Circle Trisection

circle3.rb

#!/usr/bin/env ruby

# ruby circle3.rb
#   creates Quadrant3.png, Circle3L.png, Circle3C.png, and Circle3Lrotate.png.

# rmagick required (gem install rmagick)

require "rmagick"
require "bigdecimal"
require "bigdecimal/newton"

BigDecimal.limit(6)

# http://ipafont.ipa.go.jp/#en
$font = ENV.key?("FONT") ? ENV["FONT"] : "ipaexg.ttf"

class Arcsolver
  # Solve \frac{1}{2}x + \frac{1}{4}\sin 2x - c = 0
  include Newton

  def initialize(c_ = nil, x0_ = nil)
    @zero = BigDecimal.new("0.0")
    @one = BigDecimal.new("1.0")
    @two = BigDecimal.new("2.0")
    @ten = BigDecimal.new("10.0")
    @eps = BigDecimal.new("1.0e-6")
    @c = c_ ? BigDecimal.new(c_, 6) : BigDecimal.new(Math::PI, 6) / 4 / 2
    @x0 = x0_ || @one / @two
    puts "@c = #{@c}" if $DEBUG

    solve
  end
  attr_reader :c, :x0
  attr_reader :zero, :one, :two, :ten, :eps

  def values(x)
    [(x[0] + Math.sin(@two * x[0]) / @two) / @two - @c]
  end

  def solve
    a = [@x0]
    nlsolve(self, a)
    @rad = a[0]
    @deg = @rad * 180 / Math::PI
    @x = BigDecimal(Math.sin(@rad.to_f), 6)

    self
  end
  attr_reader :rad, :deg, :x
end

class Quadrant3
  include Magick

  def initialize(opt = {})
    @arc1 = opt[:a1] || Arcsolver.new(BigDecimal.new(Math::PI, 6) / 12, BigDecimal.new("0.5"))
    @arc2 = opt[:a2] || Arcsolver.new(BigDecimal.new(Math::PI, 6) / 6, BigDecimal.new("0.5"))

    @width = 320
    @height = 320
    @margin = @margin2 = 20
    @origin_x = @margin + @margin2
    @origin_y = @height - @margin - @margin2
    @unitlen = @width - @margin * 2 - @margin2 * 2
    @filename = opt[:filename] || self.class.to_s + ".png"
    @opt_nocolor = opt[:nocolor]
  end

  def start
    puts "making #{@filename}..."

    nc = @opt_nocolor
    @image = Image.new(@width, @height) do
      self.quality = 95
      self.background_color = nc ? "white" : "#f0f0ff"
    end

    x1 = @arc1.x
    y1 = Math.sqrt(1 - x1 * x1)
    x2 = @arc2.x
    y2 = Math.sqrt(1 - x2 * x2)
    rr = [@unitlen, @unitlen].join(",")

    unless @opt_nocolor
      d = Draw.new
      d.fill = "#ccccff"
      d.stroke = "none"
      d.path("M#{u(0, 0)} L#{u(x1, 0)} L#{u(x1, y1)} A#{rr},0,0,0,#{u(0, 1)} Z")
      d.draw(@image)

      d = Draw.new
      d.fill = "#ccffcc"
      d.stroke = "none"
      d.path("M#{u(x1, 0)} L#{u(x2,0)} L#{u(x2,y2)} A#{rr},0,0,0,#{u(x1,y1)} Z")
      d.draw(@image)

      d = Draw.new
      d.fill = "#ffcccc"
      d.stroke = "none"
      d.path("M#{u(x2,0)} L#{u(1,0)} A#{rr},0,0,0,#{u(x2,y2)} Z")
      d.draw(@image)
    end

