Line Bresenhams and Circle Algorithms
Introduction. Graphics images consist of individual Picture Elements (Pixels), which are a single point in the image. For color images, each pixel has color components for the Red, Green, and Blue parts of the color, which are generally specied individually. To draw a line or a circle on a graphical display, we need an algorithm to do scan conversion. That is, for a line we are just given the two endpoints as a pair of (x,y) coordinates, and we must determine which pixels are part of the specied line and set those pixels to the appropriate color. Similarly, for circles we are just given the center point (x,y) and the radius. Here, we will discuss two algorithms for scan converting lines and circles. For further details, both of these algorithms are described in Computer Graphics by Jim Foley and Andries van Dam. Bresenhams MidPoint Line Algorithm. We could easily design an algorithm to draw a line, using oating point values for the slope of the line, and then rounding to an integer to set the appropriate pixel. However, oating point computation in a CPU is substantially more complex (and takes longer) than integer arithmetic. Given this, J. E. Bresenham developed an algorithm for line drawings that uses integer arithmetic only. The algorithm is given below: void MidpointLine(int x0, int y0, int x1, int y1) { int dx = x1 - x0; int dy = y1 - y0; int d = 2 * dy - dx; int incrE = 2 * dy; // East increment int incrNE = 2 * (dy - dx); // Northeast increment int x = x0; // Initial int x y = y0; // Initial y SetPixel(x, y); // Initial line point while(x < x1) { if (d <= 0) { d += incrE; // Move East } else { d += incrNE; // Move Northeast y++; // and advance y } x++; SetPixel(x, y); // Next pixel in line } } It is IMPORTANT to note that this algorithm only works for lines where x1 > x0 and for slope (dy/dx) between 0 and 1.0. You must modify the algorithm to work for the other cases as well. Further, take care to handle the case of innite slope lines (x1 == x0).
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Bresenhams Midpoint Circle Algorithm. For drawing circles, we could easily develop an algorithm that makes use of trigonometric functions such as sin and cosine to nd the points on a circle. However, as in the case of line drawing, efciency is of importance, and we would like an algorithm that uses simple integer arithmetic as much as possible, and in particular does not use expensive trig functions. Bresenham developed a circle drawing algorithm that does exactly this, using mostly integer arithmetic, as follows. This algorithm assumes the circle center is at (0,0). void MidpointCircle(int r) { int x = 0; int y = radius; double d = 5.0/4.0 - radius; SetPixel(x,y); while(y > x) { if (d < 0) { d += 2.0 * x + 3.0; // Select East } else { d += 2.0 * (x-y) + 5.0; // Select SE y--; } x++; SetPixel(x, y); } } It is IMPORTANT to note that this algorithm only draws oneeighth of a circle, in the range of 0 to 45 degrees. You must modify the algorithm to draw properly the remaining seven eighths of the circle.
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