Bresenham's line algorithm on arvau

Wikipedia / Bresenham's line algorithm

Bresenham's line algorithm is a line drawing algorithm that determines the points of an n-dimensional raster that should be selected in order to form a close approximation to a straight line between two points. It is commonly used to draw line primitives in a bitmap image (e.g. on a computer screen), as it uses only integer addition, subtraction, and bit shifting, all of which are very cheap operations in historically common computer architectures. It is an incremental error algorithm, and one of the earliest algorithms developed in the field of computer graphics. An extension to the original algorithm called the midpoint circle algorithm may be used for drawing circles.

While algorithms such as Wu's algorithm are also frequently used in modern computer graphics because they can support antialiasing, Bresenham's line algorithm is still important because of its speed and simplicity. 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 graphics hardware of modern graphics cards.

The label "Bresenham" is used today for a family of algorithms extending or modifying Bresenham's original algorithm.

History

Bresenham's line algorithm is named after Jack Elton Bresenham who developed it in 1962 at IBM. In 2001, Bresenham wrote: I was working in the computation lab at IBM's San Jose development lab. A Calcomp plotter had been attached to an IBM 1401 via the 1407 typewriter console. [The algorithm] was in production use by summer 1962, possibly a month or so earlier. Programs in those days were freely exchanged among corporations so Calcomp (Jim Newland and Calvin Hefte) had copies. When I returned to Stanford in Fall 1962, I put a copy in the Stanford comp center library.

A description of the line drawing routine was accepted for presentation at the 1963 ACM national convention in Denver, Colorado. It was a year in which no proceedings were published, only the agenda of speakers and topics in an issue of Communications of the ACM. A person from the IBM Systems Journal asked me after I made my presentation if they could publish the paper. I happily agreed, and they printed it in 1965.

Method

The following conventions will be applied:

the top-left is (0,0) such that pixel coordinates increase to the right and downwards (e.g. that the pixel at (7,4) is directly above the pixel at (7,5)), and
the pixel centers have integer coordinates.

, 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 (the line has a positive slope less than 1). In this octant, for each column x, there is exactly one row y 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:

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

only depends on the endpoint coordinates and can be precomputed, and the ideal y for successive integer values of x and repeatedly adding the slope.

In practice, the algorithm does not keep track of the y coordinate, which increases by m = ∆y/∆x each time the x (due to using the pixel's center coordinates), and is incremented by m each time the x coordinate is incremented by one. If the error becomes greater than 0.5, we know that the line has moved upwards one pixel, and that we must increment our y coordinate and readjust the error to represent the distance from the top of the new pixel – which is done by subtracting one from error.

Derivation

To derive Bresenham's algorithm, two steps must be taken: first, the equation of a line is transformed from the typical slope-intercept form into an implicit equation with integer coefficients; second, this new equation is used to draw the line based on the idea of error accumulation.

Line equation

The slope-intercept form of a line is written as

is the y-intercept and. Then, using algebraic manipulation,

, it can be written as

where the constants are

are defined as integers.

. The point (2,2) is on the line

and the point (2,3) is not on the line

and neither is the point (2,1)

can be called the negative half-plane, and the other half can be called the positive half-plane. This observation is very important for the remainder of the derivation.

Algorithm

The starting point is on the line

because Bresenham's algorithm assumes integer-coordinate endpoints on the pixel grid. There is no fundamental reason, however, that a line must have integer-coordinate endpoints to be rasterized.

. If it is closer to the former then include the former point on the line, if the latter then the latter. To answer this, evaluate the line function at the midpoint between these two points:

is chosen. The value of the line function at this midpoint is the sole determinant of which point should be chosen.

The adjacent image shows the blue point (2,2) chosen to be on the line with two candidate points in green (3,2) and (3,3). The black point (3, 2.5) is the midpoint between the two candidate points.

Algorithm for integer arithmetic

Alternatively, the difference between points can be used instead of evaluating f(x,y) at midpoints. This alternative method allows for integer-only arithmetic, which is generally faster than using floating-point arithmetic. To derive the other method, define the difference to be as follows:

at the starting point. Simplifying this expression yields:

will be:

. This decision can be generalized by accumulating the error on each subsequent point.

All of the derivation for the algorithm is done. One performance issue is the 1/2 factor in the initial value of D. Since all of this is about the sign of the accumulated difference, then everything can be multiplied by 2 with no consequence.

