--- name: "Math::PlanePath::Corner" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::Corner" --- # Math::PlanePath::Corner(3pm) ## NAME Math::PlanePath::Corner -- points shaped around in a corner ## SYNOPSIS ``` use Math::PlanePath::Corner; my $path = Math::PlanePath::Corner->new; my ($x, $y) = $path->n_to_xy (123); ``` ## DESCRIPTION This path puts points in layers working outwards from the corner of the first quadrant. ``` 5 | 26--... | 4 | 17--18--19--20--21 | | 3 | 10--11--12--13 22 | | | 2 | 5-- 6-- 7 14 23 | | | | 1 | 2-- 3 8 15 24 | | | | | | Y=0 | 1 4 9 16 25 +--------------------- X=0 1 2 3 4 ``` The horizontal 1,4,9,16,etc along Y=0 is the perfect squares. This is since each further row/column "gnomon" added to a square makes a one-bigger square, ``` 10 11 12 13 5 6 7 5 6 7 14 2 3 2 3 8 2 3 8 15 1 4 1 4 9 1 4 9 16 2x2 3x3 4x4 ``` N=2,6,12,20,etc on the diagonal X=Y-1 up from X=0,Y=1 is the pronic numbers k\*(k+1) which are half way between the squares. Each gnomon is 2 longer than the previous. This is similar to the `"PyramidRows"`, `"PyramidSides"` and `"SacksSpiral"` paths. The `"Corner"` and the `"PyramidSides"` are the same but `"PyramidSides"` is stretched to two quadrants instead of one for the `"Corner"` here. ### Wider An optional `"wider => $integer"` makes the path wider horizontally, becoming a rectangle. For example ``` wider => 3 4 | 29--30--31--... | 3 | 19--20--21--22--23--24--25 | | 2 | 11--12--13--14--15--16 26 | | | 1 | 5---6---7---8---9 17 27 | | | | Y=0 | 1---2---3---4 10 18 28 | ----------------------------- ^ X=0 1 2 3 4 5 6 ``` Each gnomon has the horizontal part `"wider"` many steps longer. Each gnomon is still 2 longer than the previous since this widening is a constant amount in each. ### N Start The default is to number points starting N=1 as shown above. An optional `"n_start"` can give a different start with the same shape etc. For example to start at 0, ``` n_start => 0 5 | 25 ... 4 | 16 17 18 19 20 3 | 9 10 11 12 21 2 | 4 5 6 13 22 1 | 1 2 7 14 23 Y=0 | 0 3 8 15 24 ----------------- X=0 1 2 3 ``` In Nstart=0, the squares are on the Y axis and the pronic numbers are on the X=Y leading diagonal. ## FUNCTIONS See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes. - "$path = Math::PlanePath::Corner-\>new ()" - "$path = Math::PlanePath::Corner-\>new (wider =\> $w, n\_start =\> $n)" Create and return a new path object. - "($x,$y) = $path-\>n\_to\_xy ($n)" Return the X,Y coordinates of point number `$n` on the path. For `"$n < n_start()-0.5"` the return is an empty list. There's an extra 0.5 before Nstart, but nothing further before there. - "$n = $path-\>xy\_to\_n ($x,$y)" Return the point number for coordinates `"$x,$y"`. `$x` and `$y` are each rounded to the nearest integer, which has the effect of treating each point as a square of side 1, so the quadrant x\>=-0.5 and y\>=-0.5 is entirely covered. - "($n\_lo, $n\_hi) = $path-\>rect\_to\_n\_range ($x1,$y1, $x2,$y2)" The returned range is exact, meaning `$n_lo` and `$n_hi` are the smallest and biggest in the rectangle. ## FORMULAS ### N to X,Y Counting d=0 for the first L-shaped gnomon at Y=0, then the start of the gnomon is ``` StartN(d) = d^2 + 1 = 1,2,5,10,17,etc ``` The current `"n_to_xy()"` code extends to the left by an extra 0.5 for fractional N, so for example N=9.5 is at X=-0.5,Y=3. With this the starting N for each gnomon d is ``` StartNfrac(d) = d^2 + 0.5 ``` Inverting gives the gnomon d number for an N, ``` d = floor(sqrt(N - 0.5)) ``` Subtracting