In vain your image comes to meet me And does not enter me where I am who only shows it Turning towards me you can find On the wall of my gaze only your dreamt-of shadow I am that wretch comparable with mirrors Than can reflect but cannot see Like them my eye is empty and like them inhabited By your absence which makes them blind. 3 - Contre-chant, Louis Aragon

What is the difference between a regular perspectival painting and an anamorphic one if both contain distortions and both are profitably viewed from the side? In the case of linear perspective, an artist is usually intent on rendering an entire scene spatially. Here distortion of certain parts is an undesired by product.
The term anamorphosis is often used loosely to refer to distortion in general. It is more appropriately used to refer to those cases where distortion is a desired effect. In anamorphosis an artist usually limits himself to a single object and deliberately projects it (or changes its scale) so that it is deformed or even unrecognizable when viewed from directly in front. Sometimes such objects are integrated into an otherwise normal scene, as for example, with Holbein's skull in The French Ambassadors. When viewed from directly in front as in Figure 5, the paining seems perfectly normal, except for a strange, elongated object, which seems to float above the floor. If we change our viewpoint to one at an oblique angle at the extreme right of the painting as in Figure 6, the object is seen as a human skull. The main point is that there is no single correct viewpoint for this painting.

Figure 5, Hans Holbein the Younger, Jean de Dinteville and Georges de Selve (The French Ambassadors), oil on oak, 81X82 in. 1533. National Gallery, London.

Frequently the deformed object dominates and is surrounded by regular perspectival landscapes. These theoretical considerations not only exemplify a wider anamorphic intrigue of The French Ambassadors, but also serve here as first steps to considering anamorphosis itself as a not only a way of seeing, but also the dynamic embedded in the theoretical decoy. For primarily we see that The French Ambassadors embodies a relation between the recognizability of trompe l'eoil effect and meaningless stain.
The French Ambassadors then is an example of the intersection between the two pictorial denominations of representation and abstraction. Holbein's Ambassadors conflates two points-of-view into a single surface in a manner that also visually highlights their separation and distinction. It anticipates the traditional perpendicularity, filling as it does, the front-on pictorial space with all manner of precisely rendered representations and symbols. But The French Ambassadors also disrupts that clarity with an unrecognizable "stain".

This visually incorrect disruption is one location of the anamorphic potential, and as such it is correct point-of-view is hidden elsewhere.
Once found, this elsewhere reveals its pictorial intentionality. That is, once the beholder correctly locates the second axial point-of-view, the un-interpretable stain emerges into visibility. What was hidden within unclear distortion now appears in clarity—the stain appears as a skull. This second position puts the viewer approximately seven feet to the left of the painting's frame, and virtually parallel with the wall. Now, looking slightly down to the "intransigent" mark, it appears to lift off the surface into pictorial legibility.
Anamorphosis started being considered and alternative to linear perspective. In the exploration of anamorphosis in cases involving complex projection planes, practical incentives again played a role. Artists tended to paint landscapes, scenes and buildings in accordance with linear perspective and then paint individual objects within this context in accordance with optical adjustments methods.

Although linear perspective theoretically introduces the possibility of a uniform treatment of space, painting practice continues to be much more complex. Compromises are made in the case of objects below, above and off to the side. Sometimes these compromises are made in accordance with a theory of vision and sometimes not. There is no necessary connection between theory of representation, practice of representation, theory of vision and actual visual experience.
Figure 6, the side view from the viewpoint
for the skull. From Mark Frantz and Annalisa
Crannel, Viewpoints, Mathematical
Perspective and Fractal Geometry in Art. 2011

It is certainly possible to impune certain motives for sketching the arguments above---as well as discussing certain works and not others. While anamorphosis serves as the formal device that links a heterogeneous body of material, two assumptions have guided this eccentric observer in the construction of art experiences which are at once aesthetic and critically based: first, to render an observer self-aware and conscious of the processes by which meaning is constructed; and, second, to acknowledge the central role the identity and capacities of the observer have in determining the nature of the art experience. It is to these goals that the concept of anamorphosis must be subordinated.

The importance of anamorphosis, then, is not as an alternative to the way we experience works--that is, like the latest video game or clothes for the emperor--but as a metaphor for accepting information from unfamiliar places and unexpected sources. The eccentric observer is one who acknowledges the limitations of a static, homolographic world-view and embraces instead a dynamic unfolding process encompassing a field of light and texture shot through with expressions of personality and specificity.
This process could serve at least two purposes: to undermine a certain ideology of vision that continues to dominate the way we look at artwork and to celebrate (rather than suppress) the idiosyncratic nature of our own point of view.