<< Chapter < Page Chapter >> Page >
This module is part of the collection, A First Course in Electrical and Computer Engineering . The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.


Fundamentals of Interactive Computer Graphics by J. D. Foley and A. Van Dam, ©1982 Addison-Wesley Publishing Company,Inc., Reading, Massachusetts, was used extensively as a reference book during development of this chapter. Star locations were obtained from the share-ware program “Deep Space” by David Chandler, who obtained them from the “Skymap” database of the National Space Science Data Center.

Notes to teachers and students:

In this chapter we introduce matrix data structures that may be used to represent two- and three-dimensional images. The demonstration program shows students how to create a function file for creating images from these data structures. We then show how to use matrix transformations for translating, scaling, and rotating images. Projections areused to project three-dimensional images onto two-dimensional planes placed at arbitrary locations. It is precisely such projections that we use to get perspective drawings on a two-dimensional surface of three-dimensional objects. The numerical experiment encourages students to manipulate a star field and view it from several points in space.

Once again we consider certain problems essential to the chapter development. For this chapter be sure not to miss the following exercises: Exercise 2 in "Two-Dimensional Image Transformations" , Exercise 1 in "Homogeneous Coordinates" , Exercise 2 in "Homogeneous Coordinates" , Exercise 5 in "Three-Dimensional Homogeneous Coordinates" , and Exercise 2 in "Projections" .


Pictures play a vital role in human communication, in robotic manufacturing, and in digital imaging. In a typical application of digital imaging, a CCD camera records a digital picture frame that is read into the memory of a digital computer. The digital computer then manipulates this frame (or array) of data in order to crop, enlarge or reduce, enhance or smooth, translateor rotate the original picture. These procedures are called digital picture processing or computer graphics . When a sequence of picture frames is processed and displayed at video frame rates (30 frames per second), then we have an animated picture.

In this chapter we use the linear algebra we developed in The chapter on Linear Algebra to develop a rudimentary set of tools for doing computer graphics on line drawings. We begin with an example: the rotation of a single point in the ( x , y ) plane.

Point P has coordinates ( 3 , 1 ) in the ( x , y ) plane as shown in Figure 1 . Find the coordinates of the point P ' , which is rotated π 6 radians from P .

Figure one is a two-dimensional cartesian graph with two line segments, P, and P' drawn out into the first quadrant. The horizontal axis is labeled x and the vertical axis is labeled y. Line segment P reaches point (3, 1). The angle from the x-axis to line segment P is labeled θ. Line segment P' reaches point (x', y'), which is a higher y-value than P and a lower x-value than P. The angle between P and P' is measured as π/6. Both lines are labeled as length r. Figure one is a two-dimensional cartesian graph with two line segments, P, and P' drawn out into the first quadrant. The horizontal axis is labeled x and the vertical axis is labeled y. Line segment P reaches point (3, 1). The angle from the x-axis to line segment P is labeled θ. Line segment P' reaches point (x', y'), which is a higher y-value than P and a lower x-value than P. The angle between P and P' is measured as π/6. Both lines are labeled as length r.
Rotating a Single Point in the ( x , y ) Plane

To solve this problem, we can begin by converting the point P from rectangular coordinates to polar coordinates. We have

r = x 2 + y 2 = 10 θ = tan - 1 ( y x ) 0 . 3217 r a d i a n .

The rotated point P ' has the same radius r , and its angle is θ + π 6 . We now convert back to rectangular coordinates to find x ' and y ' for point P ' :

x ' = r c o s ( θ + π 6 ) 10 cos ( 0 . 8453 ) 2 . 10 y ' = r s i n ( θ + π 6 ) 10 sin ( 0 . 8453 ) 2 . 37 .

So the rotated point P ' has coordinates (2.10, 2.37).

Now imagine trying to rotate the graphical image of some complex object like an airplane. You could try to rotate all 10,000 (or so) points in thesame way as the single point was just rotated. However, a much easier way to rotate all the points together is provided by linear algebra. In fact, with asingle linear algebraic operation we can rotate and scale an entire object and project it from three dimensions to two for display on a flat screen or sheetof paper.

In this chapter we study vector graphics , a linear algebraic method of storing and manipulating computer images. Vector graphics is especiallysuited to moving, rotating, and scaling (enlarging and reducing) images and objects within images. Cropping is often necessary too, although it is a littlemore difficult with vector graphics. Vector graphics also allows us to store objects in three dimensions and then view the objects from various locationsin space by using projections.

In vector graphics, pictures are drawn from straight lines.A curve can be approximated as closely as desired by a series of short, straight lines.Clearly some pictures are better suited to representation by straight lines than are others. For example, we can achieve a fairly good representation ofa building or an airplane in vector graphics, while a photograph of a forest would be extremely difficult to convert to straight lines. Many computer-aided design (CAD) programs use vector graphics to manipulate mechanical drawings. It is possible to extend these techniques to deal with some types of curves, but we will consider only straight lines for the sake of simplicity.

When the time comes to actually display a vector graphics image, it may be necessary to alter the representation to match the display device. Personal computer display screens are divided into thousands of tiny rectanglescalled picture elements , or pixels . Each pixel is either off (black) or on (perhaps with variable intensity and/or color). With a CRT display, the electron beam scans the rows of pixels in a raster pattern. To draw a line on a pixeldisplay device, we must first convert the line into a list of pixels to be illuminated. Dot matrix and laser printers are also pixel display devices, while pen plotters and a few specialized CRT devices can display vector graphicsdirectly. We will let MATLAB do the conversion to pixels and automatically handle cropping when necessary.

We begin our study of vector graphics by representing each point in an image by a vector. These vectors are arranged side-by-side into a matrix G containing all the points in the image. Other matrices will be used asoperators to perform the desired transformations on the image points. For example, we will find a matrix R , which functions as a rotation: the matrix product R G represents a rotated version of the original image G .

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'A first course in electrical and computer engineering' conversation and receive update notifications?