By decoding the 5-tile code sequence of a polygonal chain, we would
obtain a n-simplex
sequence representation of the chain, which accurately
reproduces the local structure. In the following, we permit translation
during the approximation process to keep the length of the line
segments
of a polygonal chain equal. (See PROGRAMS / TetraGenSeq for
the program.)
Shown on the left in figure A is a polygonal chain with 5-tile
code sequence 9HHIA81.
As you see, simple
approximation of the chain by a triangle sequence
results in a folded tetrahedron sequence whose 5-tile code sequence,
9GHIB81, is
different from that of the original chain. And some of the
local features of the original chain are missed in the representation.
Fig.
A:
Simple approximation of a shape
On the other hand, if we consider decoding of its 5-tile
code sequence, we obtain another representation of the
polygonal chain as shown in figure B. Since the obtained
representation has the same 5-tile code sequence as the original chain,
no local
structures are damaged during the approximation process.
Fig.
B: Decoding of a 5-tile code sequence
(2) Polygonal chains with the same 5-tile code sequence
Figure C shows variation of backbone
of folded triangle sequences which share the same 5-tile code sequence.
For example, the pink and red polygonal chains have the same 5-tile
code sequence of 8GHI
(left).
Fig.
C: Polygonal chains with the same 5-tile code sequence
Figure D shows the structural difference between backbones of folded
triangle fragments of different 5-tile code sequences. For
example, in the left figure, the red polygonal chain of 8GHI is
compared with the black polygonal chain of 9GHI, where the circle denotes the
location of the 5-tile code disagreement.
Fig.
D: Polygonal chains with different 5-tile code sequences
