NAME
Math::PlanePath::MathImagePeanoHalf -- 9-segment self-similar spiral
SYNOPSIS
use Math::PlanePath::MathImagePeanoHalf;
my $path = Math::PlanePath::MathImagePeanoHalf->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
In progress ...
This is an integer version of a 9-segment self-similar curve by ...
4
3
2
1
<- Y=0
-1
-2
-3
-4
-4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9 10 11 12
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***************************Arms
The optional arms => 2 parameter can give a second copy of the spiral rotated 180 degrees. With two arms all points of the plane are covered.
93--91 81--79--77--75 57--55 45--43--41--39 122-124 ..
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95 89 83 69--71--73 59 53 47 33--35--37 120 126 132
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97 87--85 67--65--63--61 51--49 31--29--27 118 128-130
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99-101-103 22--20 10-- 8-- 6-- 4 13--15 25 116-114-112
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109-107-105 24 18 12 1 0-- 2 11 17 23 106-108-110
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111-113-115 26 16--14 3-- 5-- 7-- 9 19--21 104-102-100
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129-127 117 28--30--32 50--52 62--64--66--68 86--88 98
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131 125 119 38--36--34 48 54 60 74--72--70 84 90 96
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.. 123-121 40--42--44--46 56--58 76--78--80--82 92--94 The first arm is the even numbers N=0,2,4,etc and the second arm is the odd numbers N=1,3,5,etc.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::MathImagePeanoHalf->new ()-
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.
FORMULAS
X,Y to N
The correspondence to Wunderlich's 3x3 serpentine curve can be used to turn X,Y coordinates in base 3 into an N. Reckoning the innermost 3x3 as level=1 then the smallest abs(X) or abs(Y) in a level is
Xlevelmin = (3^level + 1) / 2
eg. level=2 Xlevelmin=5which can be reversed as
level = log3floor( max(abs(X),abs(Y)) * 2 - 1 )
eg. X=7 level=log3floor(2*7-1)=2An offset can be applied to put X,Y in the range 0 to 3^level-1,
offset = (3^level-1)/2
eg. level=2 offset=4Then a table can give the N base-9 digit corresponding to X,Y digits
Y=2 4 3 2 N digit
Y=1 -1 0 1
Y=0 -2 -3 -4
X=0 X=1 X=2A current rotation maintains the "S" part directions and is updated by a table
Y=2 0 +3 0 rotation when descending
Y=1 +1 +2 +1 into sub-part
Y=0 0 +3 0
X=0 X=1 X=2The negative digits of N represent backing up a little in some higher part. If N goes negative at any state then X,Y was off the main curve and instead on the second arm. If the second arm is not of interest the calculation can stop at that stage.
It no doubt would also work to take take X,Y as balanced ternary digits 1,0,-1, but it's not clear that would be any faster or easier to calculate.