A New Concept and Method for Line Clipping • 5
Solving these inequalities is actually a max/min problem, which can be seen
as follows. Recall that t _> qi/pi for all i such that pi < 0, and also that t _> 0.
Thus,
t >_ max({qdp~lp~ < 0, i = 1, 2, 3, 4} U {0}). (2.10)
Analogously, t <_ qdP~ for all i such that p~ > 0, and also t _< 1. Thus,
t <_ min({qdpilp, > 0, i = 1, 2, 3, 4} U {1}). (2.11)
Finally, if Pi = 0 in the ith inequality for some i, then there are two possibilities.
If q~ _ 0, then there is no useful information to be gleaned from this inequality,
and this inequality can be discarded. If qi < 0, then this is a trivial reject case,
and the clipping problem is solved with no further computation needed.
• The right-hand sides of inequalities (2.10) and (2.11) are the values of the
parameter t corresponding to the beginning and end of the visible segment,
respectively (assuming there is a visible segment).
Denoting these parametric values as to and tl,
to = max({qdp~lPi < 0, i = 1, 2, 3, 4} U {0}) (2.12)
tl ----
min({qdp,[p~ > 0, i = 1, 2, 3, 4} U {1}).
If there is a visible segment, it correspnds to the parametric interval
to -< t _< tl. (2.13)
Hence, a necessary condition for a line segment to be at least partially visible is
to -< t~. (2.14)
This is not a sufficient condition because it ignores the possibility of a trivial
reject due to p~ = 0 with qi < 0. Nonetheless, this yields a sufficient condition for
rejection; specifically, if to > tl, then this is another reject case. The algorithm
checks if pi -- 0 with qi < 0 or if to > tl, in which case the line segment is
immediately rejected without further computation.
In the algorithm, to and t~ are intialized to 0 and 1, respectively. Then each
ratio qdPi is considered successively. Ifpi < 0, the ratio is first compared with t~,
and if it is greater, then to will exceed t~ and this must be a reject case; otherwise,
it is compared with to, and if it is greater, then to must be updated with this new
value. Ifp~ > 0, the ratio is first compared with to, and if it is less, then this is a
reject case; otherwise it is compared with t~, and if it is less, then tl is updated.
Finally, if pi - 0 and qi < 0, then this is a reject case. At the last stage of the
algorithm, if the line segment was not rejected, the parametric values to and t~
are used to compute the corresponding points. However, if to = 0, then the
endpoint is Vo and need not be computed; similarly, if t~ -- 1, then the endpoint
is V~.
The geometric meaning of this process is that the line segment is "squeezed"
down by considering where the extended line intersects each boundary line. More
specifically, each endpoint of the given line segment VoV~ is used as an initial
value for an endpoint of the visible segment V~V~. Then the point of intersection
of the extended line with each boundary line is considered. (This visibility
ACM Transactions on Graphics, Vol. 3, No. 1, January 1984.
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