Solution

The correct answer is [ 2 0 0 0; 0 3/2 0 5/2; 0 0 6/5 4/5; 0 0 0 1 ] .

Key Points

• Window → Viewport transformation in 3D uses scaling followed by translation.
• General transformation: Xv = Sx·Xw + Tx, Yv = Sy·Yw + Ty, Zv = Sz·Zw + Tz.
• Scaling: S = (Viewport range)/(Window range).

Step-by-Step Solution

• Step 1: Window limits
  Wmin = (1,1,1), Wmax = (4,5,6)
• Step 2: Viewport limits
  Vmin = (2,4,2), Vmax = (8,10,8)
• Step 3: Compute scaling
  Sx = (8−2)/(4−1) = 6/3 = 2
  Sy = (10−4)/(5−1) = 6/4 = 3/2
  Sz = (8−2)/(6−1) = 6/5
• Step 4: Compute translation
  Tx = xvmin − Sx·xwmin = 2 − 2(1) = 0
  Ty = yvmin − Sy·ywmin = 4 − (3/2)(1) = 5/2
  Tz = zvmin − Sz·zwmin = 2 − (6/5)(1) = 4/5
• Step 5: Final normalized matrix
  \(\begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & \frac{3}{2} & 0 & \frac{5}{2} \\ 0 & 0 & \frac{6}{5} & \frac{4}{5} \\ 0 & 0 & 0 & 1 \end{bmatrix}\)

Additional Information

• Scaling adjusts size; translation shifts position.
• Homogeneous matrices combine both operations efficiently.