Confuse about the Jacobian matrix

[SleepTight-Cat]

Hi, I have some questions regarding the Jacobian matrix used in the paper. In the Appendix, the authors mention that "𝐽 is the Jacobian of the affine approximation of the projective transformation." My understanding is that a Jacobian generally approximates a non-linear function. However, isn't the projective transformation already linear? From what I understand, the projective transformation maps 3D camera coordinates to 2D image coordinates, typically represented by a 3x4 matrix in homogeneous coordinates. Could you help clarify what exactly 𝐽 is approximating in this context? Or is there any mistake in my understanding?

Thank you!