Page 145 - DCAP504_Computer Graphics

Computer Graphics

The Liang-Barsky algorithm combines the above parametric line equations with the point clipping
conditions to obtain the inequalities as below:

xw min ≤x 0 + u ∆ x ≤xw max
yw min ≤y 0 + u ∆ y ≤yw max
The above equation can be split into four separate equations:

xw min ≤ x 0 + u ∆ x
xw max ≥x 0 + u ∆ x

yw min ≤y 0 + u ∆ y
yw max ≥y 0 + u ∆ y
Each of these inequalities specifies the limits set for each clipping boundary. The first equation
corresponds to the left hand boundary, the second equation corresponds to the right hand boundary,
the third equation corresponds to the lower boundary, and the fourth equation corresponds to the
upper boundary. These four inequalities can be expressed as:
u.p k ≤ q k where k = 1, 2, 3, 4
The parameters p and q can be defined as:
k = 1 p 1 = - ∆ x q 1 = x 0 - xw min

k = 2 p 2 = ∆ x q 2 = xw max - x 0
k = 3 p 3 = - ∆ y q 3 = y 0 - yw min

k = 4 p 4 = ∆ y q 4 = yw max - y 0
Consider k =1, the parameters defined for this value are p 1 = - ∆ x and q 1 = x 0 - xw min. Now insert
these into the equation u.p k ≤ q k . The result obtained is:
- ∆ x ≤ x 0 - xw min
Now rearranging the equation you will get:

xw min ≤ x 0 + u ∆ x
This indicates that k = 1 corresponds to the limits set by the left hand clipping boundary.
The line parallel to one of the clipping window edges has p k = 0. Here, the value of k corresponds to
that boundary where k = 1, 2, 3, and 4 corresponding to left, right, bottom and upper boundaries
respectively. For that value of k, if q k < 0 then the line lies outside the clipping boundary. If q k > 0, then
the line is inside the clipping boundary.

If p k < 0, then the line extends from outside to inside the infinite extension of the clipping window edge.
If p k > 0, then the line extends from inside to outside. If p k value is nonzero, then the value of u, which
corresponds to the point where the extended line intersects the extension of the clipping window edge,
can be calculated using the below equation:
u = q k / p k

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