# NAME

Math::PlanePath::TriangularHypot -- points of triangular lattice in order of hypotenuse distance

# SYNOPSIS

`````` use Math::PlanePath::TriangularHypot;
my \$path = Math::PlanePath::TriangularHypot->new;
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This path visits X,Y points on a triangular "A2" lattice in order of their distance from the origin 0,0 and anti-clockwise around from the X axis among those of equal distance.

``````             58    47    39    46    57                 4

48    34    23    22    33    45              3

40    24    16     9    15    21    38           2

49    25    10     4     3     8    20    44        1

35    17     5     1     2    14    32      <- Y=0

50    26    11     6     7    13    31    55       -1

41    27    18    12    19    30    43          -2

51    36    28    29    37    54             -3

60    52    42    53    61                -4

^
-7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7``````

The lattice is put on a square X,Y grid using every second point per "Triangular Lattice" in Math::PlanePath. Scaling X/2,Y*sqrt(3)/2 gives equilateral triangles of side length 1 making a distance from X,Y to the origin

``````    dist^2 = (X/2^2 + (Y*sqrt(3)/2)^2
= (X^2 + 3*Y^2) / 4``````

For example N=19 at X=2,Y=-2 is sqrt((2**2+3*-2**2)/4) = sqrt(4) from the origin. The next smallest after that is X=5,Y=1 at sqrt(7). The key part is X^2 + 3*Y^2 as the distance measure to order the points.

## Equal Distances

Points with the same distance are taken in anti-clockwise order around from the X axis. For example N=14 at X=4,Y=0 is sqrt(4) from the origin, and so are the rotated X=2,Y=2 and X=-2,Y=2 etc in other sixth segments, for a total 6 points N=14 to N=19 all the same distance.

Symmetry means there's a set of 6 or 12 points with the same distance, so the count of same-distance points is always a multiple of 6 or 12. There are 6 symmetric points when on the six radial lines X=0, X=Y or X=-Y, and on the lines Y=0, X=3*Y or X=-3*Y which are midway between them. There's 12 symmetric points for anything else, ie. anything in the twelve slices between those twelve lines. The first set of 12 equal is N=20 to N=31 all at sqrt(28).

There can also be further ways for the same distance to arise, as multiple solutions to X^2+3*Y^3=d^2, but the 6-way or 12-way symmetry means there's always a multiple of 6 or 12 in total.

## Odd Points

Option `points => "odd"` visits just the odd points, meaning sum X+Y odd, which is X,Y one odd the other even.

``````    points => "odd"
69                              5
66    50    45    44    49    65               4
58    40    28    25    27    39    57            3
54    32    20    12    11    19    31    53         2
36    16     6     3     5    15    35            1
46    24    10     2     1     9    23    43    <- Y=0
37    17     7     4     8    18    38           -1
55    33    21    13    14    22    34    56        -2
59    41    29    26    30    42    60           -3
67    51    47    48    52    68              -4
70                             -5

^
-6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6``````

## All Points

Option `points => "all"` visits all integer X,Y points.

``````    points => "all"

64 59 49 44 48 58 63                  3
69 50 39 30 25 19 24 29 38 47 68            2
51 35 20 13  8  4  7 12 18 34 46            1
65 43 31 17  9  3  1  2  6 16 28 42 62    <- Y=0
52 36 21 14 10  5 11 15 23 37 57           -1
70 53 40 32 26 22 27 33 41 56 71           -2
66 60 54 45 55 61 67                 -3

^
-6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6``````

## Hex Points

Option `points => "hex"` visits X,Y points making a hexagonal grid,

``````    points => "hex"

50----42          49----59                    5
/        \        /        \
51----39          27----33          48                 4
/        \        /        \        /
43          22----15          21----32                    3
\        /        \        /        \
28----16           6----11          26----41           2
/        \        /        \        /        \
52----34           7---- 3           5----14          47        1
/        \        /        \        /        \        /
60          23----12           1-----2          20----38      <- Y=0
\        /        \        /        \        /        \
53----35           8---- 4          10----19          58       -1
\        /        \        /        \        /
29----17           9----13          31----46          -2
/        \        /        \        /
44          24----18          25----37                   -3
\        /        \        /        \
54----40          30----36          57                -4
\        /        \        /
55----45          56----61                   -5

^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9``````

N=1 is at the origin X=0,Y=0, then N=2,3,4 are all at X^2+3Y^2=4 away from the origin, etc. The joining lines drawn above show the grid pattern but points are in order of distance from the origin.

The points are all integer X,Y with X+3Y mod 6 == 0 or 2. This is a subset of the default "even" points in that X+Y is even but with 1 of each 3 points skipped to make the hexagonal outline.

