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"As I tell my students, algebra is cosmetics, not surgery:
change the appearance but not the substance".

Walter Whitely --The Decline and Rise of Geometry
in 20th Century North America, 1999 [125].


 

A perpendicularity property

Besides the Labeling algorithm, there is still another very elegant theorem used for the recognition of the line drawings, which is called the Perpendicularity Condition. This theorem is based on the following remarkable property: in the 2D-space, i.e. xy plane, an arbitrary line can be represented by equation ax+by+c = 0, this line is perpendicular to the vector (a, b), as showed in the following figure:


Perpendicularity in the Gradient Space

With this remarkable geometric property, we can conclude the Perpendicularity Condition. Firstly we need definitions of several new concepts. Let P  be a polyhedron fixed in the space, and  fj   be the  jth face of P. Let equation

ajx+bjy+z+cj= 0
denote the planar surface on which  fj  lies. Note: this equation cannot represent a plane that is parallel to the z axis, but there is no problem if we take Assumption W1. The equation has three parameters aj, bj, and cj . Then we have:
Gradient:  the order pair Gj = (aj, bj) is called the gradient of the surface.
Gradient Space:  a 2D space with the (a, b) coordinate system. Obviously, Gj is a point in this gradient space.
Let f1 and f2 be two faces that share a common edge. The orthographic projection of this edge onto the xy plane is represented by the equation:
(a1-a2)x+(b1-b2)y+(c1-c2) = 0
From the property we mentioned above, we know that this projected edge is perpendicular to the vector (a1-a2, b1-b2). In other words, if we superimpose the gradient space upon the picture plane in such a way that the x and y axes are respectively parallel to the a and b axes, then the line connecting a pair of the gradients of faces that share a common edge is perpendicular to the image of the edge.
 


Perpendicularity Condition

We can use the above perpendicularity property to check the realizability problem, that is:

Theorem:  For a type-PK problem, a polyhedron is realizable only when the faces can be mapped to distinct points in the picture plane in such a way that, if a pair of faces share a common edge, the line connecting the corresponding pair of points is perpendicular to the image of the edge.

Now we can use this theorem to check that tricky Truncated Pyramid Question. As shown in the (a) of the following figure:


Figure:  Perpendicularity Condition.




We can plot the first point G1 arbitrarily and the second point G2 in an arbitrary distance from G1. If we proceed plotting G1, G2, G3 in this order using the Perpendicularity Condition stated in the Theorem, we find that the last point G4 cannot be plotted. Thus, we can recognize the inconsistency in interpreting the picture as a truncated pyramid.
 


Discussion

However, this condition is not sufficient. For example, a labeled line drawing in the (b) of the above figure represents the interpretation of the picture as an object that is obtained from a truncated pyramid by digging a hole so that a triangular top face is replaced with a hole penetrating the object. This interpretation is likewise incorrect, but we can plot the gradients of all the three visible faces without violating the Perpendicularity Condition, as shown in (b). Thus, this condition is necessary but not sufficient even if there is no overlap among visible faces.
Well, let's move on toward our aim ! In the following part, we are going to introduce another interesting condition which is called the Reciprocity Condition.
 
 

Labeling SchemeHomeReciprocity Condition