Space-filling curves have been found useful for many applications in diverse fields. A space-filling curve is a path in a 2(r)×2(r) raster domain, which visits each location exactly once. In mathematical terms, space-filling curves linearize a 2D integer space, bijectively mapping the space to the integer line. An algorithm is presented, which generates a large number of space-filling curves/spatial orders. Functions are derived such that the code of each location can be calculated from its coordinates and, conversely, a location code can be decoded to yield the coordinates. The algorithm first generates generate 4×4 spatial orders; they subsequently may be scaled up to any desired domain of size 2(r)×2(r) . The underlying theory of the algorithm, the processes for scaling up, encoding, and decoding are described in detail. The curves are generated as a set of incongruent curves, followed, if required, by the sets of associated congruent curves. A number of space-filling curves are illustrated.

Spatial orders such as the Morton (Z) order, U-order, or X-order have applications in matrix manipulation, graphic rendering and data encryption. It is shown that these spatial orders are single examples of entire classes of spatial orders which can be defined in arbitrary numbers of dimensions and base values. Secondly, an algorithm is proposed which can be used to transform between these spatial orders and Cartesian coordinates. It is shown that the efficiency of the algorithm improves with a larger base value. By choosing a base value that corresponds to the available memory page size, the computational effort required to perform operations such as matrix multiplication can be optimized.

Ordering a set of multi-dimensional data is an essential task in modern spatial data processing systems such as raster geographical information systems. This can effectively be carried out by a transformation called spatial ordering. It has been reported that the H ilbert order exhibits desirable performance. However, its encoding or decoding process in the n -dimensional space (n 3) is complicated and would place a considerable computational burden on the application problems. In this paper, we propose a new spatial order with efficient algorithms. The performance evaluation suggests that this new order is competitive with the Hilbert order.

Spatial ordering is a transformation relating an n-dimensional voxel to a set of natural numbers. Interest in the Hilbert order (H-order) has been high because of its desirable performance when applied to image processing. In this correspondence, an encoding/decoding algorithm for the three-dimensional (3-D) Hilbert order is presented.

There are many domains in which graphical programming languages can be applied. Use of a particular graphical programing language will be limited if the domains in which it can be applied are restricted. High-level graphical languages provide simply expressed constructs for the definition, manipulation, inquiry, and external representation of graphical data. These capabilities permit the application to various domains of high-level graphical languages. Five specific domains are discussed; an example of an application of the graphical language LIG is presented in each domain.

The definition, manipulation and display of graphical data in an applications program can occur at different levels if graphical data are viewed as a hierarchy of output primitives. This paper discusses concepts of how graphical objects may be defined and desribes how these concepts are reflected in the design of the high-level graphics programming language LIG/P.

Based on the concept of treating graphical data as a data type in its own right with associated language parts and constructs such as graphical constants, graphical variables and graphical expressions, the semantics of the assignment statement of a high-level graphical programming language are investigated in some detail.It is assumed that the graphical data are defined by the areal external representation model. In addition, the semantics of binary operators, the superposition and the deletion operators, are considered.