Cohen-Sutherland Line-Clipping Algorithm

I will simply refer to the Cohen-Sutherland line clipping algorithm as "the algorithm" as I am documenting it. For a formal treatment refer to "Computer Graphics Principles and Practice" by Foly, van Dam, Feiner and Hughes second edition page 113.

Firstly clipping is an end state in the sense that other intersection points could be first calculated. If the algorithm cuts a line (calculates an intersection point) it does so on lines generated from the regions edges extending infinity.

Indeed the algorithm divides area into 9 regions by extending the rectangular regions edges infinitely. A point is classified into one of these regions. Points on the boundary belong to either region and the algorithm exploits this.

Let $(x_{0}, y_{0})$ and $(x_{1}, y_{1})$ be the region's lower and upper coordinates. The region is classified by four comparison tests stored into a number.

$n = 0$
$n = n + (y > y_{1})$
$n = n + 2 (y_{0} > y)$
$n = n + 4 (x > x_{0})$
$n = n + 8 (x_{1} > x)$

Define a function that given any point returns the region number.

$reg (X)$

Global view

The algorithm processes lines. A line is a pair of points. Let there be three containers : U for the initial set of pairs of points, T for trashed pairs of points, and A for accepted pairs of points.

The algorithm processes a pair of points by considering one pair at a time. Three possibilities emerge. 1. clearly reject 2. clearly accept 3. further processing .

Clearly Reject Test

Clearly Accept Test

Undecided

Here's where it gets interesting. The line is divided in two. The current point is removed from U and trashed (goes to T). But from it two new lines are created and added to U, and the algorithm goes on.

Non-uniqueness of a Point

Ok - wheres the excitement? The algorithm divides the line on a region boundary say at a new point K. Since K is in both regions and the are two possibilities of K - say K- and K+ for opposite regions.

Lets think of the consequences of the split. If (A,B) intersected a regions edge it would split the line at the edge and the line outside the region must be rejected on the next algorithm pass.

If the split line doesn't intersect a regions edge it intersects the boundary that it was split on. Since this boundary deletes a line if its on one side and one of the new lines is, it will be deleted.

The above two cases are all the possibilities - excluding lines parallel to the x and y axes on borders.

Clipping Order

Cohen-Sutherland algorithm chooses an ordering on how the line is split. At least one of the points is outside the clipping square with opcode 0. Call this point p1 and the other p2.

void undecided (point & p1, point const & p2)

{

if isleft (p1)

Chop against left edge, return.

if isright (p1)

Chop against right edge, return.

if isabove (p1)

Chop against top edge, return.

if isbottom (p1)

Chop against bottom edge, return.

}

p1 is a point inside(on border) of clipping region. eg has opcode 0.