Given a point, a line, and a distance, calculate the two points on that line that are that distance away.

calculate_points_at_distance_on_line.rb

# encoding: utf-8

# Given a point, a line, and a distance, calculate the two points on that line that are that distance away.


# Relevant equations
#   eqn for line (m here is our gslope, b is the y-intercept)
#     y = mx + b
#   pythagorean theorem from origin (r, the radius, will be our gdist, the distance between the points)
#     r**2 = x**2 + y**2


# Make floats less irritating to look at
class Float
  def inspect
    sprintf('%f', self).gsub(/\.?0+$/, '')  # => "6", "2", "6", "8", "1.333333", "12", "10", "0", "-6", "10", "10", "10", "12", "10", "0", "-6", "10", "12", "10", "0", "-6", "10", "12", "10", "0", "-6", "10", "12", "10", "0", "-6", "10", "6", "8", "12", "10", "0", "-6", "6", "8", "12", "10", "0", "-6", "6", "8", "6", "8", "12", "10", "0", "-6", "6", "8", "12", "10", "0", "-6", "6", "8", "12", "10", "0", "-6"
  end                                       # => :inspect
end                                         # => :inspect


# make math functions more convenient
include Math  # => Object
alias √ sqrt

# GIVENS (indicated by starting the var name w/ a g)
gx1, gy1 = 6.0, 2.0   # => [6, 2]
g∆x, g∆y = 6.0, 8.0   # => [6, 8]
gslope   = g∆y / g∆x  # => 1.333333
gdist    = 10         # => 10
gx2, gy2 = 12, 10     # => [12, 10]
gx3, gy3 =  0, -6     # => [0, -6]


# An assertion function to explode if we get it wrong
define_method :assert do |named_values|
  named_values.each do |name, calculated|                                                      # => {:gx2=>12, :gy2=>10, :gx3=>0, :gy3=>-6, :gdist=>10}, {:"g\u2206x"=>6, :"g\u2206y"=>8, :gx2=>12, :gy2=>10, :gx3=>0, :gy3=>-6}
    expected = binding.local_variable_get name                                                 # => 12,   10,   0,    -6,   10,   6,    8,    12,   10,   0,    -6
    next if calculated == expected                                                             # => true, true, true, true, true, true, true, true, true, true, true
    raise "Expected #{name} = #{expected.inspect} but it calculated to #{calculated.inspect}"
  end                                                                                          # => {:gx2=>12, :gy2=>10, :gx3=>0, :gy3=>-6, :gdist=>10}, {:"g\u2206x"=>6, :"g\u2206y"=>8, :gx2=>12, :gy2=>10, :gx3=>0, :gy3=>-6}
end                                                                                            # => :assert

# Show we can find calculate the distance
lambda do
  assert(
    gx2:   x2 = gx1 + g∆x,  # => 12
    gy2:   y2 = gy1 + g∆y,  # => 10

    gx3:   x3 = gx1 - g∆x,  # => 0
    gy3:   y3 = gy1 - g∆y,  # => -6

    gdist: √( (x2 - gx1)**2 + (y2 - gy1)**2 ),  # => 10
    gdist: √( (x3 - gx1)**2 + (y3 - gy1)**2 ),  # => 10
    gdist: √(      (g∆x)**2 +      (g∆y)**2 ),  # => 10
  )                                             # => {:gx2=>12, :gy2=>10, :gx3=>0, :gy3=>-6, :gdist=>10}
end.call                                        # => {:gx2=>12, :gy2=>10, :gx3=>0, :gy3=>-6, :gdist=>10}


# Show we can find (x2, y2) and (x3, y3)
lambda do
  # Points on that line will satisfy
  #   y = gslope * x + b

  # The points will be gdist from (gx1, gy1),
  # thus they make a circle around that point
  
  # We'll translate (x1, y1) to the origin to solve
  # Thus there is a circle of possible values given by
  #   gdist**2 = x**2 + y**2
  # Solve for x:
  #   gdist**2 = x**2 + (gslope * x)**2
  #            = x**2 + gslope**2 * x**2
  #            = x**2 (1 + gslope**2)
  #   x**2 = gdist**2 / (1 + gslope**2)
  #   x    = ±√(gdist**2 / (1 + gslope**2))
  # Since we translated x1 to the origin, the x we found is ∆x
  #   x    = (x2 - x1)
  #        = ∆x
  # Thus
  #   ∆x   = ±√(gdist**2 / (1 + gslope**2))
  # And the rest is trivial

  assert(
    g∆x: ∆x = √(gdist**2 / (gslope**2 + 1)),  # => 6
    g∆y: ∆y = gslope * ∆x,                    # => 8
    gx2: gx1 + ∆x,                            # => 12
    gy2: gy1 + ∆y,                            # => 10
    gx3: gx1 - ∆x,                            # => 0
    gy3: gy1 - ∆y,                            # => -6
  )                                           # => {:"g\u2206x"=>6, :"g\u2206y"=>8, :gx2=>12, :gy2=>10, :gx3=>0, :gy3=>-6}
end.call                                      # => {:"g\u2206x"=>6, :"g\u2206y"=>8, :gx2=>12, :gy2=>10, :gx3=>0, :gy3=>-6}