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Assuming the object positioned in front of the mirror at some point on the principal axis before the focus, each point of that object emanates light in all directions, so what happens to the rays of light that do not fall parallel to the principal axis and those that pass through the focus, do they not converge at some point in space? Do they diverge, not forming a sharp image? Why is a real image formed in a concave mirror considered only the two rays of light, the one that falls parallel to the principal axis and the one that passes through the focus?

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All the rays emanated by a point on the object and hitting the mirror are converging at the image of that specific point. This property is called stigmatism and is also formulated as "the image of a point is a point as well". Spherical mirrors are aproximately stigmatic in the paraxial approximation. If the system is not stigmatic the image is blurred, the image of a point is not a point but some extended spot. Some eyes are like this and to correct the vision problem (astigmatism) special lenses are devised and mounted in eyglasses. Now, if you want to construct geometrically the image of a point of the object for the stigmatic case (which is the one assumed in introductory optics) you only need two rays, because you know that they all converge in the same point. You can take any other rays and the reflected ones will pass through the same point. So, for practical purposes, we use some rays whose path is the easyest to track. These are what you called "remarkable" rays.

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