## Hex Rotated Points

Option `points => "hex_rotated"` is the same hexagonal points but rotated around so N=2 is at +60 degrees instead of on the X axis.

``````    points => "hex_rotated"

60----50          42----49                             5
/        \        /        \
51          33----27          38----48                    4
\        /        \        /        \
34----22          15----21          41                 3
/        \        /        \        /
43----28          12-----6          14----26                    2
/        \        /        \        /        \
52          16-----7           2-----5          32----47           1
\        /        \        /        \        /        \
39----23           3-----1          11----20          59   <- Y=0
/        \        /        \        /        \        /
53          17-----8           4----10          37----58          -1
\        /        \        /        \        /
44----29          13-----9          19----31                   -2
\        /        \        /        \
35----24          18----25          46                -3
/        \        /        \        /
54          36----30          40----57                   -4
\        /        \        /
61----55          45----56                            -5

^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9``````

Points are still numbered from the X axis clockwise. The sets of points at equal hypotenuse distances are the same as plain "hex" but the numbering is changed by the rotation.

The points visited are all integer X,Y with X+3Y mod 6 == 0 or 4. This grid can be viewed either as a +60 degree or a +180 degree rotation of the plain hex.

## Hex Centred Points

Option `points => "hex_centred"` is the same hexagonal grid as hex above, but with the origin X=0,Y=0 in the centre of a hexagon,

``````    points => "hex_centred"

46----45                              5
/        \
39----28          27----38                     4
/        \        /        \
47----29          16----15          26----44            3
/        \        /        \        /        \
48          17-----9           8----14          43         2
\        /        \        /        \        /
30----18           3-----2          13----25            1
/        \        /        \        /        \
40          10-----4     .     1-----7          37    <- Y=0
\        /        \        /        \        /
31----19           5-----6          24----36           -1
/        \        /        \        /        \
49          20----11          12----23          54        -2
\        /        \        /        \        /
50----32          21----22          35----53           -3
\        /        \        /
41----33          34----42                    -4
\        /
51----52                             -5

^
-8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9``````

N=1,2,3,4,5,6 are all at X^2+3Y^2=4 away from the origin, then N=7,8,9,10,11,12, etc. The points visited are all integer X,Y with X+3Y mod 6 == 2 or 4.

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::TriangularHypot->new ()`
`\$path = Math::PlanePath::TriangularHypot->new (points => \$str)`

Create and return a new hypot path object. The `points` option can be

``````    "even"          only points with X+Y even (the default)
"odd"           only points with X+Y odd
"all"           all integer X,Y
"hex"           hexagonal X+3Y==0,2 mod 6
"hex_rotated"   hexagonal X+3Y==0,4 mod 6
"hex_centred"   hexagonal X+3Y==2,4 mod 6``````

Create and return a new triangular hypot path object.

`(\$x,\$y) = \$path->n_to_xy (\$n)`

Return the X,Y coordinates of point number `\$n` on the path.

For `\$n < 1` the return is an empty list as the first point at X=0,Y=0 is N=1.

Currently it's unspecified what happens if `\$n` is not an integer. Successive points are a fair way apart, so it may not make much sense to say give an X,Y position in between the integer `\$n`.

`\$n = \$path->xy_to_n (\$x,\$y)`

Return an integer point number for coordinates `\$x,\$y`. Each integer N is considered the centre of a unit square and an `\$x,\$y` within that square returns N.

For "even" and "odd" options only every second square in the plane has an N and if `\$x,\$y` is a position not covered then the return is `undef`.

# OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include,

``````    points="even" (the default)
A003136  norms (X^2+3*Y^2)/4 which occur
A004016  count of points of norm==n
A035019    skipping zero counts
A088534    counting only in the twelfth 0<=X<=Y``````

The counts in these sequences are expressed as norm = x^2+x*y+y^2. That x,y is related to the "even" X,Y on the path here by a -45 degree rotation,

``````    x = (Y-X)/2           X = 2*(x+y)
y = (X+Y)/2           Y = 2*(y-x)

norm = x^2+x*y+y^2
= ((Y-X)/2)^2 + (Y-X)/2 * (X+Y)/2 + ((X+Y)/2)^2
= (X^2 + 3*Y^2) / 4``````

The X^2+3*Y^2 is the dist^2 described above for equilateral triangles of unit side. The factor of /4 scales the distance but of course doesn't change the sets of points of the same distance.

``````    points="all"
A092572  norms X^2+3*Y^2 which occur
A158937  norms X^2+3*Y^2 which occur, X>0,Y>0 with repeats
A092573  count of points norm==n for X>0,Y>0

A092574  norms X^2+3*Y^2 which occur for X>0,Y>0, gcd(X,Y)=1
A092575  count of points norm==n for X>0,Y>0, gcd(X,Y)=1
ie. X,Y no common factor``````
``````    points="hex"
A113062  count of points norm=X^2+3*Y^2=4*n (theta series)
A113063   divided by 3

points="hex_centred"
A217219  count of points norm=X^2+3*Y^2=4*n (theta series)``````

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