Liang-Barsky Algorithm

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:

1. The parametric equation of the line.
2. 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 p_k < 0, as t increase line goes from the outside to inside (entering).
When p_k > 0, line goes from inside to outside (exiting).
When p_k = 0 and q_k < 0 then line is trivially invisible because it is outside view window.
When p_k = 0 and q_k > 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.
When,

1. p_k < 0, maximum(0, q_k/p_k) is taken.
2. p_k > 0, minimum(1, q_k/p_k) 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 –

1. Set t_min=0, t_max=1.
2. Calculate the values of t (t(left), t(right), t(top), t(bottom)),
   (i) If t < t_min 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 t_min = t; if t is existing value, set t_max = t.
3. If t_min < t_max, draw a line from (x₁ + t_min(x₂-x₁), y₁ + t_min(y₂-y₁)) to (x₁ + t_max(x₂-x₁), y₁ + t_max(y₂-y₁))
4. If the line crosses over the window, (x₁ + t_min(x₂-x₁), y₁ + t_min(y₂-y₁)) and (x₁ + t_max(x₂-x₁), y₁ + t_max(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 point
if (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 point
if (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 edge
if(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