S2T2

Jim Randell 4:27 pm on 22 September 2023 Permalink Reply
Tags: by: John Owen ( 32 )   

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  Jim Randell 4:55 pm on 22 September 2023 Permalink | Reply

  If we suppose the apparent radius of the two discs is R, and we consider the part of the circumference of the sun that is obscured, then it forms the arc of a circular sector that subtends an angle of 2θ at the centre of the sun’s disc. The length of the obscured circumference is then: 2Rθ.

  The area of the part of the sun’s disc that is obscured A can be calculated as the sum of two equal obscured segments, each being the area of the circular sector less the area of the triangle formed by replacing the arc of the sector with a chord.

  A = 4 [(𝛑R²)(θ/2𝛑) − (R cos(θ))(R sin(θ))/2)
  A = 2R²θ − 2R²sin(θ)cos(θ)
  A = R²(2θ − sin(2θ))

  (See also: [@wikipedia])

  And so B’s measure is:

  B = 2R²θ − A
  ⇒
  B = 2R²sin(θ)cos(θ)
  B = R²sin(2θ)

  And A’s measure (the magnitude) m is given by:

  m = 2R(1 − cos(θ))

  So given an integer magnitude, we can calculate a rational value for cos(θ), and using the identity cos²(θ) + sin²(θ) = 1 we can calculate a value for sin(θ), which must also be rational if B is to have an integer value.

  This Python program runs in 61ms. (Internal runtime is 413µs).

  from enigma import (Rational, irange, is_square_q, sq, as_int, printf)
  
  Q = Rational()
  
  # apparent radius
  R = 50
  
  # consider possible magnitudes from 11 to 99
  for m in irange(11, 99):
    # calculate cos(theta), sin(theta)
    cost = 1 - Q(m, 2 * R)
    sint = is_square_q(1 - sq(cost), F=Q)
    if sint is None: continue
    # calculate B's measure
    B = 2 * sq(R) * sint * cost
    # is it an integer?
    B = as_int(B, default=None)
    if B is None: continue
    # output solution
    printf("m={m} -> B={B}")
  

  Solution: The magnitudes are: 20, 40, 72.

  And we have:

  m = 20 → B = 2400
  m = 40 → B = 2400
  m = 72 → B = 1344

  There are only 3 values because the “partner” value for m = 72 (which also gives B = 1344) is m = 4.

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  • 

    Frits 8:47 pm on 22 September 2023 Permalink | Reply

    I had to look up some formula and came to the same formula for B.

       
    # apparent radius
    R = 50
    
    # consider possible magnitudes from 11 to 99
    for m in range(11, 100):
      # calculate cos(theta)
      c = 1.0 - m / (2 * R)
      # calculate sin(theta), c^2 + s^2 = 1
      s = (1 - c**2)**.5
      # B = r^2.sin(2 * theta), sin(2x) = 2.sin(x).cos(x) 
      B = 2 * R * R * s * c
      
      # is it close to an integer?
      if abs(B - round(B)) > 1e-6: continue
     
      # output solution
      print(f"m={m} -> B={B:.2f}")
    

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  • 

    Jim Randell 2:38 pm on 23 September 2023 Permalink | Reply

    With a bit more work we can simplify the calculation of B to:

    B = (1/2)(2R − m)√(m(4R − m))

    Which gives a simpler program, that only uses integers:

    Run: [ @replit ]

    from enigma import (irange, is_square, div, printf)
    
    # apparent radius
    R = 50
    
    # consider possible magnitudes from 11 to 99
    for m in irange(11, 99):
      r = is_square(m * (4 * R - m))
      if r is None: continue
      B = div((2 * R - m) * r, 2)
      if B is None: continue
      # output solution
      printf("m={m} -> B={B}")
    

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