This results in an algorithm that uses only integer arithmetic:

plotLine(x0, y0, x1, y1)
    dx = x1 - x0
    dy = y1 - y0
    D = 2*dy - dx
    y = y0

    for x from x0 to x1
        plot(x, y)
        if D > 0
            y = y + 1
            D = D + (2 * (dy - dx))
        else
            D = D + 2*dy
        end if

from (0,1) to (6,4) yields the following differences with dx=6 and dy=3:

D=2*3-6=0
Loop from 0 to 6
 * x=0: plot(0, 1), D≤0: D=0+6=6
 * x=1: plot(1, 1), D>0: D=6-12=-6, y=1+1=2, D=-6+6=0
 * x=2: plot(2, 2), D≤0: D=0+6=6
 * x=3: plot(3, 2), D>0: D=6-12=-6, y=2+1=3, D=-6+6=0
 * x=4: plot(4, 3), D≤0: D=0+6=6
 * x=5: plot(5, 3), D>0: D=6-12=-6, y=3+1=4, D=-6+6=0
 * x=6: plot(6, 4), D≤0: D=0+6=6

The result of this plot is shown to the right. The plotting can be viewed by plotting at the intersection of lines (blue circles) or filling in pixel boxes (yellow squares). Regardless, the plotting is the same.

All cases

However, as mentioned above, this only works for octant zero, that is lines starting at the origin with a slope between 0 and 1, where x increases by exactly 1 per iteration and y increases by 0 or 1.

The algorithm can be extended to cover slopes between 0 and -1 by checking whether y needs to increase or decrease (i.e. dy < 0):

plotLineLow(x0, y0, x1, y1)
    dx = x1 - x0
    dy = y1 - y0
    yi = 1
    if dy < 0
        yi = -1
        dy = -dy
    end if
    D = (2 * dy) - dx
    y = y0

    for x from x0 to x1
        plot(x, y)
        if D > 0
            y = y + yi
            D = D + (2 * (dy - dx))
        else
            D = D + 2*dy
        end if

By switching the x and y axis an implementation for positive or negative steep slopes can be written as

plotLineHigh(x0, y0, x1, y1)
    dx = x1 - x0
    dy = y1 - y0
    xi = 1
    if dx < 0
        xi = -1
        dx = -dx
    end if
    D = (2 * dx) - dy
    x = x0

    for y from y0 to y1
        plot(x, y)
        if D > 0
            x = x + xi
            D = D + (2 * (dx - dy))
        else
            D = D + 2*dx
        end if

A complete solution would need to detect whether x1 > x0 or y1 > y0 and reverse the input coordinates before drawing, thus

plotLine(x0, y0, x1, y1)
    if abs(y1 - y0) < abs(x1 - x0)
        if x0 > x1
            plotLineLow(x1, y1, x0, y0)
        else
            plotLineLow(x0, y0, x1, y1)
        end if
    else
        if y0 > y1
            plotLineHigh(x1, y1, x0, y0)
        else
            plotLineHigh(x0, y0, x1, y1)
        end if
    end if

In low level implementations which access the video memory directly, it would be typical for the special cases of vertical and horizontal lines to be handled separately as they can be highly optimized.

Some versions use Bresenham's principles of integer incremental error to perform all octant line draws, balancing the positive and negative error between the x and y coordinates:

plotLine(x0, y0, x1, y1)
    dx = abs(x1 - x0)
    sx = x0 < x1 ? 1 : -1
    dy = -abs(y1 - y0)
    sy = y0 < y1 ? 1 : -1
    error = dx + dy
    
    while true
        plot(x0, y0)
        e2 = 2 * error
        if e2 >= dy
            if x0 == x1 break
            error = error + dy
            x0 = x0 + sx
        end if
        if e2 <= dx
            if y0 == y1 break
            error = error + dx
            y0 = y0 + sy
        end if
    end while

Similar algorithms

The Bresenham algorithm can be interpreted as slightly modified digital differential analyzer (using 0.5 as error threshold instead of 0, which is required for non-overlapping polygon rasterizing).

The principle of using an incremental error in place of division operations has other applications in graphics. It is possible to use this technique to calculate the U,V co-ordinates during the raster scanning of texture-mapped polygons. The voxel heightmap software rendering engines seen in some PC games also used this principle.

Bresenham also published a Run-Slice computational algorithm: while the above described Run-Length algorithm runs the loop on the major axis, the Run-Slice variation loops the other way. This method has been represented in a number of US patents:

The algorithm has been extended to:

Draw lines of arbitrary thickness, an algorithm created by Alan Murphy at IBM;
Draw multiple kinds of curves (circles, ellipses, cubic, quadratic, and rational Bézier curves) and antialiased lines and curves; a set of algorithms by Alois Zingl.

Notes

References

"The Bresenham Line-Drawing Algorithm", by Colin Flanagan
A very optimized version of the algorithm in C and assembly for use in video games with complete details of its inner workings
, The Beauty of Bresenham's Algorithms

External links

Michael Abrash's Graphics Programming Black Book Special Edition: Chapter 35: Bresenham Is Fast, and Fast Is Good
National Institute of Standards and Technology page on Bresenham's algorithm
Calcomp 563 Incremental Plotter Information
Bresenham Algorithm in several programming languages
The Beauty of Bresenham’s Algorithm — A simple implementation to plot lines, circles, ellipses and Bézier curves