Cohen–Sutherland algorithm

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The Cohen–Sutherland algorithm is a computer graphics algorithm used for line clipping. The algorithm divides a two-dimensional space into 9 regions and then efficiently determines the lines and portions of lines that are visible in the center region of interest (the viewport).

The algorithm was developed in 1967 during flight simulator work by Danny Cohen and Ivan Sutherland.[1]

The Algorithm

The algorithm includes, excludes or partially includes the line based on whether:

• Both endpoints are in the viewport region (bitwise OR of endpoints == 0): trivial accept.
• Both endpoints share at least one non-visible region which implies that the line does not cross the visible region. (bitwise AND of endpoints != 0): trivial reject.
• Both endpoints are in different regions: In case of this nontrivial situation the algorithm finds one of the two points that is outside the viewport region (there will be at least one point outside). The intersection of the outpoint and extended viewport border is then calculated (i.e. with the parametric equation for the line) and this new point replaces the outpoint. The algorithm repeats until a trivial accept or reject occurs.

The numbers in the figure below are called outcodes. An outcode is computed for each of the two points in the line. The outcode will have four bits for two-dimensional clipping, or six bits in the three-dimensional case. The first bit is set to 1 if the point is above the viewport. The bits in the 2D outcode represent: Top, Bottom, Right, Left. For example the outcode 1010 represents a point that is top-right of the viewport. Note that the outcodes for endpoints must be recalculated on each iteration after the clipping occurs.

 1001 1000 1010 0001 0000 0010 0101 0100 0110

The Cohen–Sutherland algorithm can be used only on a rectangular clip window. For other convex polygon clipping windows, use the Cyrus–Beck algorithm.

Example C/C++ implementation

```typedef int OutCode;

const int INSIDE = 0; // 0000
const int LEFT = 1;   // 0001
const int RIGHT = 2;  // 0010
const int BOTTOM = 4; // 0100
const int TOP = 8;    // 1000

// Compute the bit code for a point (x, y) using the clip rectangle
// bounded diagonally by (xmin, ymin), and (xmax, ymax)

// ASSUME THAT xmax, xmin, ymax and ymin are global constants.

OutCode ComputeOutCode(double x, double y)
{
OutCode code;

code = INSIDE;          // initialised as being inside of [[clip window]]

if (x < xmin)           // to the left of clip window
code |= LEFT;
else if (x > xmax)      // to the right of clip window
code |= RIGHT;
if (y < ymin)           // below the clip window
code |= BOTTOM;
else if (y > ymax)      // above the clip window
code |= TOP;

return code;
}

// Cohen–Sutherland clipping algorithm clips a line from
// P0 = (x0, y0) to P1 = (x1, y1) against a rectangle with
// diagonal from (xmin, ymin) to (xmax, ymax).
void CohenSutherlandLineClipAndDraw(double x0, double y0, double x1, double y1)
{
// compute outcodes for P0, P1, and whatever point lies outside the clip rectangle
OutCode outcode0 = ComputeOutCode(x0, y0);
OutCode outcode1 = ComputeOutCode(x1, y1);
bool accept = false;

while (true) {
if (!(outcode0 | outcode1)) { // Bitwise OR is 0. Trivially accept and get out of loop
accept = true;
break;
} else if (outcode0 & outcode1) { // Bitwise AND is not 0. (implies both end points are in the same region outside the window). Reject and get out of loop
break;
} else {
// failed both tests, so calculate the line segment to clip
// from an outside point to an intersection with clip edge
double x, y;

// At least one endpoint is outside the clip rectangle; pick it.
OutCode outcodeOut = outcode0 ? outcode0 : outcode1;

// Now find the intersection point;
// use formulas y = y0 + slope * (x - x0), x = x0 + (1 / slope) * (y - y0)
if (outcodeOut & TOP) {           // point is above the clip rectangle
x = x0 + (x1 - x0) * (ymax - y0) / (y1 - y0);
y = ymax;
} else if (outcodeOut & BOTTOM) { // point is below the clip rectangle
x = x0 + (x1 - x0) * (ymin - y0) / (y1 - y0);
y = ymin;
} else if (outcodeOut & RIGHT) {  // point is to the right of clip rectangle
y = y0 + (y1 - y0) * (xmax - x0) / (x1 - x0);
x = xmax;
} else if (outcodeOut & LEFT) {   // point is to the left of clip rectangle
y = y0 + (y1 - y0) * (xmin - x0) / (x1 - x0);
x = xmin;
}

// Now we move outside point to intersection point to clip
// and get ready for next pass.
if (outcodeOut == outcode0) {
x0 = x;
y0 = y;
outcode0 = ComputeOutCode(x0, y0);
} else {
x1 = x;
y1 = y;
outcode1 = ComputeOutCode(x1, y1);
}
}
}
if (accept) {
// Following functions are left for implementation by user based on
// their platform (OpenGL/graphics.h etc.)
DrawRectangle(xmin, ymin, xmax, ymax);
LineSegment(x0, y0, x1, y1);
}
}
```

Notes

1. ^ Principles of Interactive Computer Graphics p.124 and p.252, by Bob Sproull and William M. Newman, 1973, McGraw–Hill Education, International edition, ISBN 0-07-085535-8