The Liang-Barsky algorithm is a line clipping algorithm. This algorithm is more efficient than Cohen–Sutherland line clipping algorithm and can be extended to 3-Dimensional clipping. This algorithm is considered to be the faster parametric line-clipping algorithm. The following concepts are used in this clipping:
- The parametric equation of the line.
- The inequalities describing the range of the clipping window which is used to determine the intersections between the line and the clip window.
The parametric equation of a line can be given by,
X = x1 + t(x2-x1) Y = y1 + t(y2-y1)
Where, t is between 0 and 1.
Then, writing the point-clipping conditions in the parametric form:
xwmin <= x1 + t(x2-x1) <= xwmax ywmin <= y1 + t(y2-y1) <= ywmax
The above 4 inequalities can be expressed as,
tpk <= qk
Where k = 1, 2, 3, 4 (correspond to the left, right, bottom, and top boundaries, respectively).
The p and q are defined as,
p1 = -(x2-x1), q1 = x1 - xwmin (Left Boundary) p2 = (x2-x1), q2 = xwmax - x1 (Right Boundary) p3 = -(y2-y1), q3 = y1 - ywmin (Bottom Boundary) p4 = (y2-y1), q4 = ywmax - y1 (Top Boundary)
When the line is parallel to a view window boundary, the p value for that boundary is zero.
When pk < 0, as t increase line goes from the outside to inside (entering).
When pk > 0, line goes from inside to outside (exiting).
When pk = 0 and qk < 0 then line is trivially invisible because it is outside view window.
When pk = 0 and qk > 0 then the line is inside the corresponding window boundary.
Using the following conditions, the position of line can be determined:
|Condition||Position of line|
|pk = 0||parallel to the clipping boundaries|
|pk = 0 and qk < 0||completely outside the boundary|
|pk = 0 and qk >= 0||inside the parallel clipping boundary|
|pk < 0||line proceeds from outside to inside|
|pk > 0||line proceeds from inside to outside|
Parameters t1 and t2 can be calculated that define the part of line that lies within the clip rectangle.
- pk < 0, maximum(0, qk/pk) is taken.
- pk > 0, minimum(1, qk/pk) is taken.
If t1 > t2, the line is completely outside the clip window and it can be rejected. Otherwise, the endpoints of the clipped line are calculated from the two values of parameter t.
- Set tmin=0, tmax=1.
- Calculate the values of t (t(left), t(right), t(top), t(bottom)),
(i) If t < tmin ignore that and move to the next edge.
(ii) else separate the t values as entering or exiting values using the inner product.
(iii) If t is entering value, set tmin = t; if t is existing value, set tmax = t.
- If tmin < tmax, draw a line from (x1 + tmin(x2-x1), y1 + tmin(y2-y1)) to (x1 + tmax(x2-x1), y1 + tmax(y2-y1))
- If the line crosses over the window, (x1 + tmin(x2-x1), y1 + tmin(y2-y1)) and (x1 + tmax(x2-x1), y1 + tmax(y2-y1)) are the intersection point of line and edge.
This algorithm is presented in the following code. Line intersection parameters are initialised to the values t1 = 0 and t2 = 1.
Reference: Computer Graphics by Donald Hearn, M.Pauline Baker
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