    d = Draw.new
    d.fill = "none"
    d.stroke = "black"
    d.stroke_width = 2
    d.line(*t(-0.1, 0, 1.1, 0))
    d.polyline(*t(1.1 - 0.03, -0.02, 1.1, 0, 1.1 - 0.03, 0.02))
    d.line(*t(0, -0.1, 0, 1.1))
    d.polyline(*t(-0.02, 1.1 - 0.03, 0, 1.1, 0.02, 1.1 - 0.03))
    d.arc(*t(-1, -1, 1, 1), 270, 360)
    d.line(*t(x1, 0, x1, y1))
    d.line(*t(x2, 0, x2, y2))
    d.draw(@image)

    d = Draw.new
    d.font($font)
    d.pointsize(18)
    d.fill = "black"
    d.text(*t(x1 - 0.03, -0.08), '"x1"')
    d.text(*t(x2 - 0.03, -0.08), '"x2"')
    d.text(*t(-0.07, -0.08), '"O"')
    d.text(*t(1.07, 0.03), '"x"')
    d.text(*t(0.03, 1.06), '"y"')
    d.text(*t(0.97, -0.08), '"1"')
    d.text(*t(-0.05, 0.96), '"1"')
    d.draw(@image)

    @image.write(@filename)
    @image.destroy!
    self
  end

  def t(*param)
    a = []

    while param.length >= 2
      i = param.shift
      j = param.shift
      x = i * @unitlen + @margin + @margin2
      y = @height - @margin - @margin2 - j * @unitlen
      a << x << y
    end

    return a
  end

  def u(*param); t(*param).join(","); end
end

class Circle3L
  include Magick

  def initialize(opt = {})
    @arc1 = opt[:a1] || Arcsolver.new(BigDecimal.new(Math::PI, 6) / 12, BigDecimal.new("0.5"))
    @arc2 = opt[:a2] || Arcsolver.new(BigDecimal.new(Math::PI, 6) / 6, BigDecimal.new("0.5"))

    @width = 320
    @height = 320
    @margin = 16
    @origin_x = @width / 2
    @origin_y = @height / 2
    @unitlen = (@width - @margin * 2) / 2
    @filename = opt[:filename] || self.class.to_s + ".png"
    @opt_nocolor = opt[:nocolor]
  end

  def start
    puts "making #{@filename}..."

    nc = @opt_nocolor
    @image = Image.new(@width, @height) do
      self.quality = 95
      self.background_color = nc ? "white" : "#f0f0ff"
    end

    y1 = @arc1.x
    x1 = Math.sqrt(1 - y1 * y1)
    x2, y2 = 0, -1
    rr = [@unitlen, @unitlen].join(",")

    unless @opt_nocolor
      d = Draw.new
      d.fill = "#ffcccc"
      d.stroke = "none"
      d.path("M#{u(x1,y1)} A#{rr},0,0,0,#{u(-x1,y1)} Z")
      d.draw(@image)

      d = Draw.new
      d.fill = "#ccffcc"
      d.stroke = "none"
      d.path("M#{u(0, y1)} L#{u(-x1,y1)} A#{rr},0,0,0,#{u(0,y2)} Z")
      d.draw(@image)

      d = Draw.new
      d.fill = "#ccccff"
      d.stroke = "none"
      d.path("M#{u(0,y1)} L#{u(0, y2)} A#{rr},0,0,0,#{u(x1,y1)} Z")
      d.draw(@image)
    end

    d = Draw.new
    d.fill = "none"
    d.stroke = "black"
    d.stroke_width = 2.5
    d.arc(*t(-1, -1, 1, 1), 0, 360)
    d.line(*t(-x1, y1, x1, y1))
    d.line(*t(0, -1, 0, y1))
    d.draw(@image)

    @image.write(@filename)
    @image.destroy!
    self
  end

  def t(*param)
    a = []

    while param.length >= 2
      i = param.shift
      j = param.shift
      x = @origin_x + i * @unitlen
      y = @origin_y - j * @unitlen
      a << x << y
    end

    return a
  end

  def u(*param); t(*param).join(","); end
end

class Circle3C
  include Magick

  def initialize(opt = {})
    @arc1 = opt[:a1] || Arcsolver.new(BigDecimal.new(Math::PI, 6) / 12, BigDecimal.new("0.5"))
    @arc2 = opt[:a2] || Arcsolver.new(BigDecimal.new(Math::PI, 6) / 6, BigDecimal.new("0.5"))

