Why does translation make no impact in projection for a certain dimension?

Question

Because the camera is off the floor (z=1.25m), points with z=0 should never project to the center of the image (only at infinity).

However, for z=0 and x=0 all points are projected to exactly the center of the image. Why is that?

Minimal example, with a camera image 512x512 in size (so, cx=cy=256), and arbitrary focal lenght: This

r = np.asarray([1.57079633, 0.,   0.])  # 90 degrees 
K = np.asarray([[128.,   0., 256.], 
                [  0., 128., 256.], 
                [  0.,   0.,   1.]])
position = np.asarray([0.,   0.,   1.25])  # 1.25m off the ground

points3d = np.asarray([(0, y, 0) for y in range(1,10)]).astype(float)
points2d = np.squeeze(cv2.projectPoints(points3d, rvec=r, tvec=position, cameraMatrix=K, distCoeffs=None)[0]).round().astype(int)

print(points3d)
print(points2d)

outputs

points3d
[[0. 1. 0.]
 [0. 2. 0.]
 [0. 3. 0.]
 [0. 4. 0.]
 [0. 5. 0.]
 [0. 6. 0.]
 [0. 7. 0.]
 [0. 8. 0.]
 [0. 9. 0.]]
points2d
[[256 256]
 [256 256]
 [256 256]
 [256 256]
 [256 256]
 [256 256]
 [256 256]
 [256 256]
 [256 256]]

Answer 1

So it seems cv2.projectPoints(..., rvec, tvec first rotates camera by rvec, and then translates camera (in that rotated frame) tvec following

Thus, if one wants to represent the camera being off the ground, given this specific rvec=[1.57079633, 0., 0.] one needs to alter the 2nd coordinate of tvec instead of the third

Namely, setting position = np.asarray([0., 0., 1.25]) outputs

points3d
[[0. 1. 0.]
 [0. 2. 0.]
 [0. 3. 0.]
 [0. 4. 0.]
 [0. 5. 0.]
 [0. 6. 0.]
 [0. 7. 0.]
 [0. 8. 0.]
 [0. 9. 0.]]
points2d
[[256 416]
 [256 336]
 [256 309]
 [256 296]
 [256 288]
 [256 283]
 [256 279]
 [256 276]
 [256 274]]

that is the expected "projected points starting at the bottom of the image and moving towards the center, the further the 3D point is"