Rotating Parent to Make Child Face Opposite Direction of Target's Forward

Question

Scenario:

• Parent P1 has Child C1
• Target exists
• All items' forward vectors face perfectly parallel to the X-Z Plane (pseudo 2D)
• Positions of all objects are arbitrary
• Child will never move on its own within Parent's space

Goal:

Rotate P1 exclusively about the Y-Axis to have C1 Face the opposite direction of target's forward vector

Note: Y-Axis Positive would face reader, Negative would go into screen

Implementation:

P1.rotation = Quaternion.LookRotation(-target.forward) * Quaternion.Euler(0, -C1.localEulerAngles.y, 0);

The above seems to work most of the time but breaks randomly and sometimes generates a Zero Vector as the forward. I am inexperienced with Euler Angles and Quaternions so my apologies in advance.

Answer 1

Using some quaternion algebra:

Given this in Unity terms/c#:

Desired child world = Quaternion.LookRotation(-target.forward)
Child local = C1.localRotation`

Given the equation for parent-child rotations:

Desired parent world * Child local = Desired child world 

We want to calculate desired parent world (P1.rotation). Take the above equation and multiply inverse(Child local) on right of both sides and simplify:

DPW * child local * inv(child local) = Desired child world * inv(Child local) 
                Desired parent world = Desired child world * inv(Child local)

Expressing using Unity/C#, you get:

P1.rotation = Quaternion.LookRotation(-target.forward) 
        * Quaternion.Inverse(C1.localRotation);