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; "peanosfc.scm": Peano space filling mapping
; Copyright (C) 2005, 2006 Aubrey Jaffer
;
;Permission to copy this software, to modify it, to redistribute it,
;to distribute modified versions, and to use it for any purpose is
;granted, subject to the following restrictions and understandings.
;
;1. Any copy made of this software must include this copyright notice
;in full.
;
;2. I have made no warranty or representation that the operation of
;this software will be error-free, and I am under no obligation to
;provide any services, by way of maintenance, update, or otherwise.
;
;3. In conjunction with products arising from the use of this
;material, there shall be no use of my name in any advertising,
;promotional, or sales literature without prior written consent in
;each case.
(require 'array)
;;@code{(require 'peano-fill)}
;;@ftindex peano-fill
;;; A. R. Butz.
;;; Space filling curves and mathematical programming.
;;; Information and Control, 12:314-330, 1968.
(define (natural->trit-array scalar rank)
(do ((trits '() (cons (modulo scl 3) trits))
(scl scalar (quotient scl 3)))
((zero? scl)
(let ((depth (quotient (+ (length trits) rank -1) rank)))
(define tra (make-array (A:fixZ8b 0) rank depth))
(set! trits (reverse trits))
(do ((idx (+ -1 depth) (+ -1 idx)))
((negative? idx))
(do ((rdx 0 (+ 1 rdx)))
((>= rdx rank))
(cond ((null? trits))
(else (array-set! tra (car trits) rdx idx)
(set! trits (cdr trits))))))
tra))))
(define (trit-array->natural tra)
(define rank (car (array-dimensions tra)))
(define depth (cadr (array-dimensions tra)))
(define val 0)
(do ((idx 0 (+ 1 idx)))
((>= idx depth) val)
(do ((rdx (+ -1 rank) (+ -1 rdx)))
((negative? rdx))
(set! val (+ (array-ref tra rdx idx) (* 3 val))))))
(define (trit-array->natural-coordinates tra)
(define depth (cadr (array-dimensions tra)))
(do ((rdx (+ -1 (car (array-dimensions tra))) (+ -1 rdx))
(lst '() (cons (do ((idx 0 (+ 1 idx))
(val 0 (+ (array-ref tra rdx idx) (* 3 val))))
((>= idx depth) val))
lst)))
((negative? rdx) lst)))
(define (natural-coordinates->trit-array coords)
(define depth (do ((scl (apply max coords) (quotient scl 3))
(dpt 0 (+ 1 dpt)))
((zero? scl) dpt)))
(let ((tra (make-array (A:fixN8b 0) (length coords) depth)))
(do ((rdx 0 (+ 1 rdx))
(cds coords (cdr cds)))
((null? cds))
(do ((idx (+ -1 depth) (+ -1 idx))
(scl (car cds) (quotient scl 3)))
((negative? idx))
(array-set! tra (modulo scl 3) rdx idx)))
tra))
(define (peano-flip! tra)
(define parity 0)
(define rank (car (array-dimensions tra)))
(define depth (cadr (array-dimensions tra)))
(define rra (make-array (A:fixN8b 0) (car (array-dimensions tra))))
(do ((idx 0 (+ 1 idx)))
((>= idx depth))
(do ((rdx (+ -1 rank) (+ -1 rdx)))
((negative? rdx))
(let ((v_ij (array-ref tra rdx idx)))
(if (odd? (+ parity (array-ref rra rdx)))
(array-set! tra (- 2 v_ij) rdx idx))
(set! parity (modulo (+ v_ij parity) 2))
(array-set! rra (modulo (+ v_ij (array-ref rra rdx)) 2) rdx)))))
;;@body
;;Returns a list of @2 nonnegative integer coordinates corresponding
;;to exact nonnegative integer @1. The lists returned by @0 for @1
;;arguments 0 and 1 will differ in the first element.
(define (natural->peano-coordinates scalar rank)
(define tra (natural->trit-array scalar rank))
(peano-flip! tra)
(trit-array->natural-coordinates tra))
;;@body
;;Returns an exact nonnegative integer corresponding to @1, a list of
;;nonnegative integer coordinates.
(define (peano-coordinates->natural coords)
(define tra (natural-coordinates->trit-array coords))
(peano-flip! tra)
(trit-array->natural tra))
;;@body
;;Returns a list of @2 integer coordinates corresponding to exact
;;integer @1. The lists returned by @0 for @1 arguments 0 and 1 will
;;differ in the first element.
(define (integer->peano-coordinates scalar rank)
(define nine^rank (expt 9 rank))
(do ((edx 1 (* edx nine^rank))
(cdx 1 (* cdx 9)))
((>= (quotient edx 2) (abs scalar))
(let ((tra (natural->trit-array (+ scalar (quotient edx 2)) rank))
(offset (quotient cdx 2)))
(peano-flip! tra)
(map (lambda (k) (- k offset))
(trit-array->natural-coordinates tra))))))
;;@body
;;Returns an exact integer corresponding to @1, a list of integer
;;coordinates.
(define (peano-coordinates->integer coords)
(define nine^rank (expt 9 (length coords)))
(define cobs (apply max (map abs coords)))
(let loop ((edx 1) (cdx 1))
(define offset (quotient cdx 2))
(if (>= offset cobs)
(let ((tra (natural-coordinates->trit-array
(map (lambda (elt) (+ elt offset))
coords))))
(peano-flip! tra)
(- (trit-array->natural tra)
(quotient edx 2)))
(loop (* nine^rank edx) (* 9 cdx)))))
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