Liang-Barsky Algorithm

18 Mar, 2019
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 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:
Parameters t1 and t2 can be calculated that define the part of line that lies within the clip rectangle.When,
pk < 0, maximum(0, qk/pk) is taken.pk > 0, minimum(1, qk/pk) is taken.
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.
Algorithm –
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.
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.
#include"graphics.h"#define ROUND(a) ((int)(a+0.5))int clipTest (float p,float q, float * tl, float * t2){float r ;int retVal = TRUE;  //line entry pointif (p < 0.0) {              r = q /p ;          // line portion is outside the clipping edge    if ( r > *t2 )                             retVal = FALSE;          else    if (r > *t1 )    *tl = r; }  else  //line leaving pointif (p>0.0) {                                 r = q/p ;          // line portion is outside         if ( r<*t1 )                                 retVal = FALSE;                  else i f (r<*t2)        *t2 = r;}  // p = 0, so line is parallel to this clipping edge else      // Line is outside clipping edge if (q<0.0)                                 retVal = FALSE;  return ( retVal ) ;}  void clipLine (dcPt winMin, dcPt winMax, wcPt2 pl , wcPt2 p2)   { float t1 = 0.0, t2 = 1.0, dx = p2.x-p1.x, dy;   // inside test wrt left edgeif(clipTest (-dx, p1.x - winMin.x, &t1, &t2))       // inside test wrt right edge if(clipTest (dx, winMax.x - p1.x, &t1, &t2))   {                    dy = p2.y - p1.y;          // inside test wrt bottom edge     if(clipTest (-dy, p1.y - winMin.y, &t1, &t2))              // inside test wrt top edge         if(clipTest (dy, winMax.y - p1.y, &t1, &t2)) {                      if(t2 < 1.0) {            p2.x = p1.x + t2*dx;            p2.y = p1.y + t2*dy;        }                  if(t1 > 0.0) {            p1.x += t1*dx;            p1.y += t1*dy;        }                  lineDDA ( ROUND(p1.x), ROUND(p1.y), ROUND(p2.x), ROUND(p2.y) );                  }}  } 
Reference: Computer Graphics by Donald Hearn, M.Pauline Baker
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