NAME
Math::PlanePath::MathImageQuintetCentres -- self-similar "plus" shape centres
SYNOPSIS
use Math::PlanePath::MathImageQuintetCentres;
my $path = Math::PlanePath::MathImageQuintetCentres->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This a self-similar curve tracing out a "+" shape like the QuintetCurve but taking the centre of each square visited by that curve.
92 12
/ |
124-... 93 91--90 88 11
| \ \ / \
122-123 120 102 94 82 89 86--87 10
\ / | / | / / | |
121 119 103 101-100 95 81 83--84--85 9
\ \ \ \ \
114-115-116 118 104 32 99--98 96 80 78 8
| |/ / / | |/ |/ \
112-113 110 117 105 31 33--34 97 36 79 76--77 7
\ / \ \ \ \ / \ |
111 109-108 106 30 42 35 38--37 75 6
|/ / / | | /
107 29 43 41--40--39 74 5
\ \ |
24--25--26 28 44 46 72--73 70 68 4
| |/ |/ \ \ / \ / \
22--23 20 27 18 45 48--47 71 56 69 66--67 3
\ / \ / \ | / \ |
21 6 19 16--17 49 54--55 58--57 65 2
/ \ | | \ | /
4-- 5 8-- 7 15 50--51 53 59 64 1
\ | / |/ | \
0-- 1 3 9 14 52 60--61 63 <- Y=0
|/ | \ |/
2 10--11 13 62 -1
|/
12 -2
^
-1 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13The base figure is "+" shape of the initial N=0 to N=4,
.....
. .
. 4 .
. \.
........\....
. | .\ .
. 0---1 . 3 .
. | | ./ .
......|./....
. |/.
. 2 .
. .
.....Arms
The optional arms parameter can give up to four copies of the curve, each advancing successively. For example arms=>4 is as follows. Notice the N=4*k points are the plain curve, and N=4*k+1, N=3*k+2 and N=3*k+3 are rotated copies of it.
69 ... 7
/ | \
121 113 73 65--61 53 120 6
/ \ / \ \ \ / \ /
... 117 105-109 77 29 57 45--49 116 5
| / / | | \
101 81 25 33--37--41 96-100-104 112 4
| \ \ | |/
50 97--93 85 21 13 88--92 80 108 72 3
/ | |/ |/ \ \ / \ / \
54 46--42 89 10 17 5-- 9 84 24 76 64--68 2
\ | / | | / \ |
58 38 14 6-- 2 1 16--20 32--28 60 1
/ | \ \ | /
62 30--34 22--18 3 0-- 4 12 36 56 <- Y=0
| \ / | |/ | \
70--66 78 26 86 11-- 7 19 8 91 40--44 52 -1
\ / \ / \ \ / | / | |/
74 110 82 94--90 15 23 87 95--99 48 -2
/ | | \ \ |
114 106-102--98 43--39--35 27 83 103 -3
\ | |/ / |
118 51--47 59 31 79 111-107 119 ... -4
/ \ / \ \ \ / \ /
122 55 63--67 75 115 123 -5
\ |/
... 71 -6
^
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6The pattern an ever expanding "+" shape with first cell N=0 at the origin. The further parts are effectively as follows, though with wiggly spiralling sides. Four parts mesh together and fill the plane.
+---+
| |
+---+--- +---+
| | |
+---+ +---+ +---+
| 2 | 1 | |
+---+ +---+---+ +---+
| | 3 | 0 |
+---+ +---+ +---+
| | |
+---+ +---+---+
| |
+---+FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::MathImageQuintetCentres->new ()-
Create and return a new path object.
($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.Fractional positions give an X,Y position along a straight line between the integer positions.
$n = $path->n_start()-
Return 0, the first N in the path.