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• Calculation of usable area

  sei • 02/04/2015 at 21:16 • 2 comments

  To check if the geometry is ok I wrote a little python script to calculate the area which the plotter arms can acces:

  import math
  import matplotlib.pyplot as plt
  from matplotlib.patches import Rectangle
  from planar import Vec2
  
  al = 0.75 #arm length [m]
  f = 0.75 #position of rail a [m] 
  
  apos = [2 * x * 0.005 for x in range(1, 200)]
  bpos = [2 * x * 0.005 for x in range(1, 200)]
  
  cx = []
  cy = []
  
  for x_a in apos:
      for x_b in bpos:
          a = Vec2(x_a,f) # vector to position on arm a
          b = Vec2(x_b,0) # vector to position on arm b
  
          cl = (b-a).length #distance between a and b
  
          #calculate height of triangle which is spanned by the arms
          hl = 2*((al**2)*(al**2)+(al**2)*(cl**2)+(cl**2)*(al**2))-(al**4+al**4+cl**4)
  
          if hl > 0:
              # finish calculation of height
              hl = math.sqrt(hl)/(2*cl)
  
              #vector from a to b
              d = (b-a)
              d = d/d.length
  
              # turn d around by 90°, h points to tip of triangle
              h = Vec2(d.y,-d.x)
  
              # normalize h
              h = h/h.length
  
              # calculate distance to tip set as length of d
              d *= math.sqrt(al ** 2 - hl ** 2)
  
              # vector which points to the tip of the triangle
              c = a+d+h*hl
  
              #save coordinates of tip
              cx.append(c.x)
              cy.append(c.y)
  
  plt.plot([min(apos),max(apos)],[f,f],'r')
  plt.plot([min(bpos),max(bpos)],[0,0],'g')
  ax = plt.gca()
  
  points = plt.plot(cx,cy,'o')
  plt.setp(points,'color', 'b', 'marker', '.', 'markersize', 0.5)
  
  plt.axes().set_aspect('equal')
  
  plt.xlim([-0.7,2.7])
  plt.ylim([-0.05,0.8])
  plt.xlabel('x [m]')
  plt.ylabel('y [m]')
  plt.show()

  Running the script gives the following plot:

  The blue area is what the plotter can reach, the density of the dots is a measure for the resolution. Luckily the density is pretty homogenious, so the resolution of the plotter will be more or less constant over the whole drawin area.

  Also one might wonder what with the empty space on the right, I'll show that in a follow up.

  Here's also a little sketch which shows what the calculations in the script are based on:

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