Murray Polygon

Murray integer

Cole defined a murray integer formally as follows:

Suppose that

r_n, r_n−1, ..., r₂, r₁

are n non-negative decimal integers, which may or may not be single digits. Let

d_n, d_n−1, ..., d₂, d₁

be digits such that 0 ≤ d_i < r_i for i = 1, 2, ..., n. That is, each digit d_i is a base r_i digit. Then

d_n d_n−1 ... d₂ d₁

is defined to be a mixed radix or murray integer.

Reduced radix complement

The following is Cole's definition of a reduced radix complement.

Suppose

d = d_n d_n−1 ... d₂ d₁

is a murray integer. Then the murray integer

e = e_n e_n−1 ... e₂ e₁,

where e_i = r_i−1−d_i for i = 1, 2, ..., n is the reduced radix complement of d.

Gray coding

The Gray coding algorithm used for murray integers is modified from that of fixed-based systems. In Cole's version, he replaces d_i with r_i−1−d_i rather than r−1−d_i; using the variable radix in place of the fixed radix.

Murray algorithm

The following is Cole's outline of the full algorithm to produce a murray polygon for a given rectangle.

The algorithm to transform a decimal integer d (0 ≤ d ≤ mn−1) to the corresponding point (x, y) on a murray polygon within a rectangle with sides of length m−1, n−1 (Cole 1986, 1988) is now as follows...

1) Convert d to murray integer form

p = p_n p_n−1 ... p₂ p₁,

where n = 2j and the radix r_i (1 ≤ i ≤ 2j) associated with p_i is m_k if i = 2k−1 and n_k if i = 2k.

2) Convert p to the equivalent murray Gray-coded integer

e = e_n e_n−1 ... e₂ e₁.

3) Split e into two parts

f = e_n−1 e_n−3 ... e₃ e₁

and

g = e_n e_n−2 ... e₄ e₂.

4) De-Gray code f and g to give

x = x_j x_j−1 ... x₂ x₁

and

y = y_j y_j−1 ... y₂ y₁.

5) The corresponding vertex in murray notation is now (x, y), which may be converted back to normal notation if required.

Repeating this process for every integer p from 0 to mn−1 sequentially plots all points through which a murray polygon passes for a given rectangle.