    @width = 320
    @height = 320
    @margin = 16
    @origin_x = @width / 2
    @origin_y = @height / 2
    @unitlen = (@width - @margin * 2) / 2
    @filename = opt[:filename] || self.class.to_s + ".png"
    @opt_nocolor = opt[:nocolor]
  end

  def start
    puts "making #{@filename}..."

    nc = @opt_nocolor
    @image = Image.new(@width, @height) do
      self.quality = 95
      self.background_color = nc ? "white" : "#f0f0ff"
    end

    y1 = @arc2.x
    x1 = Math.sqrt(1 - y1 * y1)
    x2, y2 = 0, -1
    x3, y3 = 0, 1 - (1 - y1) * 2
    x4, y4 = 0, y3 + 1
    rr = [@unitlen, @unitlen].join(",")

    unless @opt_nocolor
      d = Draw.new
      d.fill = "#ffcccc"
      d.stroke = "none"
      d.path("M#{u(x1, y1)} A#{rr},0,0,0,#{u(-x1, y1)} A#{rr},0,0,0,#{u(x1, y1)} Z")
      d.draw(@image)

      d = Draw.new
      d.fill = "#ccffcc"
      d.stroke = "none"
      d.path("M#{u(0, y3)} A#{rr},0,0,1,#{u(-x1, y1)} A#{rr},0,0,0,#{u(0, y2)} Z")
      d.draw(@image)

      d = Draw.new
      d.fill = "#ccccff"
      d.stroke = "none"
      d.path("M#{u(0, y3)} A#{rr},0,0,0,#{u(x1,y1)} A#{rr},0,0,1,#{u(0, y2)} Z")
      d.draw(@image)
    end

    d = Draw.new
    d.fill = "none"
    d.stroke = "black"
    d.stroke_width = 2.5
    d.arc(*t(-1, -1, 1, 1), 0, 360)
    d.arc(*t(-1, y4 - 1, 1, y4 + 1), @arc2.deg, 180 - @arc2.deg)
    d.line(*t(0, y2, 0, y3))
    d.draw(@image)

    @image.write(@filename)
    @image.destroy!
    self
  end

  def t(*param)
    a = []

    while param.length >= 2
      i = param.shift
      j = param.shift
      x = @origin_x + i * @unitlen
      y = @origin_y - j * @unitlen
      a << x << y
    end

    return a
  end

  def u(*param); t(*param).join(","); end
end

class Circle3Lrotate
  include Magick

  def initialize(opt = {})
    @src = opt[:src] || "Circle3L.png"
    @filename = opt[:filename] || self.class.to_s + ".png"
  end

  def start
    puts "making #{@filename}..."

    @image = Image.read(@src).first
    @image.rotate!(90)
    @image.write(@filename)
    @image.destroy!
    self
  end
end

if __FILE__ == $0
  pi = BigDecimal.new(Math::PI, 6)
  h = BigDecimal.new("0.5")
  arc1 = Arcsolver.new(pi / 12, h)
  arc2 = Arcsolver.new(pi / 6, h)
  puts "x1 = #{arc1.x.to_s('f')}, theta1 = #{arc1.rad.to_s('f')} (#{arc1.deg.to_s('f')} deg.)"
  puts "x2 = #{arc2.x.to_s('f')}, theta2 = #{arc2.rad.to_s('f')} (#{arc2.deg.to_s('f')} deg.)"
  nc = false

  Quadrant3.new(:a1 => arc1, :a2 => arc2, :nocolor => nc).start
  Circle3L.new(:a1 => arc1, :a2 => arc2, :nocolor => nc).start
  Circle3C.new(:a1 => arc1, :a2 => arc2, :nocolor => nc).start
  Circle3Lrotate.new(:src => "Circle3L.png").start
end