1Math::NumSeq::PlanePathUCsoeorrdC(o3n)tributed Perl DocuMmaetnht:a:tNiuomnSeq::PlanePathCoord(3)
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NAME

6       Math::NumSeq::PlanePathCoord -- sequence of coordinate values from a
7       PlanePath module
8

SYNOPSIS

10        use Math::NumSeq::PlanePathCoord;
11        my $seq = Math::NumSeq::PlanePathCoord->new
12                    (planepath => 'SquareSpiral',
13                     coordinate_type => 'X');
14        my ($i, $value) = $seq->next;
15

DESCRIPTION

17       This is a tie-in to make a "NumSeq" sequence giving coordinate values
18       from a "Math::PlanePath".  The NumSeq "i" index is the PlanePath "N"
19       value.
20
21       The "coordinate_type" choices are as follows.  Generally they have some
22       sort of geometric interpretation or are related to fractions X/Y.
23
24           "X"            X coordinate
25           "Y"            Y coordinate
26           "Min"          min(X,Y)
27           "Max"          max(X,Y)
28           "MinAbs"       min(abs(X),abs(Y))
29           "MaxAbs"       max(abs(X),abs(Y))
30           "Sum"          X+Y sum
31           "SumAbs"       abs(X)+abs(Y) sum
32           "Product"      X*Y product
33           "DiffXY"       X-Y difference
34           "DiffYX"       Y-X difference (negative of DiffXY)
35           "AbsDiff"      abs(X-Y) difference
36           "Radius"       sqrt(X^2+Y^2) radial distance
37           "RSquared"     X^2+Y^2 radius squared (norm)
38           "TRadius"      sqrt(X^2+3*Y^2) triangular radius
39           "TRSquared"    X^2+3*Y^2 triangular radius squared (norm)
40           "IntXY"        int(X/Y) division rounded towards zero
41           "FracXY"       frac(X/Y) division rounded towards zero
42           "BitAnd"       X bitand Y
43           "BitOr"        X bitor Y
44           "BitXor"       X bitxor Y
45           "GCD"          greatest common divisor X,Y
46           "Depth"        tree_n_to_depth()
47           "SubHeight"    tree_n_to_subheight()
48           "NumChildren"  tree_n_num_children()
49           "NumSiblings"  not including self
50           "RootN"        the N which is the tree root
51           "IsLeaf"       0 or 1 whether a leaf node (no children)
52           "IsNonLeaf"    0 or 1 whether a non-leaf node (has children)
53                            also called an "internal" node
54
55   Min and Max
56       "Min" and "Max" are the minimum or maximum of X and Y.  The geometric
57       interpretation of "Min" is to select X at any point above the X=Y
58       diagonal or Y for any point below.  Conversely "Max" is Y above and X
59       below.  On the X=Y diagonal itself X=Y=Min=Max.
60
61           Max=Y      / X=Y diagonal
62           Min=X   | /
63                   |/
64                ---o----
65                  /|
66                 / |     Max=X
67                /        Min=Y
68
69       Min and Max can also be interpreted as counting which gnomon shaped
70       line the X,Y falls on.
71
72           | | | |     Min=gnomon           2 ------------.  Max=gnomon
73           | | | |                          1 ----------. |
74           | | | |      ...                 0 --------o | |
75           | | |  ------ 1                 -1 ------. | | |
76           | | o-------- 0                 ...      | | | |
77           |  ---------- -1                         | | | |
78            ------------ -2                         | | | |
79
80   MinAbs
81       MinAbs = min(abs(X),abs(Y)) can be interpreted geometrically as
82       counting gnomons successively away from the origin.  This is like Min
83       above, but within the quadrant containing X,Y.
84
85                | | | | |          MinAbs=gnomon counted away from the origin
86                | | | | |
87           2 ---  | | |  ---- 2
88           1 -----  |  ------ 1
89           0 -------o-------- 0
90           1 -----  |  ------ 1
91           2 ---  | | |  ---- 2
92                | | | | |
93                | | | | |
94
95   MaxAbs
96       MaxAbs = max(abs(X),abs(Y)) can be interpreted geometrically as
97       counting successive squares around the origin.
98
99           +-----------+       MaxAbs=which square
100           | +-------+ |
101           | | +---+ | |
102           | | | o | | |
103           | | +---+ | |
104           | +-------+ |
105           +-----------+
106
107       For example Math::PlanePath::SquareSpiral loops around in squares and
108       so its MaxAbs is unchanged until it steps out to the next bigger
109       square.
110
111   Sum and Diff
112       "Sum"=X+Y and "DiffXY"=X-Y can be interpreted geometrically as
113       coordinates on 45-degree diagonals.  Sum is a measure up along the
114       leading diagonal and DiffXY down an anti-diagonal,
115
116           \           /
117            \   s=X+Y /
118             \       ^\
119              \     /  \
120               \ | /    v
121                \|/      * d=X-Y
122              ---o----
123                /|\
124               / | \
125              /  |  \
126             /       \
127            /         \
128           /           \
129
130       Or "Sum" can be thought of as a count of which anti-diagonal stripe
131       contains X,Y, or a projection onto the X=Y leading diagonal.
132
133                  Sum
134           \     = anti-diag
135            2      numbering          / / / /   DiffXY
136           \ \       X+Y            -1 0 1 2   = diagonal
137            1 2                     / / / /      numbering
