Jump to content

Generalized balanced ternary

From Wikipedia, the free encyclopedia

Generalized balanced ternary is a generalization of the balanced ternary numeral system to represent points in a higher-dimensional space. It was first described in 1982 by Laurie Gibson and Dean Lucas.[1] It has since been used for various applications, including geospatial[2] and high-performance scientific[3] computing.

General form

[edit]

Like standard positional numeral systems, generalized balanced ternary represents a point as powers of a base multiplied by digits .

Generalized balanced ternary uses a transformation matrix as its base . Digits are vectors chosen from a finite subset of the underlying space.

One dimension

[edit]

In one dimension, generalized balanced ternary is equivalent to standard balanced ternary, with three digits (0, 1, and -1). is a matrix, and the digits are length-1 vectors, so they appear here without the extra brackets.

Addition table

[edit]

This is the same addition table as standard balanced ternary, but with replacing T. To make the table easier to read, the numeral is written instead of .

Addition
+ 0 1 2
0 0 1 2
1 1 12 0
2 2 0 21

Two dimensions

[edit]
The 2D points addressable by three generalized balanced ternary digits. Each point is addressed by its path from the origin; the six colors correspond to the six non-zero digits.

In two dimensions, there are seven digits. The digits are six points arranged in a regular hexagon centered at .[4]

Addition table

[edit]

As in the one-dimensional addition table, the numeral is written instead of (despite e.g. having no particular relationship to the number 2).

Addition[4]
+ 0 1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 1 12 3 34 5 16 0
2 2 3 24 25 6 0 61
3 3 34 25 36 0 1 2
4 4 5 6 0 41 52 43
5 5 16 0 1 52 53 4
6 6 0 61 2 43 4 65

If there are two numerals in a cell, the left one is carried over to the next digit. Unlike standard addition, addition of two-dimensional generalized balanced ternary numbers may require multiple carries to be performed while computing a single digit.[4]

See also

[edit]

References

[edit]
  1. ^ Gibson, Laurie; Lucas, Dean (1982). "Spatial Data Processing Using Generalized Balanced Ternary". Proceedings of the IEEE Computer Society Conference on Pattern Recognition and Image Processing: 566–571.
  2. ^ Sahr, Kevin (2011-01-01). "Hexagonal Discrete Global Grid Systems for Geospatial Computing" (PDF). Archives of Photogrammetry, Cartography and Remote Sensing. 22: 363. Bibcode:2011ArFKT..22..363S.
  3. ^ de Kinder, R. E. Jr.; Barnes, J. R. (August 1997). "The Generalized Balanced Ternary (GBT) Applied to High-Performance Computational Algorithms". APS Meeting Abstracts. Bibcode:1997APS..CPC..C409D.
  4. ^ a b c van Roessel, Jan W. (1988). "Conversion of Cartesian coordinates from and to Generalized Balanced Ternary addresses" (PDF). Photogrammetric Engineering and Remote Sensing. 54: 1565–1570.
[edit]