#!/usr/bin/gnuplot
# Copyright 2012 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
# set terminal png
# set terminal xterm
n_to_xy(n) = \
(n <= 1 ? n : nd_to_xy(n, n_to_d(n)))
nd_to_xy(n,d) = \
drem_to_xy(d, n - d_to_base(d))
n_to_d(n) = int ((2 + sqrt(int(4*n))) / 4)
d_to_base(d) = 4 * d**2
# right vertical
# top horizontal
# left vertical
# bottom horizontal
#
drem_to_xy(d,r) = \
(r < -2*d ? d + (r+3*d)*{0,1} \
: r < 0 ? -r + d*{-1,1} \
: r < 2*d ? -d + (d-r)*{0,1} \
: r-2*d + d*{-1,-1})
print n_to_d(0)
print n_to_d(1)
print n_to_d(2)
print n_to_d(3)
print n_to_d(4)
print n_to_d(5)
print ''
print n_to_xy(0)
print n_to_xy(1)
print n_to_xy(2)
print n_to_xy(3)
print n_to_xy(4)
print n_to_xy(5)
print n_to_xy(6)
print n_to_xy(7)
print n_to_xy(8)
print n_to_xy(9)
# set xrange [0:36]
# plot real(n_to_xy(x))
# pause 100
# set xrange [0:36]
# plot x-d_to_base(n_to_d(x))
# pause 100
length=49
set trange [0:length]
set samples 5*length+1
set parametric
set key off
plot real(n_to_xy(t)),imag(n_to_xy(t)) with labels
pause 100
# # Return the position of the highest 1-bit in n.
# # The least significant bit is position 0.
# # For example n=13 is binary "1101" and the high bit is pos=3.
# # If n==0 then the return is 0.
# # Arranging the test as n>=2 avoids infinite recursion if n==NaN (any
# # comparison involving NaN is always false).
# #
# high_bit_pos(n) = (n>=2 ? 1+high_bit_pos(int(n/2)) : 0)
#
# # Return 0 or 1 for the bit at position "pos" in n.
# # pos==0 is the least significant bit.
# #
# bit(n,pos) = int(n / 2**pos) & 1
#
# # dragon(n) returns a complex number which is the position of the
# # dragon curve at integer point "n". n=0 is the first point and is at
# # the origin {0,0}. Then n=1 is at {1,0} which is x=1,y=0, etc. If n
# # is not an integer then the point returned is for int(n).
# #
# # The calculation goes by bits of n from high to low. Gnuplot doesn't
# # have iteration in functions, but can go recursively from
# # pos=high_bit_pos(n) down to pos=0, inclusive.
# #
# # mul() rotates by +90 degrees (complex "i") at bit transitions 0->1
# # or 1->0. add() is a vector (i+1)**pos for each 1-bit, but turned by
# # factor "i" when in a "reversed" section of curve, which is when the
# # bit above is also a 1-bit.
# #
# dragon(n) = dragon_by_bits(n, high_bit_pos(n))
# dragon_by_bits(n,pos) \
# = (pos>=0 ? add(n,pos) + mul(n,pos)*dragon_by_bits(n,pos-1) : 0)
#
# add(n,pos) = (bit(n,pos) ? (bit(n,pos+1) ? {0,1} * {1,1}**pos \
# : {1,1}**pos) \
# : 0)
# mul(n,pos) = (bit(n,pos) == bit(n,pos+1) ? 1 : {0,1})
#
# # Plot the dragon curve from 0 to "length" with line segments.
# # "trange" and "samples" are set so the parameter t runs through
# # integers t=0 to t=length inclusive.
# #
# # Any trange works, it doesn't have to start at 0. But must have
# # enough "samples" that all integers t in the range are visited,
# # otherwise vertices in the curve would be missed.
# #