138           \ \ \                  -1 0 1 2         X-Y
139            0 1 2                   / / /
140             \ \ \                 0 1 2
141
142   DiffYX
143       "DiffYX" = Y-X is simply the negative of DiffXY.  It's included to give
144       positive values on paths which are above the X=Y leading diagonal.  For
145       example DiffXY is positive in "CoprimeColumns" which is below X=Y,
146       whereas DiffYX is positive in "CellularRule" which is above X=Y.
147
148   SumAbs
149       "SumAbs" = abs(X)+abs(Y) is similar to the projection described above
150       for Sum or Diff, but SumAbs projects onto the central diagonal of
151       whichever quadrant contains the X,Y.  Or equivalently it's a numbering
152       of anti-diagonals within that quadrant, so numbering which diamond
153       shape the X,Y falls on.
154
155                |
156               /|\       SumAbs = which diamond X,Y falls on
157              / | \
158             /  |  \
159           -----o-----
160             \  |  /
161              \ | /
162               \|/
163                |
164
165       As an example, the "DiamondSpiral" path loops around on such diamonds,
166       so its SumAbs is unchanged until completing a loop and stepping out to
167       the next bigger.
168
169       SumAbs is also a "taxi-cab" or "Manhattan" distance, being how far to
170       travel through a square-grid city to get to X,Y.
171
172           SumAbs = taxi-cab distance, by any square-grid travel
173
174           +-----o       +--o          o
175           |             |             |
176           |          +--+       +-----+
177           |          |          |
178           *          *          *
179
180       If a path is entirely X>=0,Y>=0 in the first quadrant then Sum and
181       SumAbs are identical.
182
183   AbsDiff
184       "AbsDiff" = abs(X-Y) can be interpreted geometrically as the distance
185       away from the X=Y diagonal, measured at right-angles to that line.
186
187            d=abs(X-Y)
188                  ^    / X=Y line
189                   \  /
190                    \/
191                    /\
192                   /  \
193                 |/    \
194               --o--    \
195                /|       v
196               /           d=abs(X-Y)
197
198       If a path is entirely below the X=Y line, so X>=Y, then AbsDiff is the
199       same as DiffXY.  Or if a path is entirely above the X=Y line, so Y>=X,
200       then AbsDiff is the same as DiffYX.
201
202   Radius and RSquared
203       Radius and RSquared are per "$path->n_to_radius()" and
204       "$path->n_to_rsquared()" respectively (see "Coordinate Methods" in
205       Math::PlanePath).
206
207   TRadius and TRSquared
208       "TRadius" and "TRSquared" are designed for use with points on a
209       triangular lattice as per "Triangular Lattice" in Math::PlanePath.  For
210       points on the X axis TRSquared is the same as RSquared but off the axis
211       Y is scaled up by factor sqrt(3).
212
213       Most triangular paths use "even" points X==Y mod 2 and for them
214       TRSquared is always even.  Some triangular paths such as "KochPeaks"
215       have an offset from the origin and use "odd" points X!=Y mod 2 and for
216       them TRSquared is odd.
217
218   IntXY and FracXY
219       "IntXY" = int(X/Y) is the quotient from X divide Y rounded to an
220       integer towards zero.  This is like the integer part of a fraction, for
221       example X=9,Y=4 is 9/4 = 2+1/4 so IntXY=2.  Negatives are reckoned with
222       the fraction part negated too, so -2 1/4 is -2-1/4 and thus IntXY=-2.
223
224       Geometrically IntXY gives which wedge of slope 1, 2, 3, etc the point
225       X,Y falls in.  For example IntXY is 3 for all points in the wedge
226       3Y<=X<4Y.
227
228                                      X=Y    X=2Y   X=3Y   X=4Y
229           *  -2  *  -1  *   0  |  0   *  1   *  2   *   3  *
230              *     *     *     |     *     *     *     *
231                 *    *    *    |    *    *    *    *
232                    *   *   *   |   *   *   *   *
233                       *  *  *  |  *  *  *  *
234                          * * * | * * * *
235                             ***|****
236           ---------------------+----------------------------
237                              **|**
238                            * * | * *
239                          *  *  |  *  *
240                        *   *   |   *   *
241                      *    *    |    *    *
242                2   *  1  *  0  |  0  * -1  *  -2
243
244       "FracXY" is the fraction part which goes with IntXY.  In all cases
245
246           X/Y = IntXY + FracXY
247
248       IntXY rounds towards zero so the remaining FracXY has the same sign as
249       IntXY.
250
251   BitAnd, BitOr, BitXor
252       "BitAnd", "BitOr" and "BitXor" treat negative X or negative Y as
253       infinite twos-complement 1-bits, which means for example X=-1,Y=-2 has
254       X bitand Y = -2.
255
256           ...11111111    X=-1
257           ...11111110    Y=-2
258           -----------
259           ...11111110    X bitand Y = -2
260
261       This twos-complement is per "Math::BigInt" (which has bitwise
262       operations in Perl 5.6 and up).  The code here arranges the same on
263       ordinary scalars.
264
265       If X or Y are not integers then the fractional parts are treated
266       bitwise too, but currently only to limited precision.
267

