Editing Optical aberration (section)

===Distortion of the image===
[[File:Barrel distortion.svg|thumb|right|'''Figure 3a''': Barrel distortion]]
[[File:Pincushion distortion.svg|thumb|right|'''Figure 3b''': Pincushion distortion]]
{{main|Distortion (optics)}}
Even if the image is sharp, it may be distorted compared to ideal [[pinhole projection]].  In pinhole projection, the magnification of an object is inversely proportional to its distance to the camera along the optical axis so that a camera pointing directly at a flat surface reproduces that flat surface. Distortion can be thought of as stretching the image non-uniformly, or, equivalently, as a variation in magnification across the field. While "distortion" can include arbitrary deformation of an image, the most pronounced modes of distortion produced by conventional imaging optics is "barrel distortion", in which the center of the image is magnified more than the perimeter ('''Figure 3a'''). The reverse, in which the perimeter is magnified more than the center, is known as "pincushion distortion" ('''Figure 3b'''). This effect is called lens distortion or [[image distortion]], and there are algorithms to correct it.

Systems free of distortion are called ''orthoscopic'' ({{Transliteration|grc|orthos}}, right; {{Transliteration|grc|skopein}}, to look) or ''rectilinear'' (straight lines).

[[File:ABERR3rev.svg|left|frame|'''Figure 4''']]
This aberration is quite distinct from that of the sharpness of reproduction; in unsharp, reproduction, the question of distortion arises if only parts of the object can be recognized in the figure. If, in an unsharp image, a patch of light corresponds to an object point, the ''center of gravity'' of the patch may be regarded as the image point, this being the point where the plane receiving the image, e.g., a focusing screen, intersects the ray passing through the middle of the stop. This assumption is justified if a poor image on the focusing screen remains stationary when the aperture is diminished; in practice, this generally occurs. This ray, named by Abbe a ''principal ray'' (not to be confused with the ''principal rays'' of the Gaussian theory), passes through the center of the entrance pupil before the first refraction, and the center of the exit pupil after the last refraction.  From this it follows that correctness of drawing depends solely upon the principal rays; and is independent of the sharpness or curvature of the image field. Referring to '''Figure 4''', we have {{math|1= ''{{prime|O}}{{prime|Q}}''/''OQ'' = (''{{prime|a}}'' tan ''{{prime|w}}'')/(''a'' tan ''w'') = 1/''N''}}, where {{mvar|N}} is the ''scale'' or magnification of the image. For {{mvar|N}} to be constant for all values of {{mvar|w}}, {{math|(''{{prime|a}}'' tan ''{{prime|w}}'')/(''a'' tan ''w'')}} must also be constant. If the ratio {{math|''{{prime|a}}''/''a''}} be sufficiently constant, as is often the case, the above relation reduces to the ''condition of [[George Biddell Airy|Airy]],'' i.e. {{math|tan ''{{prime|w}}''/tan ''w''}} is a constant.<ref>{{cite journal |last= Airy |first= George Biddell |date= 1830 |title= On the Spherical Aberration of the Eye-Pieces of Telescopes |journal= Transactions of the Cambridge Philosophical Society |volume= 3 |pp= 1–58 |quote= It is evident that an object will be seen without distortion if its image, exactly similar to the object, be formed on a plane; and then the trigonometrical tangent of the angle, made with the axis of the lens by the axis of the pencil after refraction, will bear to the tangent of the angle before refraction a constant ratio: if the ratio be not constant, its difference from a constant ratio will indicate the degree of distortion. |url= https://archive.org/details/transactionsofca03camb/page/4/ |access-date= 2025-03-25}}</ref> This simple relation is fulfilled in all systems which are symmetrical with respect to their diaphragm (briefly named ''symmetrical or holosymmetrical objectives''), or which consist of two like, but different-sized, components, placed from the diaphragm in the ratio of their size, and presenting the same curvature to it (hemisymmetrical objectives); in these systems {{math|1=tan ''{{prime|w}}'' / tan ''w'' = 1}}.<ref name=EB1911/>

The constancy of {{math|''{{prime|a}}''/''a''}} necessary for this relation to hold was pointed out by R. H. Bow (Brit.  Journ.  Photog., 1861), and Thomas Sutton (Photographic Notes, 1862); it has been treated by O. Lummer and by M. von Rohr (Zeit. f.  Instrumentenk., 1897, 17, and 1898, 18, p.&nbsp;4). It requires the middle of the aperture stop to be reproduced in the centers of the entrance and exit pupils without spherical aberration. M. von Rohr showed that for systems fulfilling neither the Airy nor the Bow-Sutton condition, the ratio {{math|(''{{prime|a}}'' cos ''{{prime|w}}'')/(''a'' tan ''w'')}} will be constant for one distance of the object.  This combined condition is exactly fulfilled by holosymmetrical objectives reproducing with the scale 1, and by hemisymmetrical, if the scale of reproduction be equal to the ratio of the sizes of the two components.<ref name=EB1911/>