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; "peanospc.scm": Peano space filling mapping
; Copyright (C) 2005 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->tet-array scalar rank)
(do ((tets '() (cons (modulo scl 3) tets))
(scl scalar (quotient scl 3)))
((zero? scl)
(let* ((len (length tets))
(depth (quotient (+ len rank -1) rank)))
(define tra (make-array (A:fixZ8b 0) rank depth))
(set! tets (reverse tets))
(do ((idx (+ -1 depth) (+ -1 idx)))
((negative? idx))
(do ((rdx 0 (+ 1 rdx)))
((>= rdx rank))
(cond ((null? tets))
(else (array-set! tra (car tets) rdx idx)
(set! tets (cdr tets))))))
tra))))
(define (tet-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 (tet-array->coordinates tra)
(define rank (car (array-dimensions tra)))
(define depth (cadr (array-dimensions tra)))
(do ((rdx (+ -1 rank) (+ -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 (coordinates->tet-array coords)
(define depth (do ((scl (apply max coords) (quotient scl 3))
(dpt 0 (+ 1 dpt)))
((zero? scl) dpt)))
(define rank (length coords))
(let ((tra (make-array (A:fixN8b 0) rank 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)))
(do ((idx 0 (+ 1 idx)))
((>= idx depth))
(do ((rdx (+ -1 rank) (+ -1 rdx)))
((negative? rdx))
(let ((v_ij (array-ref tra rdx idx))
(tpar parity))
(do ((idx (+ -1 idx) (+ -1 idx)))
((negative? idx))
(set! tpar (+ (array-ref tra rdx idx) tpar)))
(array-set! tra (if (odd? tpar) (- 2 v_ij) v_ij) rdx idx)
(set! parity (modulo (+ parity v_ij) 2))))))
;;@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->tet-array scalar rank))
(peano-flip! tra)
(tet-array->coordinates tra))
;;@body
;;Returns an exact nonnegative integer corresponding to @1, a list of
;;nonnegative integer coordinates.
(define (peano-coordinates->natural coords)
(define tra (coordinates->tet-array coords))
(peano-flip! tra)
(tet-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 three^rank (expt 3 rank))
(do ((edx 1 (* edx three^rank))
(m 0 (+ 1 m)))
((>= (quotient edx 2) (abs scalar))
(let ((tra (natural->tet-array (+ scalar (quotient edx 2)) rank))
(offset (quotient (expt 3 m) 2)))
(peano-flip! tra)
(map (lambda (k) (* (if (odd? m) -1 1) (- k offset)))
(tet-array->coordinates tra))))))
;;@body
;;Returns an exact integer corresponding to @1, a list of integer
;;coordinates.
(define (peano-coordinates->integer coords)
(define cobs (apply max (map abs coords)))
(let loop ((xpo 1))
(define offset (quotient (expt 3 xpo) 2))
(if (>= offset cobs)
(let ((tra (coordinates->tet-array
(map (lambda (elt) (+ elt offset))
coords))))
(peano-flip! tra)
((if (odd? xpo) - +)
(- (tet-array->natural tra)
(quotient (expt 3 (* (length coords) xpo)) 2))))
(loop (+ 1 xpo)))))
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