Before tackling this one, take a look at this one.

     +-------------D
    /|            /|
   / |           / |
  /  |          /  |
 /   |         /   |
C-------------+    |
|    |        |    |
|    B--------|----+
|   /         |   /
|  /          |  /
| /           | /
|/            |/
+-------------A

A, B, C and D are non-adjacent vertices of a cube. There is a point P in space such that PA=3, PB=5, PC=7, and PD=6. Find the distance from P to the other four vertices and find the length of the edge of the cube.
There are two answers, one with P outside the cube and one with P inside the cube.

(In reply to PD can't be 9. Correct Restrictions on PD. by np_rt)

Fortunately I chose my coordinate system the same as np_rt. My program chooses random values, within reason, for x, y, z and the size of the cube.  It starts with a certain increment and figures the total discrepancies from the distances given.  It does this for all possible locations with x, y, z and size incremented or decremented by the increment (delta), or not incremented -- but at least one must be incremented or decremented.  Whichever produces the best result is taken as the new starting point.  I tried various automatic schemes for decreasing the increment for finer results, but eventually had to make this a human intervention.

The program, using 8 as the goal for PD is:

DEFDBL A-Z
size = 6
x = 5
y = 3
z = 1

  aGoal = 3: bGoal = 5: cGoal = 7: dGoal = 8
DO
  x = RND(1) * 11
  y = RND(1) * 4
  z = RND(1) * 4
  size = RND(1) * 4
  delta = .1
  CLS
  GOSUB evalPos
  aOff = da - 3
  bOff = db - 5
  cOff = dc - 7
  dOff = dd - 11

  DO
    FOR itr = 1 TO 48
     leastTot = 9999
     FOR dx = -delta TO delta STEP delta
       xTr = x + dx
     FOR dy = -delta TO delta STEP delta
       yTr = y + dy
     FOR dz = -delta TO delta STEP delta
       zTr = z + dz
     FOR dS = -delta TO delta STEP delta
       sizeTr = size + dS
     IF dx <> 0 OR dy <> 0 OR dz <> 0 OR dS <> 0 THEN
       GOSUB evalTrPos
       IF totTr < leastTot THEN
        dxP = dx: dyP = dy: dzP = dz: dSP = dS
        leastTot = totTr
       END IF
     END IF
     NEXT dS
     NEXT dz
     NEXT dy
     NEXT dx
     xTr = x + 2 * dxP: yTr = y + 2 * dyP: zTr = z + 2 * dzP
     sizeTr = size + 2 * dSP
     GOSUB evalTrPos
     'IF totTr > leastTot THEN delta = delta * .8
     x = x + dxP: y = y + dyP: z = z + dzP: size = size + dSP
     GOSUB evalPos
     PRINT USING "#.### #.### #.###; ##.#### ##.#### ##.#### ##.#### ##.####"; x; y; z; size; da; db; dc; dd
    NEXT itr
    DO: a$ = INKEY$: LOOP UNTIL a$ > ""
  IF a$ = "-" THEN delta = delta / 2
  LOOP UNTIL a$ = CHR$(27)
LOOP
END

evalTrPos:
  daTr = SQR((xTr - sizeTr) ^ 2 + (yTr - sizeTr) ^ 2 + zTr ^ 2)
  dbTr = SQR(xTr ^ 2 + yTr ^ 2 + zTr ^ 2)
  dcTr = SQR((xTr - sizeTr) ^ 2 + (zTr - sizeTr) ^ 2 + yTr ^ 2)
  ddTr = SQR((yTr - sizeTr) ^ 2 + (zTr - sizeTr) ^ 2 + xTr ^ 2)
  totTr = ABS(daTr - aGoal) + ABS(dbTr - bGoal) + ABS(dcTr - cGoal) + ABS(ddTr - dGoal)
RETURN

evalPos:
  da = SQR((x - size) ^ 2 + (y - size) ^ 2 + z ^ 2)
  db = SQR(x ^ 2 + y ^ 2 + z ^ 2)
  dc = SQR((x - size) ^ 2 + (z - size) ^ 2 + y ^ 2)
  dd = SQR((y - size) ^ 2 + (z - size) ^ 2 + x ^ 2)
  totTr = ABS(da - aGoal) + ABS(db - bGoal) + ABS(dc - cGoal) + ABS(dd - dGoal)
RETURN

After a few iterations we get the following line:

4.043 2.530 -1.502;  4.9600  3.0000  5.0000  7.0000  8.0000

Indicating the cube side is 4.9600 with x=4.043, y=2.530, z=-1.502.  The negative z indicates the point is external to the cube, below the bottom as shown.

I never seem to get a point within the cube.

Posted by Charlie on 2004-07-18 19:58:24