K. Koh et al., RECONSTRUCTION OF POLYHEDRA BY A MECHANICAL THEOREM-PROVING METHOD, IEICE transactions on information and systems, E76D(4), 1993, pp. 437-445
In this paper we propose a new application of Wu's mechanical theorem
proving method to reconstruct polyhedra in 3-D space from their projec
tion image. First we set up three groups of equations. The first group
is of the geometric relations expressing that vertices are on a plane
segment, on a line segment, and forming angle in 3-D space. The secon
d is of those relations on image plane. And the rest is of the relatio
ns between the vertices in 3-D space and their correspondence on image
plane. Next, we classify all the groups of equations into two sets, a
set of hypotheses and a conjecture. We apply this method to seven cas
es of models. Then, we apply Wu's method to prove that the hypotheses
follow the conjecture and obtain pseudodivided remainders of the conje
ctures, which represent relations of angles or lengths between 3-D spa
ce and their projected image. By this method we obtained new geometric
al relations for seven cases of models. We also show that, in the regi
on in image plane where corresponding spatial measures cannot reconstr
ucted, leading coefficients of hypotheses polynomials approach to zero
. If the vertex of an image angle is in such regions, we cannot calcul
ate its spatial angle by direct manipulation of the hypothesis polynom
ials and the conjecture polynomial. But we show that by stability anal
ysis of the pseudodivided remainder the spatial angles can be calculat
ed even in those regions.