# Points And Lines In Computer Graphics Pdf

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- Vector graphics
- Computer Graphics - Tutorials Point
- Handwritten Computer Graphics Notes PDF Download BCA, BSc
- CAD/CAM Machine Drawing and Computer Graphics 3(1+2)

*Vector graphics are computer graphics images that are defined in terms of points on a Cartesian plane , which are connected by lines and curves to form polygons and other shapes. Vector graphics have the unique advantage over raster graphics in that the points, lines, and curves may be scaled up or down to any resolution with no aliasing.*

## Vector graphics

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. It is an incremental error algorithm. It is one of the earliest algorithms developed in the field of computer graphics.

An extension [ which? 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 means 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 graphics hardware of modern graphics cards.

The label "Bresenham" is used today for a family of algorithms extending or modifying Bresenham's original algorithm. In Bresenham wrote: [1]. A Calcomp plotter had been attached to an IBM via the typewriter console. Programs in those days were freely exchanged among corporations so Calcomp Jim Newland and Calvin Hefte had copies. A description of the line drawing routine was accepted for presentation at the 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. I happily agreed, and they printed it in Bresenham's algorithm has been extended to produce circles, ellipses, cubic and quadratic bezier curves, as well as native anti-aliased versions of those.

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:. If the error becomes greater than 0. In the following pseudocode sample, plot x,y function plots the pixel centered at coordinates x,y and abs returns absolute value :.

Note that this pseudocode only works for the specific case described above, where the line goes down and to the right, and where the change in x is greater than the change in y. To derive Bresenham's algorithm, two steps must be taken. The first step is transforming the equation of a line from the typical slope-intercept form into something different; and then using this new equation to draw a line based on the idea of accumulation of error. This is a function of only x and it would be useful to make this equation written as a function of both x and y.

Everything about this form involves only integers if x and y are integers since the constants are necessarily integers. The point 2,2 is on the line. 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. 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.

This observation is very important in the remainder of the derivation. 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:. This observation is crucial to understand! 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. 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 alternative method, define the difference to be as follows:. Simplifying this expression yields:.

This decision can be generalized by accumulating the error on each subsequent point. All of the derivation for the algorithm is done.

Since all of this is about the sign of the accumulated difference, then everything can be multiplied by 2 with no consequence. 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. 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.

The algorithm can be extended to cover gradients between 0 and -1 by checking whether y needs to increase or decrease i. By switching the x and y axis an implementation for positive or negative steep gradients can be written as.

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.

The Bresenham algorithm can be interpreted as slightly modified digital differential analyzer using 0. 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. Bresenham also published a Run-Slice as opposed to the Run-Length computational algorithm. From Wikipedia, the free encyclopedia. Selects raster points to form a close approximation to a straight line segment.

Retrieved 20 December Retrieved Categories : Computer graphics algorithms Digital geometry. Hidden categories: Articles with short description Short description is different from Wikidata All articles with specifically marked weasel-worded phrases Articles with specifically marked weasel-worded phrases from November All Wikipedia articles needing clarification Wikipedia articles needing clarification from May Commons category link is on Wikidata Articles with example pseudocode.

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## Computer Graphics - Tutorials Point

In any 2-Dimensional plane if we connect two points x0, y0 and x1, y1 , we get a line segment. For using graphics functions, our system output screen is treated as a coordinate system where the coordinate of the top-left corner is 0, 0 and as we move down our y-ordinate increases and as we move right our x-ordinate increases for any point x, y. Now, for generating any line segment we need intermediate points and for calculating them we can use a basic algorithm called DDA Digital differential analyzer line generating algorithm. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute. See your article appearing on the GeeksforGeeks main page and help other Geeks. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

In object-oriented programming, a class that is meant to be used only as a basis for subclasses. Objects can be created from the subclasses, but not from the abstract class itself. The purpose of an abstract class is to define the properties and behaviors that all of its subclasses have in common. A transform that preserves parallel lines. That is, when the transform is applied to a pair of lines that are parallel, then the resulting transformed lines are also parallel. An affine transform, T, has the property that the transform of the line segment between a point x1,y1 and a point x2,y2 is the line between the points T x1,y1 and T x2,y2. Effectively, the transform of a line segment can be computed just by transforming its two endpoints.

## Handwritten Computer Graphics Notes PDF Download BCA, BSc

Computer Graphics are used to simplify this process. Various algorithms and techniques are used to generate Graphics in computers. This tutorial will help you understand how all these are processed by the Computer to give a rich visual experience to the user. Audience This tutorial has been prepared for students who don't know how Graphics are used in computers. It explains the basics of Graphics and how they are implemented in computers to generate various visuals.

A computer Graphics can be anything like beautiful scenery, images, terrain, trees, or anything else that we can imagine, however all these computer graphics are made up of the most basic components of Computer Graphics that are called Graphics Output Primitive or simply primitive. The Primitives are the simple geometric functions that are used to generate various Computer Graphics required by the User. Some most basic Output primitives are point-position pixel , and a straight line. However different Graphic packages offers different output primitives like a rectangle, conic section, circle, spline curve or may be a surface. Once it is specified what picture is to be displayed, various locations are converted into integer pixel positions within the frame buffer and various functions are used to generate the picture on the two dimensional co ordinate system of output display.

We develop a unified framework for combining point and line samples in direct lighting calculations. While line samples have proven beneficial in a variety of rendering contexts, their application in direct lighting has been limited due to a lack of formulas for evaluating advanced BRDFs along a line and performance tied to the orientation of occluders in the scene. We lift these limitations by elevating line samples to a shared higher-dimensional space with point samples. Our key insight is to separate the probability distribution functions of line samples and points that lie along a line sample.

*We provide complete computer graphics pdf. Computer Graphics lecture notes include computer graphics notes, computer graphics book, computer graphics courses, computer graphics syllabus , computer graphics question paper, MCQ, case study, computer graphics interview questions and available in computer graphics pdf form.*

### CAD/CAM Machine Drawing and Computer Graphics 3(1+2)

This algorithm is used for scan converting a line. It was developed by Bresenham. It is an efficient method because it involves only integer addition, subtractions, and multiplication operations. These operations can be performed very rapidly so lines can be generated quickly.

In programming, primitives are the basic operations supported by the programming language. To creative any drawing in the computer these primitives form a part of the software and the type of display to store these in the form of data is important. Graphics Primitives. Graphics is constructed from three basic elements, as opposed to the great variety of graphics applications. The most basic of these elmental structures is the pixel, short for picture element. A pixel is a point of light.

It explains the basics of graphics and how they are implemented in computers content and graphics published in this e-book are the property of Tutorials Point (I) Random-scan displays are designed to draw all the component lines of a.

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. It is an incremental error algorithm.

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These notes emphasize the basic principles needed to design, use, and understand computer graphics systems. Sc, B. Tech CSE, M. Tech branch to enhance more knowledge about the subject and to score better marks in the exam. Overview of computer graphics, storage tube graphics display, Raster scan display.

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It is difficult to display an image of any size on the computer screen.

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In computer graphics , a line drawing algorithm is an algorithm for approximating a line segment on discrete graphical media, such as pixel -based displays and printers.

Vijay M. Shekhat, CE Department | – Computer Graphics. Points and Lines. • Point plotting is done by converting a single coordinate position furnished.