Calculate center of rotation from quaternion and two points

Given is a point P in a plane, a rotation (quaternion) and a point P’ resulting from the rotation. Is there an elegant way or a function in Unity to calculate the center of the rotation? Thanks in advance.

I don’t believe there are any built in functions to do such a thing, but we can get there with some trigonometry and vector maths;

// - rotation = input quaternion
// - p0 = starting point
// - p1 = point resulting from rotating vector[p0 - centre]
// - planeNormal = the relative 'up' vector of the plane

//Find the total angle of rotation (in radians)
float theta = Quaternion.Angle (Quaternion.identity, rotation) * Mathf.Deg2Rad;

//Find the vector between p0 and p1
Vector3 p01 = p1 - p0;

//Find the distance^2 between p0 and p1
float dist2 = p01.sqrMagnitude;

//Form a triangle with vertices [p0, p1, centre], where sides [p0-centre] and [p1-centre] are equal in length
//Use the cosine rule to find the length^2 (A2) of said sides;
// - c^2 = a^2 + b^2 - 2ab cos (C)
//    - where c = dist, b = a;
// - dist^2 = 2a^2 - 2a^2 cos (theta)
// - dist^2 = 2a^2 (1 - cos (theta))
// - dist^2 / (1 - cos (theta)) = 2a^2
// - a^2 = dist^2 / 2(1 - cos (theta))
float sideA2 = dist2 / (2f * (1f - Mathf.Cos (theta)));

//Find the height of the triangle using Pythagoras' theorem
float height = Mathf.Sqrt (sideA2 - 0.25f * dist2);

//Find the midpoint between p0 and p1
Vector3 midpoint = (p0 + p1) * 0.5f;

//Find the direction of the centre from the midpoint (use the plane's normal to calculate the vector perpendicular to p01)
Vector3 dir = Vector3.Cross (planeNormal, p01 / Mathf.Sqrt (dist2)).normalized;

//Combine and offset to find the centre
Vector3 centre = midpoint + dir * height;

I haven’t tested the above yet, but that should theoretically do it.