Calculate the pose of a point on the sphere pointing to the center of the sphere

Background: Recently, there is a need to use a camera to take pictures of an object. In the initial state, the camera is directly above the object, and then the coordinates of the camera change according to a certain rule, requiring its field of view center to always be fixed on the object. This problem can be understood as: establish a coordinate system with the item, and the attitude of the camera in this space is [x,y,z, $a$ ,$b$ ,$gamma$ ]. Among them, X, Y, and Z are the coordinates of the camera in the object coordinate system,$a$、$b$、$\gamma$ is the angle of rotation of the camera around the three axes. And X, Y, Z are known quantities,$a$、$b$、$\gamma$ is the quantity to be demanded.

Regarding the problem of finding the angle, it can be converted to the problem of the angle of the sphere:

For a point on a sphere, we can express it in a three-dimensional Cartesian coordinate system, but also in a spherical coordinate system. Suppose the initial coordinates of a point on the sphere are in [0,0,r], where $r$ is the radius of the sphere. Then we can turn the point to any position on the sphere by two rotations, so the representation of the point is: [$r$，$\varphi$，$\theta$]。$r$ is the distance from the spherical coordinate point to the center of the sphere;$\varphi$ is the angle measured counterclockwise starting from the x-axis on the equatorial plane (the plane determined by the x-axis and y-axis);$\theta$ is the angle between the z axis and r.

Going back to the original question, the question can be understood as: In the initial state, the camera's attitude is [0, 0, 0, 3.14, 0, 0], that is, the camera is directly above the Z axis, and the camera's Z axis is facing down . Then if we need to turn the camera to [ rr on the sphere at this time$r$，$\varphi$，$\theta$ ] position, the new angle value of the camera is [3.14+$\theta$ , 0,$\varphi$ ], so that we can know the pose of the camera relative to the object.

Then the next step is to find out $\varphi$，$Two\; angles\; of\; \theta$ are enough, according to the relationship between the representation method of the three-dimensional rectangular coordinate system and the representation method of the spherical coordinate system: {
$$\{\begin{array}{l}x=rsin\theta cos\phi ,\\ y=rsin\theta sin\phi ,\\ z=rcos\theta \end{array}$$

It can be seen that when the coordinates [X, Y, Z] of a point on the sphere are known, the rotation angle around the two axes can be calculated according to the above formula: { θ = arccos ( z / r ) φ =
$$\{\begin{array}{l}i=arccos(z/r)\\ Phi=arcsin(y/(rsin\theta )\end{array}$$
According to the above formula, it can be known that the camera pose at any position is: [X, Y, Z, 3.14+ $\theta$ , 0,$\varphi$ ].

Publish the results according to TF:

It can be seen that although the position of the camera is always changing, the Z axis of the camera is always pointing to the center of the circle, which can meet the initial requirements.

reference:

Mind-only Consciousness 061·A Brief Introduction to the Spherical Coordinate System