# Cyclical transformation A graphical depiction of a cyclical transformation of a body or system, on a Y,X-plane, the variables typically being pressure and volume, respectively, showing a body at an initial state A transforming along path M to state B in the expansion stroke, then transforming again along path N, in the contraction stroke, to final state A, which is also the initial state (macroscopically speaking). 
In thermodynamics, cyclical transformation is a transformation of a body or system in which the initial state and final state are the same. 

A cyclic transformation is sometimes called a cycle, for short, if the initial and final states are the same; although a cycle tends to refer to a two step cycle, i.e. an expansion stroke, followed by a contraction stroke; whereas a transformation may follow any number of intermediate paths.

A cyclical transformation, subsequently, brings the system back to its initial state. If the state of the system can be represented on a PV diagram, then a cyclical transformation can be represented by a closed curve, such as the curve AMBN, depicted adjacent. 

The pressure volume work (PV work) done by the system during the cyclical transformation, if the ordinate (vertical axis) and abscissa (horizontal axis) are pressure and volume, is represented geometrically by the area enclosed by the curve representing the cycle or AMBN, which equals the area of the region enclosed by AMBB'A'A less the area ANBB'A'A or expansion work less contraction work.

To note, if the cycle is performed in the counterclockwise direction, ANBM, the work again will be given by the enclosed region, only in this case it will have a negative value.

Mathematics
Cyclical transformations are quantified or calculated by the cycle integral.

References
1. Fermi, Enrico. (1936). Thermodynamics (pg. 6). Prentice-Hall.
2. Preston, Thomas.(1894). A Theory of Heat (section 283: Remarks on the Energy Equation: Cyclical Transformations, pgs. 597-99). MacMillan.
3. Isochoric process – Wikipedia.