Suppose an Earth-sized (convex) perfect lens is placed between the sun and the Earth (with the center of the lens on the line between the center of the Earth and the center of the sun, and the lens perpendicular to this line), how will the image of the sun that's projected on Earth look like? Now I don't know that much of optics, but I do know that the image of the sun varies with the diameter of the lens (when I played with lenses to start a fire the image was very little). I know how to construct the image of an object, but in this case, the huge scale (the distance to the sun included) confuses me.
Because of the big diameter of the lens, and the big size of the sun, there will be a big image on the Earth (because of the curvature of the Earth the image is blurred; only circles on the Earth with a diameter varying from zero to the magnitude of the Earth's diameter can be in the focal plane). Because of the big distance (between the Earth and the sun) it's hard to draw an image. How varies the magnitude of the image on the surface of the Earth (which is blurred because the surface of the Earth is for the not in the focal plane of the lens) with the strength (and thus the focus and the distance of the lens to the Earth) of the lens and can the image of the sun projected on the surface of the Earth be smaller than the Earth's diameter so in some parts it looks like night and in the other parts the sun shines more intense?