Identify what the use of homogenous co-ordinates, Computer Graphics

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1. Why are homogeneous co-ordinates utilized in computer vision? I want to identify what the use of homogenous co-ordinates makes possible in terms of camera models.

2. Consider the case where a point P = (X, Y, Z) projects to the image plane. Using the matrix below which defines the projection of this point for a camera with focal length f to prove that every other point on the line from the origin through the point P projects to the same image point

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