Swiss/French artist Felice Varini [difficult navigation on the site; choose "index général" for artwork list]  paints walls (both indoor and outdoor) in such a way that, when viewed from a particular viewpoint, perspective effects vanish and simple, flat geometrical patterns appear.

How does this happen?  Most is explained by basic projective geometry...

Consider the following artwork , which generates flat concentric circles when viewed from the correct viewpoint:

Question: Is the shape of the curves drawn on the walls analytical? Can they be described as circumferences, ellipses, cubic curves, higher order curves, or what else?

Solution: what the camera sees from the correct viewpoint is a number of circles: consider just one of them from now on.  Its interpretation surface is a cone, with circular base (this is not true if the camera is not natural, but this is not relevant in this context);  the curve drawn on the wall is the intersection between this interpretation surface and the wall.

Since the wall is flat, the curve is a conic section .  Note: this is not the same as answering "an ellipse": for example, the curves drawn on the floor could be parts of an hyperbola.

Under what hypotesis the curves drawn on the floor would be parts of an hyperbola?  If the line containing both the viewpoint and the point on the image plane at the center of the projected circles was parallel to the floor.

Can you determine if the hypotesis holds in the current context?   (You can; you don't even need a ruler... Hint: the lateral walls are parallel planes; then...)

What about the curves drawn on the walls? 

Question: How would you determine the "correct" viewpoint in 3D?

Solution: There are at least a couple of ways to compute where the correct viewpoint lies.

a) Consider the curve drawn on a single wall; this is a part of a conic section we'll call w from now on.  Since a conic section has 5 parameters, 5 points on the curve (theoretically even very close toghether) allow us to completely parametrize w.  This allows to enforce some constraints on the position and orientation of the camera such that w projects to a circle (provided that  the camera is natural).  By considering curves on multiple walls, a sufficient number of constraints should completely define the viewpoint position and orientation (i.e. the camera parameters).

b) An easier and more intuitive solution is the following: consider the intersection between a lateral wall and the ceiling (or the floor).  When seen from the correct viewpoint, the projections of the two curves blend toghether without slope discontinuity.  Call t_c the tangent line to the curve on the ceiling at its extreme point, and call t_w the tangent line to the related curve on the wall at its extreme point.  Note that t_c and t_w intersect; also, t_c lies on the plane of the ceiling, whereas t_w lies on the plane of the lateral wall.  In order for the ceiling and wall curve to project a single continuously derivable curve, t_c and t_w should project to the same line on the image.  That is to say, the viewpoint must be placed on the plane containing both these lines (there exists such a plane since, as we already noted, t_c and t_w do intersect).  By applying this to three different junction points between drawn curves, we fix the viewpoint (intersection of three planes).

Note: this method does not allow to derive the correct orientation of the camera; why? Because it only enforces that the curves we see are continuously derivable.  If the camera center is placed correctly but the image plane is oriented incorrectly, what do we see instead of circles? Ellipses, which are continuously derivable.

This video, of a different artwork , shows how the curves match when the viewpoint is placed correctly.

Problem: consider the following artwork by the same artist:

Question: is there a "correct" viewpoint in the room, even if you remove the (flat) mirror?

Solution: yes and no; there is still a correct viewpoint; unfortunately, it is not in the room; it is under the feet of the photographer.  Exactly, the correct viewpoint is the point symmetrical to the viewpoint of othe photo w.r.t. the plane of the mirror.