Definitions

• Convex polygon is a polygon that, if you take each segment, draw a line through it, the whole polygon must lie on one side of the line.
• A line that goes through a vertex of the polygon such that the whole polygon lies on one side of the line is called a supporting line.
• A pair of antipodals is a pair of vertices such that one can draw two parallel supporting lines through them.
• The diameter of the polygon is said to be the largest distance between a pair of its vertices. This is proven to be the largest distance between a pair of parallel supporting lines, and also concide with the largest distance between a pair of antipodals.

Rotating Calipers

• Step 0: The calipers can be drawn in anyway, it doesn't matter (of course, they must be parallel supporting lines that squeeze the polygon between them!).
• Step 1: Rotate the calipers, they touch AB first! So B-E is another pair of antipodals. Pivot A is moved to B, while pivot E is unchanged.
• Step 2: Rotate the calipers, they touch BC and EF at the same time! So in this case we have three more pairs of antipodals: B-F, C-E, and C-F. Pivots B and E are both moved to C and F, respectively.
• Step 3: Rotate the calipers, one line touches FG before the other line touches CD. We collect C-G as another pair. Pivot F is moved to G.
• Step 4: Rotate the calipers, one line touches CD before the other line touches GA. We collect D-G as another pair. Pivot C is moved to D.
• Step 5: Rotate the calipers, one line touches GA before the other line touches DE. We collect D-A as another pair. Pivot G is moved to A.
• Step 6: (not shown) Because the pivot that was at E at the beginning has travelled all the way through G and to A, if we continue to rotating, we will collect A-E as a pair again. We can stop here with the result of 8 pairs of antipodals.

The flaws

/* p[] is in standard form, ie, counter clockwise order, 
     distinct vertices, no collinear vertices.
   ANGLE(m,n) is a procedure that returns the clockwise angle 
     swept out by a ray as it rotates from a position parallel 
     to the directed segment Pm,Pm+1 to a position parallel to Pn,Pn+1
   We assume all indices are reduced to mod N (so that N+1 = 1).
*/
GetAllAntiPodalPairs(p[1..n])
  //Find first anti-podal pair by locating vertex opposite P1
  i = 1
  j = 2
  while angle(i, j) < pi
    j++
  yield i,j

  /* Now proceed around the polygon taking account of
     possibly parallel edges. Line L passes through
     Pi, Pi+1 and M passes through Pj, Pj+1
  */

  //Loop on j until all of P has been scanned
  current = i  
  while j <> n
    if angle(current, i+1) <= angle(current, j+1)
      j++
      current = j
    else
      i++
      current = i
    yield i,j

    //Now take care of parallel edges
    if angle(current, i+1) = angle(current, j+1)
      yield i+1, j
      yield i, j+1
      yield i+1, j+1
      if current = i
        j++
      else
        i++

A more precise approach

/* p[] is in standard form, ie, counter clockwise order, 
     distinct vertices, no collinear vertices.
   ANGLE(m,n) is a procedure that returns the COUNTERclockwise angle 
     swept out by a ray as it rotates from a position parallel 
     to the directed segment Pm,Pm+1 to a position parallel to Pn,Pn+1
   We assume all indices are reduced to mod N (so that N+1 = 1).
*/
GetAllAntiPodalPairs(p[1..n])
  //Find first anti-podal pair by locating vertex opposite P1
  //(Reuse this piece of code from Shamos, very beautifully written)
  i = 1
  j = 2
  while angle(i, j) < pi
    j++
  yield i,j

  //Loop on j until all of P has been scanned 
  while j <> 1
    let a = 2*pi - angle(i, j) // clockwise angle
    if (a == pi): // Pi Pi+1 and Pj Pj+1 are parallel
      yield i+1, j
      yield i, j+1
      yield i+1, j+1
      // Notice that (i, j) has been added to the result before
      // being the pivots, so no need to yield i,j
      i++
      j++
    elif (a < pi): // Will touch Pi Pi+1 first
      yield i+1, j
      i++
    else:
      yeild i, j+1 // Will touch Pj Pj+1 first
      j++

Explanation

How do I know which edge the calipers will touch first?

How do I know it will produce all pairs of antipodals?