The Cohen-Sutherland Algorithm
In this algorithm we divide the line clipping process into two phases: (1) identify those lines which intersect the clipping window and so need to be clipped and (2) perform the clipping.
All lines fall into one of the following clipping categories:

1. Visible---both endpoints of the line lie within the window.
2. Not visible---the line definitely lies outside the window. This will occur if the line from (x1, y1) to (x2, y2) satisfies anyone of the following four inequalities: x1, x2 >xmax y1, y2 >ymax x1, x2 <xmin y1, y2 <ymin
3. Clipping candidate--- the line is in neither category 1 nor 2.

Midpoint Subdivision
An alternative way to process a line in category 3 is based on binary search. The line is divided at its midpoint into two shorter line segments. The clipping categories of the two new line segments are then determined by their region codes. Each segment in category 3 is divided again into shorter segment and categorized. This bisection and categorization process continues until each line segment that spans across a window boundary (hence encompasses an intersection point) reaches a threshold for line size and all other segements are either in category 1 (visible) or in category 2 (invisible). The midpoint coordinates (xm , ym) of a line joining (x1 , y1) and (x2, y2) are given by xm=x1+x22 ym=y1+y22

The Liang-Barsky algorithm for finding the visible portion of the line, if any, can be stated as a four-step process:
1. If pk = 0 and qk < 0 for any k, eliminate the line and stop. Otherwise proceed to the next step.
2. For all k such that pk < 0, calculate rk = qk /pk. Let u1 be the maximum of the set containing 0 and the calculated r values.
3. For all k such that pk > 0, calculate rk = qk/pk. Let u2 be the minimum of the set containing 1 and the calculated r values.
4. If u1>u2, eliminate the line since it is completely outside the clipping window. Otherwise, use u1 and u2 to calculate the endpoints of the clipped line.