Transformations in 2D - Computer Graphics, Computer Science & Engineering Video Lecture - Computer Science Engineering (CSE)

welcome
everybody may continue the discussion on
2d transformations or two dimensional
transformations in the last lecture we
have been discussing the various forms
of the transformation matrix T basically
the elements and we have almost known
what are the different combination of
the values of the elements create
different type of transformations and
you have been mainly discussing four
type of transformations rotation scaling
shear and reflection okay rotation
scaling sheer reflexion for different
types and we know that the combination
of the diagonal elements create certain
type of transformation of diagonal
creates one and rotation matrices are
again orthogonal we know all those
properties we have seen in the previous
lecture and also I did mention that for
some reason I did not discuss the very
simple fundamental transformation as the
translation translation was not
discussed and it is a reason why we will
know soon will let us look into this
slide to see an example of translation
as you see here translation is a
movement of one object to another and
this slide there is an object located
yes it's again the house light blue
structure basically something like a
square and a pyramid sitting on top
which has been given a translation of
five units along x axis and one unit
along y axis the coordinates of just one
pixel value on the left bottom most
corner of the structure is given which
is 2 comma true on the left side here
and you add 5 to 2 you get 7 you add 1
to 2 you get 3 so 2 comma 2 is displaced
to 7 comma 3 and all other coordinates
are also displaced in a similar manner
okay because we are talking about body
transformation so all of these points
are the entire object undergoes the same
type of transformation like the
translation if you want to depress this
by matrix equations the translations you
will see you can easily represent by an
equation like this
remember the coordinates of a and B are
to cross one basically it's a vertical
vector a column vector and they're just
two elements and so you add T X to X and
T Y to Y and you get the coordinates of
the displaced point or for the entire
object so that's the okay and
translations basically to happen in
almost all sorts of transformations okay
it is then inherently if you do not
write even an expression and do not give
it a translation also it is there in
other type of transformations as for
example rotation rotations of any
objects which are not centered at the
origin undergo a translation is the same
for scaling when objects and lines are
not centered at the origin anything
which is not at the origin and undergo
strands undergoes rotation and scaling
those objects including lines and points
also go undergo a translation in some
form we have seen examples of rotation
earlier we will see again today and it
will be clear to you that translation
inherently happens with most other types
of transformations as well but of course
if the object is at the point is at the
origin of the line intersects the origin
there may not be in translation when you
under give it a rotation or a scaling
there may not exist Genesis things or we
say the origin by itself is invariant to
scaling reflection and shear but it is
not invariant to translation that is
indeed a very nice observation because
if you plug on the top of the expression
given on the top of the slide here and
substitute a is equal to 0 which is the
origin are the coordinates of the origin
in 2d space you get B equals TD which is
T X and T Y so the origin is shifted
based on the coordinates or the amount
of displacement vector basically the the
translational components will match if
George you should be able to shift
something to the origin and also the
origin to somewhere else but of course
you cannot scale an origin you cannot
reflect it a cannot shearing there's the
key so you see the expression which we
talked about in the last class in terms
of the transformations we use
multiplication matrix multiplication in
general to represent all transformations
in 2d in except translation we have a
multi we have a addition instead of
so we have a mathematical problem we
cannot directly represent translations
as much tricks multiplication as we can
do for say these are examples in figures
of scaling and rotation again in the
same structure like a house which is
scaled from scaled up or scaled down
that means expanded or contracted or
something you can make the house rotate
about its origin and in this case there
may be inherently a translation but when
you represent these transformations
you'd use a matrix multiplication
operation okay mathematical model but
for translation you have given the
previous slide we are forced to use an
addition how can we combine these two
because it's not easy to do that so how
can we represent translations in our
general transformation matrix which is
ABCD for parameters where do we plug in
the transformation elements for the
translation and it's it's made possible
with the help of what are called the
homogeneous coordinates we will see what
are the homogeneous coordinates or the
representation of a point in homogeneous
coordinate system we first see the the 3
cross 3 matrix are the matrix
transformation matrix in homogeneous
representation but the homogeneous
coordinates are there not to the right
of you instead of X Y we have a third
element called W in some books you use H
instead of W so you can use it
interchangeably but stick to one again H
of W and X Y W are the original
coordinates of a point before
transformation is transformed by a 3
cross 3 transformation matrix and you
get X prime Y Prime and W Prime ok what
is the significance of W and what is the
role before going that if you just take
out the two skullery linear equations
out of this matrix equations you get X
prime is X plus CY plus T X and of
course Y prime is similarly BX plus C y
plus dy and now what is these
homogeneous coordinates each point is
basically now represented by a triplet
XY W so there's a third element which
comes remember it's not a 3d space we
are still talking of 2d space XY are the
coordinates value what is the role of W
that is the homogeneous term which is
just a parameter and we will say once
given X Y W you divide the first two X
comma Y by the
w or the third parameter W and you get
