Math::PlanePath::QuadricIslands -- quadric curve rings
use Math::PlanePath::QuadricIslands; my $path = Math::PlanePath::QuadricIslands->new; my ($x, $y) = $path->n_to_xy (123);
This is concentric islands made from four sides of the QuadricCurve
,
27--26 3 | | 29--28 25 22--21 2 | | | | 30--31 24--23 20--19 1 | 4--3 | 34--33--32 | 16--17--18 <- Y=0 | 1--2 | 35--36 7---8 15--14 -1 | | | 5---6 9 12--13 -2 | | 10--11 -3 ^ -3 -2 -1 X=0 1 2 3 4
The initial figure is the square N=1,2,3,4 then for the next level each straight side expands to 4x longer and a zigzag like N=5 through N=13 and the further sides to N=36.
*---* | | *---* becomes *---* * *---* | | *---* * <------ * | ^ | | | | v | * ------> *
Counting the innermost square as level 0, each ring is
length = 4 * 8^level many points Nlevel = 1 + length[0] + ... + length[level-1] = (4*8^level + 3)/7 Xstart = - 4^level / 2 Ystart = - 4^level / 2
For example the lower partial ring shown above is level 2 starting N=(4*8^2+3)/7=37 at X=-(4^2)/2=-8,Y=-8.
The innermost square N=1,2,3,4 is on 0.5 coordinates, for example N=1 at X=-0.5,Y=-0.5. This is centred on the origin and consistent with the (4^level)/2. Points from N=5 onwards are integer X,Y.
4-------3 Y=+1/2 | | | o | | 1-------2 Y=-1/2 X=-1/2 X=+1/2
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::QuadricIslands->new ()
Create and return a new path object.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return per "Level Ranges" above,
( ( 4 * 8**$level + 3) / 7, (32 * 8**$level - 4) / 7 )
Math::PlanePath, Math::PlanePath::QuadricCurve, Math::PlanePath::KochSnowflakes, Math::PlanePath::GosperIslands
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014 Kevin Ryde
This file is part of Math-PlanePath.
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