question

[rajperimdad]

mujhe ise question ka answer chahi computer graphics ke subjects se. Q:describe the bresenhams' line drawing algorithm with its advantages over other line drawing algorithms?

Reply

[ch_ttayyab]

check this

http://en.wikipedia.org/wiki/Bresenham's_line_algorithm

[Zaara]

The Bresenham line algorithm is an algorithm which determines which points in an n-dimensional raster should be plotted in order to form a close approximation to a straight line between two given points. It is commonly used to draw lines on a computer screen, as it uses only integer addition, subtraction and bit shifting, all of which are very cheap operations in standard computer architectures. It is one of the earliest algorithms developed in the field of computer graphics. A minor extension to the original algorithm also deals with drawing circles.

While algorithms such as Wu's algorithm are also frequently used in modern computer graphics because they can support antialiasing, the speed and simplicity of Bresenham's line algorithm mean that it is still important. The algorithm is used in hardware such as plotters and in the graphics chips of modern graphics cards. It can also be found in many software graphics libraries. Because the algorithm is very simple, it is often implemented in either the firmware or the hardware of modern graphics cards.

The algorithm
Illustration of the result of Bresenham's line algorithm. (0,0) is at the top left corner.

The common conventions that pixel coordinates increase in the down and right directions (e.g. that the pixel at (1,1) is directly above the pixel at (1,2)) and that the pixel centers that have integer coordinates will be used. The endpoints of the line are the pixels at (x0, y0) and (x1, y1), where the first coordinate of the pair is the column and the second is the row.

The algorithm will be initially presented only for the octant in which the segment goes down and to the right (x0≤x1 and y0≤y1), and its horizontal projection x1 − x0 is longer than the vertical projection y1 − y0 (the line has a slope whose absolute value is less than 1 and greater than 0.) In this octant, for each column x between x0 and x1, there is exactly one row y (computed by the algorithm) containing a pixel of the line, while each row between y0 and y1 may contain multiple rasterized pixels.

Bresenham's algorithm chooses the integer y corresponding to the pixel center that is closest to the ideal (fractional) y for the same x; on successive columns y can remain the same or increase by 1. The general equation of the line through the endpoints is given by:

\frac{y - y_0}{y_1-y_0} = \frac{x-x_0}{x_1-x_0}.

Since we know the column, x, the pixel's row, y, is given by rounding this quantity to the nearest integer:

y = \frac{y_1-y_0}{x_1-x_0} (x-x_0) + y_0.

The slope (y1 − y0) / (x1 − x0) depends on the endpoint coordinates only and can be precomputed, and the ideal y for successive integer values of x can be computed starting from y0 and repeatedly adding the slope.

In practice, the algorithm can track, instead of possibly large y values, a small error value between −0.5 and 0.5: the vertical distance between the rounded and the exact y values for the current x. Each time x is increased, the error is increased by the slope; if it exceeds 0.5, the rasterization y is increased by 1 (the line continues on the next lower row of the raster) and the error is decremented by 1.0.

In the following pseudocode sample plot(x,y) plots a point and abs returns absolute value:

function line(x0, x1, y0, y1)
int deltax := x1 - x0
int deltay := y1 - y0
real error := 0
real deltaerr := deltay / deltax // Assume deltax != 0 (line is not vertical),
// note that this division needs to be done in a way that preserves the fractional part
int y := y0
for x from x0 to x1
plot(x,y)
error := error + deltaerr
if abs(error) ≥ 0.5 then
y := y + 1
error := error - 1.0

[edit] Generalization

The version above only handles lines that descend to the right. We would of course like to be able to draw all lines. The first case is allowing us to draw lines that still slope downwards but head in the opposite direction. This is a simple matter of swapping the initial points if x0 > x1. Trickier is determining how to draw lines that go up. To do this, we check if y0 ≥ y1; if so, we step y by -1 instead of 1. Lastly, we still need to generalize the algorithm to drawing lines in all directions. Up until now we have only been able to draw lines with a slope less than one. To be able to draw lines with a steeper slope, we take advantage of the fact that a steep line can be reflected across the line y=x to obtain a line with a small slope. The effect is to switch the x and y variables throughout, including switching the parameters to plot. The code looks like this:

function line(x0, x1, y0, y1)
boolean steep := abs(y1 - y0) > abs(x1 - x0)
if steep then
swap(x0, y0)
swap(x1, y1)
if x0 > x1 then
swap(x0, x1)
swap(y0, y1)
int deltax := x1 - x0
int deltay := abs(y1 - y0)
real error := 0
real deltaerr := deltay / deltax
int ystep
int y := y0
if y0 < y1 then ystep := 1 else ystep := -1
for x from x0 to x1
if steep then plot(y,x) else plot(x,y)
error := error + deltaerr
if error ≥ 0.5 then
y := y + ystep
error := error - 1.0

for more see it...

http://en.wikipedia.org/wiki/Bresenh...line_algorithm