Computer graphics class Calculate the plane coefficients (A, B, C and D) of 3 po

Question

Computer graphics class Calculate the plane coefficients (A, B, C and D) of 3 points in a plane defined by P_1, P_2 and P_3. P_1 = (60, 76,12); P_2 = (-30, 38, 78); P_3 = (35, -10, 55); Then determine if the following point P_4 is behind, in front of or on the polygon surface contained within that plane: P_4 = (10, 72, -29); Be sure to show your work. Explain why determining if Point P_4 is behind or in front of the polygon surface relevant to computer graphics applications. Please work out the problem and label which part you are solving. Thank you for your help

Explanation / Answer

P1=(60,76,12)
P2=(-30,38,78)
P3=(35,-10,55)
P4=(10,72,-29)

A= y1 (z2 – z3) + y2 (z3 – z1) + y3 (z1 – z2)
   =76(78-55) +38(55-12) + (-10)(12-78)
    =76(23) + 38(43) - 10(-66)
     =1748 + 1634 + 660
      4043

B= z1 (x2 – x3) + z2 (x3 – x1) + z3 (x1 – x2)
   =12(-30-35) +78(35-60) + 55(60 -(-30))
    =12(-65) + 78(-25) +55(90)
    =-780 - 1950+4950
    =2220

C= x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)
   =60(38-(-10)) + (-30)(-10-76) + 35(76-38)
   =60(48) -30(-86) + 35(38)
    =2880 +2580 + 1330
    =6790

D= -x1 (y2 z3 – y3 z2) - x2 (y3 z1 – y1 z3)
   = -60(38 *55 - -10 * 78) - (-30)(-10 * 12 - 76*55)
   =-60(2090 + 78) -(-30)(-120 - 4180)
   =-130080 + 121800
    =8280

For P4,
Ax + By + Cz + D
=4040(10) + 2220(72) + 6790(-29) + 8280
=40400 + 159840 - 196910 + 8280
=11610

Since P4>0, P4 lies in front of the polygon surface

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