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"The knowledge at which geometry aims is the knowledge of the eternal".

Plato --The Republic, Book VII, 427-347 BC

"Plato seems to have foreseen, on the basis of what must have been very
sparse evidence indeed at that time, that: on the one hand, mathematics
must be studied and understood for its own sake, and one must not demand
completely accurate applicability to the objects of physical experience; on the
other hand, the workings of the actual external world can ultimately be
understood only in terms of precise mathematics".

Roger Perrose --The Emperor's New Mind, Chapter 5, 1991 [74].


 

Telling whether a given line drawing correctly represents the projection of a plane-faced polyhedron in 3D-space is a long-standing challenge for Artificial Intelligence and Robotics. The availability of an efficient algorithm achieving this task would have important applications in projective geometry, image understanding, monocular vision, automatic 3D reconstruction and modelling of real environments.
The aim of this project is to find out a necessary and sufficient condition for realizability.


First Look

A polyhedron is a solid object bounded by a finite number of planar faces. It is not necessarily convex; it may be concave or even have holes.
A line drawing is made of straight line segments and junctions --points where two or more of the segments meet. A line drawing is said to be realizable or correct if it is the orthographic or perspective projection of some 3-dimensional scene of polyhedral objects, and non-realizable orincorrect otherwise. Such a scene is known as a spatial realization or interpretation of the drawing.

Here there are some examples of correct drawings:

and some other incorrect ones (can you tell why?):


Truncated Pyramid Question

The most naive type of the realization problem is the following:

Naive Problem:  Given a line drawing, judge whether a polyhedron is realizable from the line drawing.

Why do we say this question is naive ? Because without some constraints, it's impossible to answer whether a polyhedron is realizable from a line drawing. For example, let's look at that classical  Truncated Pyramid Question, we will mention it many times in this project. The question is: consider a line drawing shown in the following figure(a):


Figure:  Intuitive interpretation and tricky interpretation of a line drawing.







One may say yes, because it can usually be regarded as an illustration of a truncated pyramid seen from above. In a mathematical sense, however, it cannot be a picture of any truncated pyramid. Because the three side faces of a truncated pyramid should have a common point of intersection in a space(when extended), and hence the three lines corresponding to the pairwise intersections of the three faces should meet at a common point on the picture plane, but obviously they do not as shown in (b). So it seems that one should answer that a polyhedron is not realizable from such line drawing. We can, however, object to this answer. Indeed that line drawing cannot be a picture of any truncated pyramid, but can be a picture of other kind of polyhedrons. The object shown in (c) is an example of a polyhedron that can yield the line drawing in (a). So it seems that one should answer yes again !
From this example, we see how tricky that object and situation are ! Thus, it's nonsense to discuss whether a polyhedron is realizable from a line drawing without any constraints. For this sake, Sugihara introduced a list of assumptions on objects, on view points, on drawing rules, etc. So, let's move on to the assumptions......
 


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