The the term "mapping" shrouds the process of aligning imagery with the architectural features and proportions of a given two dimensional plane or three dimensional object utterly mysterious.
Mapping has been employed successfully on many well publicized projects. It has widely been used for product launches or opening ceremonies (recordings of which find vast audiences via the internet). The mapping of imagery to a building's architectural features has become particularly popular whether it is on the ornate walls of Carnegie Hall or the Neo-Classical columns outside numerous public buildings. This in turn has led to many more directors, designers, creative agencies and promoters to ask for "mapping." Of course, a lot of these projects had a 14 person army of specialists employed to produce the desired result. The theater in comparison is a modest environment compared to say a product launch for a large, multinational corporation. However, the technical concept behind the implementation of mapped imagery is the same.
The concept of mapping is quite old in the sense that it was employed in most of the earlier examples of large scale theater projection back in the 1960's. Back then, technicians and designers referred to it as "pre-distortion." Richard Pilbrow describes the process in vivid detail in his recent book, A Life In Theater. Together with his colleagues from Theatre Projects, he developed a method that could be employed on any project. It was possible for him to do that because he understood the limitations of projectors, optics and what it means to use these within the architectural constraints of the theater. He put a camera into a scale model of the set design and took a picture of the projection surface. The position of the camera is crucial and must be in the same position the projector is to be rigged. The lens of the camera must be equivalent or close to that of the projector to emulate the way the real projector will "look" at the real set.
I think it is worthwhile repeating the central principle: if the projector displays the same image seen through its own lens, then the projected image and reality match. In other words, if you make the virtual reality match the physical reality, then the imagery is properly mapped. Virtual reality could also be less virtual as in the case of Richards’ approach: a scale model instead of a computer model. The important criteria are that the proportions, distances and positions of all elements (notably projectors, optics and screens) are the same as they will be in the theater where the production takes place. There is much more to this in practice and depending on the particular application or project, the technical solution may vary. In general, this is the principle and idea that makes it all work.