is any 2-dimensional linear subspace of Rn. Note that in our usage, every 2-plane contains the origin 0 ∈ Rn. Definition: A 2-frame in Rn is any ordered pair of orthonormal vectors in Rn. Note that any 2-frame uniquely determines a 2-plane. Notation: Suppose v and w are orthonormal vectors in Rn. Then the 2-frame determined by v and w (in that order) will be denoted by (v,w). Notation: The 2-plane determined by the 2-frame F = (v,w) will be denoted by span(F) or span(v,w). Definition: The Grassmann manifold, or Grassmannian, G2,n of 2-planes in R n is the topological space each point of which represents a distinct 2-plane in Rn.