FUNCTIONS

269       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
270       classes.
271
272       "$seq = Math::NumSeq::PlanePathCoord->new (planepath => $name,
273       coordinate_type => $str)"
274           Create and return a new sequence object.  The options are
275
276               planepath          string, name of a PlanePath module
277               planepath_object   PlanePath object
278               coordinate_type    string, as described above
279
280           "planepath" can be either the module part such as "SquareSpiral" or
281           a full class name "Math::PlanePath::SquareSpiral".
282
283       "$value = $seq->ith($i)"
284           Return the coordinate at N=$i in the PlanePath.
285
286       "$i = $seq->i_start()"
287           Return the first index $i in the sequence.  This is the position
288           rewind() returns to.
289
290           This is "$path->n_start()" from the PlanePath, since the i
291           numbering is the N numbering of the underlying path.  For some of
292           the "Math::NumSeq::OEIS" generated sequences there may be a higher
293           i_start() corresponding to a higher starting point in the OEIS,
294           though this is slightly experimental.
295
296       "$str = $seq->oeis_anum()"
297           Return the A-number (a string) for $seq in Sloane's Online
298           Encyclopedia of Integer Sequences, or return "undef" if not in the
299           OEIS or not known.
300
301           Known A-numbers are also presented through
302           "Math::NumSeq::OEIS::Catalogue".  This means PlanePath related OEIS
303           sequences can be created with "Math::NumSeq::OEIS" by giving their
304           A-number in the usual way for that module.
305

SEE ALSO

307       Math::NumSeq, Math::NumSeq::PlanePathDelta,
308       Math::NumSeq::PlanePathTurn, Math::NumSeq::PlanePathN,
309       Math::NumSeq::OEIS
310
311       Math::PlanePath
312

HOME PAGE

314       <http://user42.tuxfamily.org/math-planepath/index.html>
315

LICENSE

317       Copyright 2011, 2012, 2013, 2014 Kevin Ryde
318
319       This file is part of Math-PlanePath.
320
321       Math-PlanePath is free software; you can redistribute it and/or modify
322       it under the terms of the GNU General Public License as published by
323       the Free Software Foundation; either version 3, or (at your option) any
324       later version.
325
326       Math-PlanePath is distributed in the hope that it will be useful, but
327       WITHOUT ANY WARRANTY; without even the implied warranty of
328       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
329       General Public License for more details.
330
331       You should have received a copy of the GNU General Public License along
332       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
333
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335
336perl v5.36.0                      2023-01-20   Math::NumSeq::PlanePathCoord(3)
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