the gnomon start gives an offset into that gnomon ``` OffStart = N - StartNfrac(d) ``` The corner point 1,3,7,13,etc where the gnomon turns down is at d+0.5 into that remainder, and it's convenient to subtract that so negative for the horizontal and positive for the vertical, ``` Off = OffStart - (d+0.5) = N - (d*(d+1) + 1) ``` Then the X,Y coordinates are ``` if (Off < 0) then X=d+Off, Y=d if (Off >= 0) then X=d, Y=d-Off ``` ### X,Y to N For a given X,Y the bigger of X or Y determines the d gnomon. If Y\>=X then X,Y is on the horizontal part. At X=0 have N=StartN(d) per the Start(N) formula above, and any further X is an offset from there. ``` if Y >= X then d=Y N = StartN(d) + X = Y^2 + 1 + X ``` Otherwise if Y\ N increasing | ----------------------- ``` Going up a column, N values are increasing away from the X=Y diagonal up or down, and all N values above X=Y are bigger than the ones below. ``` | ^ N increasing up from X=Y diagonal | | | |/ | / | /| | / | N increasing down from X=Y diagonal | / v |/ ----------------------- ``` This means the biggest N is the top right corner if that corner is Y\>=X, otherwise the bottom right corner. ``` max N at top right | / | --+ if corner Y>=X | / --+ | | / | / | | |/ | / | | | | / ----v | /| | / max N at bottom right | --+ |/ if corner Y<=X |/ ---------- ------- ``` For the smallest N, if the bottom left corner has Y\>X then it's in the "increasing" part and that bottom left corner is the smallest N. Otherwise Y\<=X means some of the "decreasing" part is covered and the smallest N is at Y=min(X,Ymax), ie. either the Y=X diagonal if it's in the rectangle or the top right corner otherwise. ``` | / | | / | | / min N at bottom left | +---- if corner Y>X | / | / |/ ---------- | / | / | | / | / | |/ min N at X=Y | / | * if diagonal crossed | / +-- min N at top left | /| | / | if corner Y= Xmin Xmin,min(Xmin,Ymax) if Ymin <= Xmin ``` ## OEIS This path is in Sloane's Online Encyclopedia of Integer Sequences as, \ (etc) ``` wider=0, n_start=1 (the defaults) A213088 X+Y sum A196199 X-Y diff, being runs -n to +n A053615 abs(X-Y), runs n to 0 to n, distance to next pronic A000290 N on X axis, perfect squares starting from 1 A002522 N on Y axis, Y^2+1 A002061 N on X=Y diagonal, extra initial 1 A004201 N on and below X=Y diagonal, so X>=Y A020703 permutation N at transpose Y,X A060734 permutation N by diagonals up from X axis A064790 inverse A060736 permutation N by diagonals down from Y axis A064788 inverse A027709 boundary length of N unit squares A078633 grid sticks of N points n_start=0 A000196 max(X,Y), being floor(sqrt(N)) A005563 N on X axis, n*(n+2) A000290 N on Y axis, perfect squares A002378 N on X=Y diagonal, pronic numbers n_start=2 A059100 N on Y axis, Y^2+2 A014206 N on X=Y diagonal, pronic+2 wider=1 A053188 abs(X-Y), dist to nearest square, extra initial 0 wider=1, n_start=0 A002378 N on Y axis, pronic numbers A005563 N on X=Y diagonal, n*(n+2) wider=1, n_start=2 A014206 N on Y axis, pronic+2 wider=2, n_start=1 A028387 N on X=Y diagonal, k*(k+3) + 1 wider=2, n_start=0 A005563 N on Y axis, (Y+1)^2-1 A028552 N on X=Y diagonal, k*(k+3) wider=3, n_start=0 A028552 N on Y axis, k*(k+3) ``` ## SEE ALSO Math::PlanePath, Math::PlanePath::PyramidSides, Math::PlanePath::PyramidRows, Math::PlanePath::SacksSpiral, Math::PlanePath::Diagonals ## HOME PAGE \ ## LICENSE Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 Kevin Ryde This file is part of Math-PlanePath. Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see \.