1Math::NumSeq::PlanePathUDseelrtaC(o3n)tributed Perl DocuMmaetnht:a:tNiuomnSeq::PlanePathDelta(3)
2
6 Math::NumSeq::PlanePathDelta -- sequence of changes and directions of
7 PlanePath coordinates
8
10 use Math::NumSeq::PlanePathDelta;
11 my $seq = Math::NumSeq::PlanePathDelta->new
12 (planepath => 'SquareSpiral',
13 delta_type => 'dX');
14 my ($i, $value) = $seq->next;
15
17 This is a tie-in to present coordinate changes and directions from a
18 "Math::PlanePath" module in the form of a NumSeq sequence.
19 The "delta_type" choices are
21 "dX" change in X coordinate
23 "dY" change in Y coordinate
24 "AbsdX" abs(dX)
25 "AbsdY" abs(dY)
26 "dSum" change in X+Y, equals dX+dY
27 "dSumAbs" change in abs(X)+abs(Y)
28 "dDiffXY" change in X-Y, equals dX-dY
29 "dDiffYX" change in Y-X, equals dY-dX
30 "dAbsDiff" change in abs(X-Y)
31 "dRadius" change in Radius sqrt(X^2+Y^2)
32 "dRSquared" change in RSquared X^2+Y^2
33 "Dir4" direction 0=East, 1=North, 2=West, 3=South
34 "TDir6" triangular 0=E, 1=NE, 2=NW, 3=W, 4=SW, 5=SE
35 In each case the value at i is per "$path->n_to_dxdy($i)", being the
37 change from N=i to N=i+1, or from N=i to N=i+arms for paths with
38 multiple "arms" (thus following the arm). i values start from the
39 usual "$path->n_start()".
40 AbsdX,AbsdY
42 If a path always steps NSEW by 1 then AbsdX and AbsdY behave as a
43 boolean indicating horizontal or vertical step,
44 NSEW steps by 1 gives
46 AbsdX = 0 vertical AbsdY = 0 horizontal
48 1 horizontal 1 vertical
49 If a path includes diagonal steps by 1 then those diagonals are a non-
51 zero delta, so the indication is then
52 NSEW and diagonals steps by 1 gives
54 AbsdX = 0 vertical AbsdY = 0 horizontal
56 1 non-vertical 1 non-horizontal
57 ie. horiz or diag ie. vert or diag
58 dSum
60 "dSum" is the change in X+Y and is also simply dX+dY since
61 dSum = (Xnext+Ynext) - (X+Y)
63 = (Xnext-X) + (Ynext-Y)
64 = dX + dY
65 The sum X+Y counts anti-diagonals, as described in
67 Math::NumSeq::PlanePathCoord. dSum is therefore a move between
68 diagonals, or 0 if a step stays within the same diagonal.
69 \
71 \ ^ dSum > 0 dSum = step dist to North-East
72 \/
73 /\
74 dSum < 0 v \
75 \
76 dSumAbs
78 "dSumAbs" is the change in the abs(X)+abs(Y) sum,
79 dSumAbs = (abs(Xnext)+abs(Ynext)) - (abs(X)+abs(Y))
81 As described in "SumAbs" in Math::NumSeq::PlanePathCoord, SumAbs is a
83 "Manhattan" or "taxi-cab" distance from the origin, or equivalently a
84 move between diamond-shaped rings.
85 For example "DiamondSpiral" follows a diamond shape ring around and so
87 has dSumAbs=0 until stepping out to each next diamond with dSumAbs=1.
88 A path might make a big X,Y jump which is only a small change in
90 SumAbs. For example "PyramidRows" in its default step=2 from the end
91 of one row to the start of the next has dSumAbs=2.
92 dDiffXY and dDiffYX
94 "dDiffXY" is the change in DiffXY = X-Y, which is also simply dX-dY
95 since
96 dDiffXY = (Xnext-Ynext) - (X-Y)
98 = (Xnext-X) - (Ynext-Y)
99 = dX - dY
100 The difference X-Y counts diagonals downwards to the south-east as
102 described in "Sum and Diff" in Math::NumSeq::PlanePathCoord. dDiffXY
103 is therefore movement between those diagonals, or 0 if a step stays
104 within the same diagonal.
105 dDiffXY < 0 /
107 \ / dDiffXY = step dist to South-East
108 \/
109 /\
110 / v
111 / dDiffXY > 0
112 "dDiffYX" is the negative of dDiffXY. Whether X-Y or Y-X is desired
114 depends on which way you want to measure diagonals, or which way around
115 to have the sign for the changes. dDiffYX is based on Y-X and so
116 counts diagonals upwards to the North-West.
117 dAbsDiff
119 "dAbsDiff" is the change in AbsDiff = abs(X-Y). AbsDiff can be
120 interpreted geometrically as distance from the leading diagonal, as
121 described in "AbsDiff" in Math::NumSeq::PlanePathCoord. dAbsDiff is
122 therefore movement closer to or further away from that leading
123 diagonal, measuring perpendicular to it.
