The Bresenham line algorithm, is an algorithm that 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.
Through a minor expansion, the original algorithm for lines can also be used to draw circles. Also this can be done with simple arithmetic operations; quadratic or trigonometric expressions can be avoided or recursively dissolved into simpler steps.
The mentioned properties make it still an important algorithm, and it is used among others in plotters, in graphics chips of modern graphics cards, and in many graphics libraries. As it is so simple, it is not only implemented in the firmware of such devices, but is also cast into hardware of those graphics chips.
To be precise, the label "Bresenham" is today often used for a whole family of algorithms, which have actually been developed by others, later, yet in succession of Bresenham and with a similar basic approach. See deeper references below.
Contents
[hide]
1 The algorithm
2 Generalization
3 Optimization
4 History
5 Similar algorithms
6 References
7 See also
8 External links
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[edit] 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 and that pixel centers 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 (in other words, the line has a slope 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:
Since we know the column, x, the pixel's row, y, is given by rounding this quantity to the nearest integer:
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) 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
Actually this first formation was done by Bresenham.This first version 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
The function now handles all lines and implements the complete Bresenham's algorithm.
[edit] Optimization
The problem with this approach is that computers operate relatively slowly on fractional numbers like error and deltaerr; moreover, errors can accumulate over many floating-point additions. Working with integers will be both faster and more accurate. The trick we use is to multiply all the fractional numbers above by deltax, which enables us to express them as integers. The only problem remaining is the constant 0.5—to deal with this, we change the initialization of the variable error. The new program looks like this:
(This implementation does not work. See article discussion.) 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) int error := -(deltax + 1) / 2 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 + deltay if error ≥ 0 then y := y + ystep error := error - deltax
[edit] History
The algorithm was developed by Jack E. Bresenham 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."
Bresenham's algorithm was later modified to produce circles, the resulting algorithm being sometimes known as either "Bresenham's circle algorithm" or Midpoint circle algorithm.
[edit] Similar algorithms
The Bresenham algorithm can be interpretated as slightly modified DDA (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 raster scan 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 (as opposed to the Run-Length) computational algorithm.

network security

Network security:
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Network security consists of the provisions made in an underlying computer network infrastructure, policies adopted by the network administrator to protect the network and the network-accessible resources from unauthorized access and the effectiveness (or lack) of these measures combined together.
Contents
1 Comparison with computer security
2 Attributes of a secure network
3 Security management
3.1 Small homes
3.2 Medium businesses
3.3 Large businesses
3.4 Government
4 References
5 Further reading
6 See also
7 External links
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[edit] Comparison with computer security
Securing network infrastructure is like securing possible entry points of attacks on a country by deploying appropriate defense. Computer security is more like providing means to protect a single PC against outside intrusion. The former is better and practical to protect the civilians from getting exposed to the attacks. The preventive measures attempt to secure the access to individual computers--the network itself--thereby protecting the computers and other shared resources such as printers, network-attached storage connected by the network. Attacks could be stopped at their entry points before they spread. As opposed to this, in computer security the measures taken are focused on securing individual computer hosts. A computer host whose security is compromised is likely to infect other hosts connected to a potentially unsecured network. A computer host's security is vulnerable to users with higher access privileges to those hosts.
[edit] Attributes of a secure network
Network security starts from authenticating any user, most likely an username and a password. Once authenticated, a stateful firewall enforces access policies such as what services are allowed to be accessed by the network users.[1] Though effective to prevent unauthorized access, this component fails to check potentially harmful contents such as computer worms being transmitted over the network. An intrusion prevention system (IPS)[2] helps detect and prevent such malware. IPS also monitors for suspicious network traffic for contents, volume and anomalies to protect the network from attacks such as denial of service. Communication between two hosts using the network could be encrypted to maintain privacy. Individual events occurring on the network could be tracked for audit purposes and for a later high level analysis.
Honeypots, essentially decoy network-accessible resources, could be deployed in a network as surveillance and early-warning tools. Techniques used by the attackers that attempt to compromise these decoy resources are studied during and after an attack to keep an eye on new exploitation techniques. Such analysis could be used to further tighten security of the actual network being protected by the honeypot.[3]

[edit] Security management
Security Management for networks is different for all kinds of situations. A small home or an office would only require basic security while large businesses will require high maintenance and advanced software and hardware to prevent malicious attacks from hacking and spamming.
[edit] Small homes
A basic firewall.
For Windows users, basic Antivirus software like McAfee, Norton AntiVirus, AVG Antivirus or Windows Defender, others may suffice if they contain a virus scanner to scan for malicious software.
When using a wireless connection, use a robust password.
[edit] Medium businesses
A fairly strong firewall
A strong Antivirus software and Internet Security Software.
For authentication, use strong passwords and change it on a bi-weekly/monthly basis.
When using a wireless connection, use a robust password.
Raise awareness about physical security to employees.
Use an optional network analyzer or network monitor.
[edit] Large businesses
A strong firewall and proxy to keep unwanted people out.
A strong Antivirus software and Internet Security Software.
For authentication, use strong passwords and change it on a weekly/bi-weekly basis.
When using a wireless connection, use a robust password.
Exercise physical security precautions to employees.
Prepare a network analyzer or network monitor and use it when needed.
Implement physical security management like closed circuit television for entry areas and restricted zones.
Security fencing to mark the company's perimeter.
Fire extinguishers for fire-sensitive areas like server rooms and security rooms.
Security guards can help to maximize security.
[edit] Government
A strong strong firewall and proxy to keep unwanted people out.
A strong Antivirus software and Internet Security Software.
Strong encryption, usually with a 256 bit key.
Whitelist authorized wireless connection, block all else.
All network hardware is in secure zones.
All host should be on a private network that is invisible from the outside.
Put all servers in a DMZ, or a firewall from the outside and from the inside.
Security fencing to mark perimeter and set wireless range to this.