back your Cartesian 2d coordinates just
what the last line says X by W y by W
how do you get this take the first 2
divided by third okay and those are
called the Cartesian coordinates of the
homogeneous point so homogeneous points
coordinates are this X Y W or XY w here
or in equation right hand side x Y W or
X prime Y prime W prime but to get the
Cartesian coordinates take the first 2
divided by third you get X by W Y Phi W
you get the actual coordinates which you
have been talking for W could be equal
to 1 there is now this is a
interpretation of homogeneous coordinate
representation any point in this vector
line which comes out of this XY W space
you can visualize this would be now a 3d
space where W could be Z but actually it
is not Z was X Y are the coordinates and
any vector which points in this
direction X y wo pH is a point in this
homogeneous space in X Y W space and it
also points as a vector along a certain
direction and any point on this line or
this vector in this direction basically
represents a single point in Cartesian
coordinate system which is this one so
the P 2d is nothing but X by W Y by W
and one that is obtained by dividing all
the X Y W triplets by the third element
W and that is what gives you the first
two of those give you the 2d coordinates
so basically the intersection of this
vector with W equal to one plane that is
you draw a plane which is parallel to XY
axis and normal or orthogonal to the W
axis the W axis is the perpendicular the
normal to this plane and it intersects
the W axis at W equal to one which is
marked here at the center so this plane
which is again I repeat parallel to XY
plane perpendicular to W axis intersects
the W X is equal to 1 this plane
intersects the corresponding vector this
pH vector at a point which will give you
the 2d coordinates of the actual point
in 2d space so this is the
interpretation of homogeneous
coordinates ok more examples on this if
you take any two homogeneous coordinate
X 1 Y 1 W 1 and X 2 y 2 W 2
they may actually represent the same
point if and only if they are multiples
of one another a typical example is 1 2
3 & 3 6 9 if you see here I did say that
you can obtain the 2d coordinates from
the homogenous one by dividing the first
two elements by the third so 1 by 2 1 by
3 and 2 by 3 at the respective Cartesian
coordinates you can get the same thing
here because 3 by 9 is 1 by 3 and 6 by 9
is also 2 by 3 or you can multiply all
the values by a constant scalar quantity
3 and you get the other one so you can
scale down or scale up is basically the
same point there is no unique
homogeneous representation of a point I
can say this is 1 2 3 is one point which
equivalent it is equivalent to a 1 by 3
2 by 3 in in Cartesian coordinate system
the same as 3 6 9 or I can take it to be
6 12 18 or I can take it to be 100 200
300 multiply this 1 2 3 by any all the
elements by a same quantity K ya all
those it present basically the same
point this is you can start to visualize
the figure which I showed you in the
previous slide their intersection of
that vector with that plane any point in
that line of that vector can be
represented by values intersection gives
a unique point all Triplets of the form
T dot X T dot y and theta W form a line
in the XY W space we talked about this
vector earlier on and you get that by
dividing of course by W T dot W if you
divide you get X by W Y by W from this
triplet and the Cartesian coordinates
are just the plane W equal to 1 in this
space we have seen in the previous slide
I am talking this again that is how you
get the Cartesian coordinates from the
homogenous coordinates by intersecting
this vector with aw equal to 1 plane or
dividing the first two by the third
element
well what about W equal to 0 and W
should be a nonzero quantity because we
are talking about W equal to 0 they're
talking about points at infinity in 2d
space so we don't talk about those in
real numbers world world numbers which
can be represented and stored in a
computer infinity cannot be stored so
you don't need to bother about that but
just the physical interpretation W equal
to 0 D represents points at infinity so
once we know what are the homogeneous
coordinates and is
station we can now relook at the
general-purpose 2d transformation matrix
in general which should be able to
handle all different five
transformations five type of
transformations and this is a
general-purpose
transformation matrix okay so there are
nine elements 3 cross 3 matrix we need a
3 cross 3 because homogeneous Cartesian
coordinate representation talks about a
3 cross 1 element column vector so in a
3 cross 3 element transformation matrix
ok the sub matrix consisting of the top
left 4 elements ABCD these are the
parameters involved in scaling rotation
reflection and shear as the previous
case though that part is unaltered we
still have that ABCD with the same
significance as we have been talking in
the previous lecture with the four
different type of transformations of
scaling rotation reflection and shear so
there's absolutely no problem with the
ABCD that's corresponding four
parameters behave in the same way we
still have five different parameters out
of those which to represent translation
is most important to us correct we need
to know what is translation well I
talked about now two different notations
a pre multiplication and post
multiplication operator to obtain the
new coordinates be from a and if you are
pre multiplying the coordinates a by T
the transformation translation
parameters are PQ as given here at the
bottom okay so the third column first to
indicate translation if you are pre
multiplying if you are post multiplying
a by T you have M and the first two
elements of the last row vector of the
last row of the matrix T indicate
transformation we talked about these pre
and post scenarios row a becomes either
a row or a column vector depending upon
whether you are pre multiplying or post
multiplying and depending on that the
corresponding translation parameters TX
and Tim I could change place do change
place in fact they could be either P or
Q or MN depending upon whether you're
using B equals T multiplied by a or a
multiplied by T okay so that's okay you
of course you have another
that if MN is a transition parameter
what happens to PQ and in PQ is a
transition what happens to MN will hold
on to that for the time being don't
worry about it and right now but you
