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A Point and a Cube (Posted on 2004-07-16) Difficulty: 4 of 5
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.

No Solution Yet Submitted by Brian Smith    
Rating: 4.0000 (5 votes)

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Solution More precise computer solution | Comment 16 of 23 |

Using a modified version of the program, the distances to the corners of the cube (including the unlabeled ones) appear below the x,y,z coordinates of the point and the edge length of the cube:

External:
x= 1.7877458 y= 3.8912615 z=-2.5810945; size  3.0900649;
  3.0000000  5.0000000  7.0000000  6.0000000
  4.8476798  5.8736701  7.1063352  3.2403704

Internal:
x= 2.8806627 y= 4.0650647 z= 0.4207511; size  5.4880020;
  3.0000000  5.0000000  7.0000000  6.0000000
  4.8476799  5.8736701  7.1063352  3.2403703

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

  aGoal = 3: bGoal = 5: cGoal = 7: dGoal = 6
  x = RND(1) * 8
  y = RND(1) * 4
  z = RND(1) * 1
  size = RND(1) * 5
  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 ##.#######;"; x; y; z; size
     PRINT USING " ##.####### ##.####### ##.####### ##.#######"; da; db; dc; dd
    NEXT itr
    DO: a$ = INKEY$: LOOP UNTIL a$ > ""
  IF a$ = "-" THEN delta = delta / 2
  LOOP UNTIL a$ = CHR$(27)
  GOSUB evalOthPos
  PRINT USING " ##.####### ##.####### ##.####### ##.#######"; de; df; dg; dh
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

evalOthPos:
  de = SQR((x - size) ^ 2 + (y) ^ 2 + z ^ 2)
  df = SQR((x - size) ^ 2 + (y - size) ^ 2 + (z - size) ^ 2)
  dg = SQR(x ^ 2 + (z - size) ^ 2 + y ^ 2)
  dh = SQR((y - size) ^ 2 + z ^ 2 + x ^ 2)
RETURN
 

 


  Posted by Charlie on 2004-07-20 08:45:54
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