I am trying to map the position of an object in one image to another. I have four points in one image with corresponding points in another image so as to bound an area say A. Now, if I have points in A from one image, how do I get the coordinate of the points in the second image.
It must be noted that the first image has oblique coordinates while the second one is orthogonal. Also, I do not have any other information such as the angle between the two coordinate systems. I only have four points in both the images that encompass all my required points (in area A) within them.
It depends on the coordinate systems and how they are related to one another. For example, if they are related by a linear transformation then you need to construct the change of basis matrix.
More work is required for other coordinate systems, e.g. changing from to polar coordinates to Cartesian coordinates. If a point has polar coordinates $(r,\theta)$ then it has Cartesian coordinates $$(x,y) = (r\cos\theta,r\sin\theta).$$
Perhaps you might like to give more information about the coordinate systems you're using. In the meantime, take a look at this chapter on Changes of Coordinates in 2D.