124 / X=Y line
126 /
127 / ^
128 / \
129 / * dAbsDiff move towards or away from X=Y line
130 |/ \
131 --o-- v
132 /|
133 /
134 When an X,Y jumps from one side of the diagonal to the other dAbsDiff
136 is still the change in distance from the diagonal. So for example if
137 X,Y is followed by the mirror point Y,X then dAbsDiff=0. That sort of
138 thing happens for example in the "Diagonals" path when jumping from the
139 end of one run to the start of the next. In the "Diagonals" case it's
140 a move just 1 further away from the X=Y centre line even though it's a
141 big jump in overall distance.
142 dRadius, dRSquared
144 "dRadius" and "dRSquared" are the change in the Radius and RSquared as
145 described in "Radius and RSquared" in Math::NumSeq::PlanePathCoord.
146 dRadius = next_Radius - Radius
148 dRSquared = next_RSquared - RSquared
149 dRadius can be interpreted geometrically as movement towards (negative
151 values) or away from (positive values) the origin, ignoring direction.
152 Notice that dRadius is not sqrt(dRSquared), since sqrt(n^2-t^2) != n-t
154 unless n or t is zero. Here would mean a step either going to or
155 coming from the origin 0,0.
156 Dir4
158 "Dir4" is the curve step direction as an angle in the range 0 <= Dir4 <
159 4. The cardinal directions E,N,W,S are 0,1,2,3. Angles in between are
160 a fraction.
161 Dir4 = atan2(dY,dX) scaled as range 0 <= Dir4 < 4
163 1.5 1 0.5
165 \ | /
166 \|/
167 2 ----o---- 0
168 /|\
169 / | \
170 2.5 3 3.5
171 If a row such as Y=-1,X>0 just below the X axis is visited then the
173 Dir4 approaches 4, without ever reaching it. The
174 "$seq->value_maximum()" is 4 in this case, as a supremum.
175 TDir6
177 "TDir6" is the curve step direction 0 <= TDir6 < 6 taken in the
178 triangular style of "Triangular Lattice" in Math::PlanePath. So
179 dX=1,dY=1 is taken to be 60 degrees which is TDir6=1.
180 2 1.5 1 TDir6
182 \ | /
183 \|/
184 3 ---o--- 0
185 /|\
186 / | \
187 4 4.5 5
188 Angles in between the six cardinal directions are fractions. North is
190 1.5 and South is 4.5.
191 The direction angle is calculated as if dY was scaled by a factor
193 sqrt(3) to make the lattice into equilateral triangles, or equivalently
194 as a circle stretched vertically by sqrt(3) to become an ellipse.
195 TDir6 = atan2(dY*sqrt(3), dX) in range 0 <= TDir6 < 6
197 Notice that angles on the axes dX=0 or dY=0 are not changed by the
199 sqrt(3) factor. So TDir6 has ENWS 0, 1.5, 3, 4.5 which is steps of
200 1.5. Verticals North and South normally don't occur in the triangular
201 lattice paths which go by unit steps, but TDir6 can be applied on any
202 path.
203 The sqrt(3) factor increases angles in the middle of the quadrants.
205 For example dX=1,dY=1 becomes TDir6=1 whereas a plain angle would be
206 only 45/360*6=0.75 in the same 0 to 6 scale. The sqrt(3) is a
207 continuous scaling, so a plain angle and a TDir6 are a one-to-one
208 mapping. As the direction progresses through the quadrant TDir6 grows
209 first faster and then slower than the plain angle.
210
212 See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
213 classes.
214 "$seq = Math::NumSeq::PlanePathDelta->new (key=>value,...)"
216 Create and return a new sequence object. The options are
217 planepath string, name of a PlanePath module
219 planepath_object PlanePath object
220 delta_type string, as described above
221 "planepath" can be either the module part such as "SquareSpiral" or
223 a full class name "Math::PlanePath::SquareSpiral".
224 "$value = $seq->ith($i)"
226 Return the change at N=$i in the PlanePath.
227 "$i = $seq->i_start()"
229 Return the first index $i in the sequence. This is the position
230 "$seq->rewind()" returns to.
231 This is "$path->n_start()" from the PlanePath.
233
235 Some path sequences don't have oeis_anum() and are not available
236 through Math::NumSeq::OEIS entry due to the path n_start() not matching
237 the OEIS "offset". Paths with an "n_start" parameter have suitable
238 adjustments applied, but those without are omitted from the
239 Math::NumSeq::OEIS mechanism presently.
240
242 Math::NumSeq, Math::NumSeq::PlanePathCoord,
243 Math::NumSeq::PlanePathTurn, Math::NumSeq::PlanePathN
244 Math::PlanePath
246
248 <http://user42.tuxfamily.org/math-planepath/index.html>
249
251 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020,
252 2021 Kevin Ryde
253 This file is part of Math-PlanePath.
255 Math-PlanePath is free software; you can redistribute it and/or modify
257 it under the terms of the GNU General Public License as published by
258 the Free Software Foundation; either version 3, or (at your option) any
259 later version.
260 Math-PlanePath is distributed in the hope that it will be useful, but
262 WITHOUT ANY WARRANTY; without even the implied warranty of
263 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
264 General Public License for more details.
265 You should have received a copy of the GNU General Public License along
267 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
268perl v5.36.0 2023-01-20 Math::NumSeq::PlanePathDelta(3)