Focus breathing is any geometric change in field-of-view with adjustment in focus distance. Cinematographers noticed these changes in their moving pictures. The visual effect reminded them of the filling and emptying of lungs or other bags of air, hence they used the term breathing.
Breathing can involve changes of angle-of-view, but it is not limited to changes of angle-of-view. Any geometric change in the field-of-view is breathing. In fact a shift from pincushion distortion to barrel distortion more closely resembles breathing than a simple change in angle-of-field.
Our eyes show little breathing when focusing. They focus by changing the curve of their fixed position squishy lens. This change of curve changes the lens’s optical power which is also described by a focal length. Focal length is the typical way to measure optical power.
Camera lenses tend to use stiff lens elements of set focal length/power. Simple unit focus prime lenses are focused by extension. This extension reduces the angle-of-view at the film plane.
It easy to believe that focal length and angle-of-view are directly related. By this view unit focused lenses seem to increase their focal length when extended. However optical power and focal length are properties which do not change with lens position. This also leads to conflating breathing with focal length changes with focus distance. This has progressed to complaints about so-called breathing of lenses with excellent mechanical breathing compensation.
Focusing by extension has become less popular for camera lenses. Most cameras lenses now use some form of internal focus. This has had effects on their breathing characteristics. It has generally not eliminated breathing.
A camera lens might designed to be as compact as possible. This is often done with large range normal zooms. These lenses can increase their angle-of-view with closer focus–the opposite breathing behavior to extending focus lenses.
If reduced angle focus breathing is considered positive breathing, increased angle breathing can be referred to as negative breathing. Remember that changing angle-of-view is only one aspect of breathing.
More camera lenses are being designed with breathing compensation to show less breathing. These tend to be longer, internal focus lenses.
Long lenses are generally used for higher magnification so breathing can be an accepted performance tradeoff, and breathing compensation can be seen as a liability. This is often the case with lenses used for portraiture or macro. An example would be a 70-200mm varifocal lens that reduces to 135 mm at the long end at closest focus. Or a 200-600mm zoom used for wildlife that reduces to 400 mm at the long end at closest focus. Extension can be used to increase magnification by recovering focal length with the loss of ability to focus at infinity.
Focus stacking post-processing has become popular leading to demands for breathing corrected lenses for use with focus bracketing for macro and other uses.
Some cameras now have electronic breathing compensation which digitally crops images as an extension of their geometric distortion compensation.