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In optics, focus, or lack thereof, is the result of transformations applied to the path of light. It occurs to me that rather than these transformations distorting both the vertical and horizontal components of the path of light, you could in theory have such transformations only in the vertical or horizontal, leaving the other unchanged.

Is such a "horizontal-only" or "vertical-only" out-of-focus condition possible in real life, as in cases in which light will naturally be out of focus only in one axis? What would it look like?

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not only is cylindrical out-of-focus possible in real life, it is a common vision defect in human eyes. Optometrists refer to it as "cylinder" and specify corrective lenses for the condition in terms of the strength of the cylindrical aberration and the angle of its principal axis.

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Yes, it is because, when different light Ray's hit the surface of the cylinder at the same angle along the horizontal the angle of reflection will be same for all.enter image description here

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