(1.1) M' is complete and its sectional curvatures are non-positive. In terms of local coordinates x = (x, . . . , x) on M and y = (y, . . . , y) on M', let the respective Riemann elements of arc-length be ds = gij dx dx\ ds' = g'a$ dy a dy& and r^-fc, T'Vy be the corresponding Christoffel symbols. When there is no danger of confusion, x (or y) will represent a point of M (or M') or its coordina...

Parallel coordinates is a well-known technique used for visualization of multivariate data. When the size of the data sets increases the parallel coordinates display results in an image far too cluttered to perceive any structure. We tackle this problem by constructing high-precision textures to represent the data. By using transfer functions that operate on the high-precision textures, it is p...

i) 8 (p;Xp) 2 TM , as p 2M ) there exists (U ; ) 2 S such that p 2 U ; i.e. (p;Xp) 2 TU , and we have TU =  1 (R) 2 . ii) If we de…ne F : TpM ! R by F (Xp) = (Xp(x); Xp(x); :::::; Xp(x)) where x; x; ::::; x are local coordinates on (U ; ), then clearly F is an isomorphism, so  (p; Xp) = ( (p); F ( Xp)); and  1 = ( 1 ; F 1 ). Now take  1 (U);  1 (V ) 2 and suppose (p; Xp) 2  1 (U)\  1 (V ...

One of the fundamental tasks for analytic activity is retrieving (i.e., reading) the value of a particular quantity in an information visualization. However, few previous studies have compared user performance in such value retrieval tasks for different visualizations. We present an experimental comparison of user performance (time and error distance) across four multivariate data visualization...

We reduce the question of local nonsolvability of the Darboux equation, and hence of the isometric embedding problem for surfaces, to the local nonsolvability of a simple linear equation whose type is explicitly determined by the Gaussian curvature. Let (M, g) be a two-dimensional Riemannian manifold. A well-known problem is to ask, when can one realize this locally as a small piece of a surfac...

We present a generic framework for animating between different visualizations using polygon morphing. The motivation for this work arises from using many different visualization techniques (e.g. scatter plots, parallel coordinates, etc.) to view complex, multi-dimensional data. When transitioning from a visual representation that the user is familiar with to one that is new or one that the user...

BigData, defined as structured and unstructured data containing images, videos, texts, audio and other forms of data collected from multiple datasets, is too big, too complex and moves too fast to analyze using traditional methods. This has given rise to a few issues that must be addressed; 1) how to analyze BigData across multiple datasets, 2) how to classify the different data forms, 3) how t...

The order and arrangement of dimensions (variates) is crucial for the effectiveness of a large number of visualization techniques such as parallel coordinates, scatterplots, recursive pattern, and many others. In this paper, we describe a systematic approach to arrange the dimensions according to their similarity. The basic idea is to rearrange the data dimensions such that dimensions showing a...

Concepts of metadata visualization have been developed in order to help users in understanding of metadata about geographic datasets. These concepts apply graphic samples and four multivariate visualization methods: scatter plot matrix, parallel coordinates plot, star plots and Chernoff faces. The concepts were tested with users of geographic metadata in the Finnish Defence Forces. The test res...

On the other hand, whether the Christoffel symbol is the only connection which can be constructed from a symmetric second–rank tensor gμν remains an open question. In this note we exhibit a constructive demonstration of the uniqueness of the Christoffel symbol. Let us start by reviewing some simple results of tensor calculus. Let M be an n–dimensional differentiable manifold. Several geometric ...