# 6.2 Vector graphics: two-dimensional image transformations

 Page 1 / 1
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.

We now turn our attention to operating on the point matrix $G$ to produce the desired transformations. We will consider

1. rotation
2. scaling
3. and translation (moving) of objects
. Rotation and scaling are done by matrix multiplication with a square transformation matrix A. If we call thetransformed point matrix ${G}_{new}$ , we have

$\left[{G}_{new}\right]=\left[A\right]\left[G\right].$

We call $A$ a matrix operator because it “operates” on $G$ through matrix multiplication. In contrast, translation must be done by matrix addition.

In a later section you will see that it is advantageous to perform all operations by matrix operators and that we can modify our image representation to allow translation to be done with a matrix operator like rotation andscaling. We will call the modified representation homogeneous coordinates .

Rotation. We saw in the chapter on linear algebra that the matrix that rotates points by an angle $\theta$ is

$A=R\left(\theta \right)=cos\theta -sin\theta sin\theta cos\theta .$

When applied to the point matrix G , this matrix operator rotates each point by the angle θ , regardless of the number of points.

We can use the rotation matrix to do the single point rotation of the example from "Vector Graphics: Introduction" . We have a point matrix consisting of only the point $\left(3,1\right)$ :

$G=31.$

The necessary transformation matrix is $R\left(\theta \right)$ with $\theta =\frac{\pi }{6}$ Then the rotated point is given by

${G}_{new}=R\left(\frac{\pi }{6}\right)G=cos\left(\frac{\pi }{6}\right)-sin\left(\frac{\pi }{6}\right)sin\left(\frac{\pi }{6}\right)cos\left(\frac{\pi }{6}\right)31\approx 2.102.37·$

Scaling. An object can be enlarged or reduced in each dimension inde- pendently. The matrix operator that scales an image by a factor of ${s}_{x}$ along the x-axis and ${s}_{y}$ along the y-axis is

$A=S\left({s}_{x},{s}_{y}\right)={s}_{x}00{s}_{y}.$

Most often we take ${s}_{x}={s}_{y}$ to scale an image by the same amount in both dimensions.

Translation. An object can be moved by adding a constant vector b to every point in the object. For example, $b=20-5$ will move an object 20 units to the right and 5 units down. We can write this in terms of the point matrix as

${G}_{new}=G+b{1}^{T}$

where 1 (read “the one-vector”) is a vector of n l's:

$1=11⋮1.$

In MATLAB, 1 may be obtained by. The outer product of $b$ with 1 in Equation 7 simply serves to make $n$ copies of $b$ so that one copy can be added to each point in $G$ . ones(n,1)

Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By OpenStax By Danielle Stephens By Danielrosenberger By OpenStax By By Stephen Voron By Rhodes By Rebecca Butterfield By OpenStax By Richley Crapo