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( Succ ( BF 1( r ))) is not link-2 visible from Succ ( r ). If the vertex BB 1( r )( BF 1( r ))
exists, we define the backward (forward) link-2 ray-shooting of r , denoted by
BB 2( r )( BF 2( r )), to be the smallest (largest) boundary point such that no point
of P ( BB 2( r ) ,BB 1( r )) ( P ( BF 1( r ) ,BF 2( r ))) is link-2 visible from Succ ( r ). The
backward (forward) link-2 component of r is then defined as the boundary chain
P [ BB 1( r ) ,BB 2( r )] ( P [ BF 2( r ) ,BF 1( r )]). See Fig. 2 (a). Note that P [ r, B ( r )] is
always contained in the backward or forward link-2 component of r . Also, if
the forward component of r is non-redundant, we analogously define its back-
ward/forward link-2 ray-shooting and the backward/forward link-2 component
P [ FB 1( r ) ,FB 2( r )]/ P [ FF 2( r ) ,FF 1( r )]. See Fig. 2 (a). (In the above notion, the
first letter “B” (or “F”) stands for the original backward (or forward) component
of r .) Note that it is possible that only one of the backward and forward link-2
ray-shootings from r is defined. See Fig. 2 (b). (Since P is not LR -visible, at least
one of the backward and forward link-2 ray-shootings from r is defined.)
BF2(a)
BF1(a)
FB2(b)
c
FB1(b)
b
a
b
a
FF1(b)
BB1(c)
BF2(b )
F F1(a)
FF2(b)
BB1(a)
BB2(c)
FF2(a)
BF1(b)
BB2(a)
(b)
(a)
Fig. 2. Link-2 ray-shootings and link-2 components.
We call the link-2 components, which are derived from the non-redundant
backward and forward components, the link-2 ʱ -components and ʲ -components,
respectively. A link-2 ʱ -component is non-redundant if it does not contain any
other link-2 ʱ -component, no matter which is the forward or backward
ʱ -component. For instance, the ʱ -component P [ BB 1( c ) ,BB 2( c )] in Fig. 2 (b)
contains the ʱ -component P [ BF 2( b ) ,BF 1( b )] and is thus redundant. (Since P is
not LR -visible, both the forward and backward link-2 ʱ -components of a vertex
r cannot be non-redundant simultaneously.) We make the analogous definition
for the non-redundant ʲ -components.
From the definition of link-2 ʱ -components, the following observation can
simply be made. (The observation on the link-2 ʲ -components can be made
analogously.)
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