*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 *f _{j}*
be the

LetGradient: the order pair G_{j}= (a_{j}, b_{j}) is called the gradient of the surface.

Gradient Space: a 2D space with the (a, b) coordinate system. Obviously, G_{j}is a point in this gradient space.

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 G_{1} arbitrarily and the second
point G_{2} in an arbitrary distance from G_{1}. If we
proceed plotting G_{1}, G_{2}, G_{3} in this order
using the Perpendicularity Condition stated
in the Theorem, we find that the last point G_{4} 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.