Given a constant rotation and target point, find position so that point will be in camera center

Question

Given a fixed rotation and a target "center" point, how to I find the position the camera should be in so that that point is the center? LookAt changes the rotation keeping the position constant. Basically, I'm trying to do the reverse, but the math escapes me :-). Thanks!

Camera is currently psuedo-isometric (orthographic with rotation from Euler angles [X:35.264389683ish°, Y:45°, Z:0°]). Might want to change these a bit or allow rotation though, so ideally a solution would work for any rotation. Can easily get the projection matrix.

EDIT: thinking about it, there would be infinite such positions that satisfy this constraint. I would want to fix the Y component of the position (camera height) and find only an X/Z.

Thanks to Mokosha's answer below, here's the final code I'm using:

    private static Vector3 reverseLookAt(Quaternion cameraRotation, float cameraHeight, Vector3 targetPosition)
    {
        Vector3 d = cameraRotation * Vector3.forward;
        Plane plane = new Plane(Vector3.up, -cameraHeight);
        Ray ray = new Ray(targetPosition, d);
        Vector3 pos = UnityUtils.intersect(ray, plane);
        return pos;
    }

Answer 1

Let's rephrase your question:

Given a rotation R, and a position p, we would like the rotated point p' to lie along the Z-axis (also known as the center of the camera). For this, we can use linear algebra:

1. Compose the rotation matrix R from your euler angles.

2. Solve problem (1) which we can rewrite as (2): $$\begin{align}Rd&=\begin{vmatrix} 0 & 0 & 1 \end{vmatrix}^T \tag 1\\ \\ d &= R^{-1}\begin{vmatrix} 0 \\ 0 \\ 1 \end{vmatrix}\tag 2\end{align}$$

3. d is now the direction in world space that corresponds to the Z-axis of the camera in eye space. Construct p' by translating p by d: p' = p + d.

4. Using p and p', construct a line P.

5. Compute the intersection of P with the plane whose normal is [0 1 0] at D distance away from the origin where D is the desired Y value of your camera in world units.

The code for this in Unity's C# would look something like this:

Quaternion rot = ...
Vector3 p = ...
float desired_y = ...

Vector3 d = rot * (new Vector3(0, 0, 1)); // Perhaps Camera.forward...?

Vector3 camPos;
if (Mathf.Abs(d.y) < Mathf.epsilon) {
  Log.Error("Camera direction parallel to XZ plane");
} else {
  Plane y_plane = new Plane(new Vector3(0, 1, 0), desired_y);
  float dist;
  y_plane.Raycast(new Ray(p, p + d), dist)
  camPos = r.GetPoint(dist);
}