Generalized balanced ternary

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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

Like standard positional numeral systems, generalized balanced ternary represents a point [math]\displaystyle{ p }[/math] as powers of a base [math]\displaystyle{ B }[/math] multiplied by digits [math]\displaystyle{ d_i }[/math].

[math]\displaystyle{ p = d_0 + B d_1 + B^2 d_2 + \ldots }[/math]

Generalized balanced ternary uses a transformation matrix as its base [math]\displaystyle{ B }[/math]. Digits are vectors chosen from a finite subset [math]\displaystyle{ \{D_0 = 0, D_1, \ldots, D_n\} }[/math] of the underlying space.

One dimension

In one dimension, generalized balanced ternary is equivalent to standard balanced ternary, with three digits (0, 1, and -1). [math]\displaystyle{ B }[/math] is a [math]\displaystyle{ 1\times 1 }[/math] matrix, and the digits [math]\displaystyle{ D_i }[/math] are length-1 vectors, so they appear here without the extra brackets.

[math]\displaystyle{ \begin{align}B &= 3 \\ D_0 &= 0 \\ D_1 &= 1 \\ D_2 &= -1\end{align} }[/math]

Addition table

This is the same addition table as standard balanced ternary, but with [math]\displaystyle{ D_2 }[/math] replacing T. To make the table easier to read, the numeral [math]\displaystyle{ i }[/math] is written instead of [math]\displaystyle{ D_i }[/math].

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

Two dimensions

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 [math]\displaystyle{ D_1, \ldots, D_6 }[/math] are six points arranged in a regular hexagon centered at [math]\displaystyle{ D_0 = 0 }[/math].[4]

[math]\displaystyle{ \begin{align} B &= \frac{1}{2}\begin{bmatrix} 5 & \sqrt{3} \\ -\sqrt{3} & 5 \end{bmatrix} \\ D_0 &= 0 \\ D_1 &= \left( 0, \sqrt{3} \right) \\ D_2 &= \left( \frac{3}{2}, -\frac{\sqrt{3}}{2} \right) \\ D_3 &= \left( \frac{3}{2}, \frac{\sqrt{3}}{2} \right) \\ D_4 &= \left( -\frac{3}{2}, -\frac{\sqrt{3}}{2} \right) \\ D_5 &= \left( -\frac{3}{2}, \frac{\sqrt{3}}{2} \right) \\ D_6 &= \left( 0, -\sqrt{3} \right) \\ \end{align} }[/math]

Addition table

As in the one-dimensional addition table, the numeral [math]\displaystyle{ i }[/math] is written instead of [math]\displaystyle{ D_i }[/math] (despite e.g. [math]\displaystyle{ D_2 }[/math] 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

References

  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". Archives of Photogrammetry, Cartography and Remote Sensing 22: 363. Bibcode2011ArFKT..22..363S. http://webpages.sou.edu/~sahrk/sqspc/pubs/sahrMMT11us.pdf. 
  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. Bibcode1997APS..CPC..C409D. 
  4. Jump up to: 4.0 4.1 4.2 van Roessel, Jan W. (1988). "Conversion of Cartesian coordinates from and to Generalized Balanced Ternary addresses". Photogrammetric Engineering and Remote Sensing 54: 1565–1570. https://www.asprs.org/wp-content/uploads/pers/1988journal/nov/1988_nov_1565-1570.pdf. 

External links