A viewer is located at the position (1, 0, 1) and is looking at a point object located at (5, 5, 5). If the up direction is along ˆy, find out the transformation axes.

To find the transformation axes, we need to calculate three vectors: the forward vector, the right vector, and the up vector.

1. Forward Vector (f): This is the direction the viewer is looking at. It is calculated by subtracting the viewer's position from the object's position.

   Object Position = (5, 5, 5) Viewer Position = (1, 0, 1)

   Forward Vector = Object Position - Viewer Position = (5-1, 5-0, 5-1) = (4, 5, 4)

   But we need to normalize this vector to get the unit vector. The normalization is done by dividing each component of the vector by its magnitude.

   Magnitude of Forward Vector = sqrt((4^2) + (5^2) + (4^2)) = sqrt(41)

   Normalized Forward Vector = Forward Vector / Magnitude = (4/sqrt(41), 5/sqrt(41), 4/sqrt(41))

2. Right Vector (r): This is perpendicular to the forward vector and the up direction. It is calculated using the cross product of the up direction and the forward vector.

   Up Direction = (0, 1, 0)

   Right Vector = Up Direction x Forward Vector = (1/sqrt(41), 0, -4/sqrt(41))

   We also need to normalize this vector.

   Magnitude of Right Vector = sqrt((1/sqrt(41))^2 + (0)^2 + (-4/sqrt(41))^2) = sqrt(17/41)

   Normalized Right Vector = Right Vector / Magnitude = (sqrt(41)/17, 0, -4*sqrt(41)/17)

3. Up Vector (u): This is perpendicular to both the forward and right vectors. It is calculated using the cross product of the forward vector and the right vector.

   Up Vector = Forward Vector x Right Vector = (20sqrt(41)/41, 17sqrt(41)/41, 20*sqrt(41)/41)

   The up vector is already normalized because it is the cross product of two normalized vectors.

So, the transformation axes are the normalized forward vector, the normalized right vector, and the up vector.