# BRESENHAM LINE DRAWING ALGORITHM DERIVATION PDF

Computer graphics – bresenham line drawing algorithm DERIVATION • Starting from the left endpoint (x0, y0) of a given line, we step to each. Assumption: Y=mX+b where b is the intercept cut by line at Y axis and m is the slope of line (0 Derivation: Initially we have plotted a. To derive Bresenham’s algorithm, two steps must be taken. and then using this new equation for a line to draw a line based on the.

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August Learn how and when to remove this template message. In order to develop a fast way of doing this, we will not be comparing these values in such a manner, instead we will create a decision variable that can be used to quickly determine which point to use. Therefore the final recursive definition for pi will be based on choice, as follows remember that the sign of pi is the same as the sign of d1 — d2: Simplifying this expression yields:.

Draw the pixel at the starting point Check decision variable: So we can simply choose subsequent pixels based on the following: It is possible to use this technique to calculate the U,V co-ordinates during raster scan of texture mapped polygons [ citation needed ]. Retrieved 20 December However, as mentioned above this is only for octant zero, that is lines starting at the origin with a gradient between 0 and 1 where x increases by exactly 1 per iteration and y increases by 0 or 1.

## derivation of bresenham line algorithm

The algorithm can be extended to cover gradients between dtawing and -1 by checking whether y needs to increase or decrease i. I happily agreed, and they printed it in Draing is one of the earliest algorithms developed in the field of computer graphics.

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An extension to the original algorithm may be used for drawing circles. Note that the loop will only include integer arithmetic. It is commonly used to draw line primitives in a bitmap image e.

Please help improve this article by adding citations to reliable sources. We need to isolate these eight choices into only two choices. Regardless, the plotting is the same. A Calcomp plotter had been attached to an IBM via the typewriter console. The Bresenham algorithm can be interpreted algofithm slightly modified digital differential analyzer using 0.

### MATH Project: The Bresenham Line-Drawing Algorithm

There are now 6 multiplications, two additions and one selection in each turn of the loop. The dX and dY values are always positive regardless of which line is chosen with slope from -1 to 1.

A line splits a plane into halves and the half-plane that has a negative f x,y can be called the negative half-plane, and the other half can be called the positive half-plane.

Alternatively, the difference between points can be used instead of evaluating f x,y at midpoints. The Figure 2 shows how is the same line approximated by “illuminating” particular pixels. Figure 1 shows the real line drawn over the pixel grid. Unsourced material may be challenged and removed.

### Bresenham’s line algorithm – Wikipedia

Notice that the points 2,1 and 2,3 are on opposite sides of the line and f x,y evaluates to positive or negative. With slopes greater than 1 or less than -1, we must take the previous implementation and swap all x and y values to “move” the calculations back into the “First Octant”. Consider a line with initial point x1y1 and terminal point x2y2 in device space. If we now evaluate a new decision variable P kwe get: Fig-3 To find the best “next pixel”, first we must find the distances to the two available choices from the ideal location of the real line.

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In Bresenham wrote: Therefore the final recursive definition for pi will be based on choice, as follows remember that the sign of pi is the same as the sign of d1 — d This observation is very important in the remainder of the derivation. This decision can be generalized by accumulating the error. The general equation of the line through the endpoints is given by:.

Since we know the column, xthe pixel’s row, yis given by rounding this quantity to the nearest integer:. The algorithm is used in hardware such as plotters and in the graphics chips of modern graphics cards.

This process is called rasterization. In other projects Wikimedia Commons. These pixels represent the one just to the right and xrawing one to the right and one up pixel, respectively as shown. To derive Bresenham’s algorithm, two steps must be taken.

Demo implementation of the Bresenham Algorithm implements algorithm for the lines in all directions. Distance between pixel-to-right and ideal pixel is:.