Question

(4) The technique of triangulation in surveying is to locate a position in R3 if the distance to 3 fixed points is known. This is similar to how global position systems (GPS) work. A GPS unit measures the time differences taken for a signal to travel from each of 4 satellites to a receiver on Earth. This is then converted to a difference in the distances from each satellite to the receiver, and this can then be used to calculate the distance to 4 satellites in known positions Let P (2,-1,4), P2 (3,4,-3), P (4,-2,6), P (6,4, 12) We wish to find a point P-(xy:) with r, 20 satisfying P is distance Δ from P. P is distance (Δ-12+ 9V3) from P2, P is distance A - 1 from Ps, and P i Pa s distance A-9 from

Answer 1

P (x,y,z)

distance from P1 (2, -1,4)

² = (x-2)² + (y+1)² + (z-4)²

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

P2(3,4,-3) : (-12+9 $\sqrt{3}$ )² = (x-3)² + (y-4)² + (z+3)²

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

P3(4,-2,6) ; (-1)² = (x-4)² + (y+2)² + (z-6)²

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

P4(6,4,12) : (-9)² = (x-6)² + (y-4)² + (z-12)²  

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

expanding the terms

² = (x² +y² + z²) -4x +2y - 8z + 21

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

² + (144 +81*3 -216 $\sqrt{3}$ ) - (24-18 $\sqrt{3}$ ) $\Delta$ = (x² +y² + z²) -6x -8y + 6z + 34

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -
² - 2
P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -
+1 = (x² +y² + z²) -8x +4y - 12z

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -
² - 18
P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -
+81 = (x² +y² + z²) - 12x - 18y - 24z

re-arrange the terms and put s = ² - (x² +y² + z²)

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

-4x +2y - 8z + 0 = s-21

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

-6x - 8y +6z + (24-18 $\sqrt{3}$ ) $\Delta$ = s +353 - 216 $\sqrt{3}$

-8x +4y - 12z + 2 = s-55

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

-12x -8y - 24z + 18 = s-115

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -

We have 4 variables and 4 equations, solving them we get

x= -0.185s +37.22

y = 0.53s +16.39

z= 0.1s +2.07

= 0.3s +7.64

P (x, y, z) distance from P1 (2, -1,4) delta^2 = (x - 2)^2 + (y + 1)^2 + (z - 4)^2 P2(3, 4, -3): (delta - 12 + 9 squareroot 3)^2 = (x - 3)^2 + (y - 4)^2 + (z + 3)^2 P3(4, -2, 6): (delta - 1)^2 = (x - 4)^2 + (y + 2)^2 + (z - 6)^2 P4(6, 4, 12): (delta - 9)^2 = (x - 6)^2 + (y - 4)^2 + (z - 12)^2 expanding the terms delta^2 = (x^2 + y^2 + z^2) - 4x + 2y - 8z + 21 delta^2 + (144 + 81 * 3 - 216 squareroot 3) - (24 - 18 squareroot 3) delta = (x^2 + y^2 + z^2) - 6x -8y + 6z + 34 delta^2 - 2 delta + 1 = (x^2 + y^2 + z^2) - 8x + 4y - 12z delta^2 - 18 delta + 81 = (x^2 + y^delta^2 + z^2) - 12x -