Lune-Based Neighbourhoods
We difine N(x,y,) to be the intersection of the two circles of
radius
d(x,y)/2 centered at the points (1-
/2)x+(
/2)y and
(
/2)x + (1-
/2)y, respectively.
When =1, N(x,y,
) corresponds exactly to
the Gabriel neighbourhood
of x and y. When
=2, we get the ``relative neighbourhood'' of the RNG.
As
approaches
, the neighbourhood of x and y approximates the
infinite strip formed by translating the line segment (x, y) normal to itself.
We difine N(x,y,) to be the intersection of the two circles of radius
d(x,y)/(2
) passing through both x and y. When
=1, this is
consistent with the definition abvove. As
approaches 0, N(x,y,
)
approximates the line segment joining x and y. Thus, except in degenerate
situations (three or more points colinear), all point pairs are
-neighbours under this scheme for
sufficiently small.