still have a ninth parameter which is
the s can you almost guess what
s could be doing well I leave it as an
exercise for you to try out and write
equations by keeping s as nonzero and a
and B to be one and all other elements
that means basically protect this is an
identity matrix with s equal equal to
two and all other elements see M and n
PQ B these are all off diagonal elements
to be 0 a and D diagonal elements to be
1 and s to eat about 2 or 3 you see I'm
sure if you don't make a mistake you
will find that the S gets involved also
in uniform scaling very interesting
coming back to composite transformations
because now we know to handle all
different five different transformations
that means reflection rotation scale
shear and translation of course we
didn't discuss necessarily in this order
but there are five different
transformations which are possible for
by manipulating the different elements
of the transformation matrix and what
happens now when we want to apply two
different transformations either
together or one after the other
that is known as composite
transformations which states that if we
want to apply a series of
transformations t1 then t2 and then t3
not in some sense together but in a
sequence the sequence is very important
we will come to that in a moment from
now but you have these three
transformations applying let us say one
after the another effect is the same as
if it is applied together maybe and it
is applied to a set of points or a
single point we can do it in two ways
number one we can calculate from the
vector P which is the coordinates of the
point P of course we are all discussing
homogeneous coordinate representation
and the T is also a three cross the
transformation matrix which could
represent any transformation we and we
have in this case we are taking pre
multiplication of P by T so we multiply
P by T one you get P Prime then from P
prime you can
pre multiplied by t2 to get P double
prime which means that you are first
applying the transformation of t1 on P
to get P Prime
then you apply transformation t2 on P
prime to get P double prime and
ultimately on P double prime you apply a
transformation t3 you should be able to
get P triple prime are basically the
resultant coordinates of the point P
after three successive transformations
we can do three different such
multiplications one after the another or
you can do the other way you can
calculate T as a product of all the
three transformations and then apply and
then obtain P prime as T multiplied by P
where T is given by this this you can
easily derive from the number one step
we start substituting P double prime
here and then P prime there you will
find that the T is nothing but a
sequence of mathematics multiplication
of T 1 T 2 T 3 and then you get P prime
by a single matrix multiplication
actually in most scenarios you have this
multiplication matrix form known and if
it is known actually you don't have to
compute for a given problem the
resultant matrix T 4/3 transfer so in
fact you are saving a lot of time in in
this process and Method 2 is optimal
that means it saves a large number of
additions and multiplications and it
gives you low computational cost or low
computational time the complexity the
computational complexity of the second
step operation becomes really low
computational complexity is really low
as it needs approximately one third of
as many operations as in step number one
therefore we concatenate or compose the
matrices that is get these T 1 T 2 T C
all of them together put it to one final
transformation matrix T and then apply
to that to all the points or to a single
point to get the final transformed point
okay just to take few simple examples as
I was talking if you want to apply two
translations one after the another that
means you need to translate the points
by T X 1 comma T y 1 x and y indicate
the translational components along x and
y direction
effectively and then you want to
translate it by again by DX 2 - why - I
leave it as an exercise for you to find
out that the overall translational
matrix which is a multiplication of the
two translational components we are
taking pre multiplication in this case
is given on the right hand side ok so
that's very simple and I leave it as an
exercise for you to find out that the
scaling effect is also similar to that
of translations and these skill link
coefficients will come along the
diagonal elements of the matrix T
rotation is a little bit tricky but
there are good methods to solve it you
need to rotate the problem says that you
need to rotate by and amount or an angle
theta1 and then you need to rotate it by
an angle theta
- what instead of applying two different
transformations one 42 and then 42 in
succession you put in the transformation
matrix for rotation and angle theta 1
plus theta 2 itself in place of theta in
the original transformation matrix where
you remember you had a cos theta minus
sine theta sine theta cos theta you
instead replace that theta by theta 1
plus theta 2 and then you better do that
instead of doing the second stage which
will be cos T that means calculate T 1
for T 2 1 T 1 2 T 2 and then multiply
and if you multiply them the effect
which you get basically is the same is
the same by replacing theta by theta 1
plus theta 2 take it is an exercise that
both step 1 and step 2 give the same
result of the rotation of the matrix
okay you will come into that form often
you need to rotate about an arbitrary
point in space because if you think of
the rotation matrix which we discussed
in the previous lecture of
transformation it was basically rotation
with respect to the origin that means
you're rotating with respect to the
origin so we have the transformation
matrix which helps you to rotate about
the origin what about trying to rotate
about any arbitrary point of space what
do you mean by that
okay let us take the revolution or
rotation of the earth or spins about sin
on axis so the transformation matrix
which you know for rotation you can
apply if we are applying the rotation
about the center point about the earth
that is fine but let us say
the case when you let us take the case
of the moon okay so the the moon there
are points which are rotating on the
moon about the center of the moon so you
are not the points actually
of course the moon revolves around the
earth like the earth also moves around
the Sun but if you are taking if you are
if you are trying to move an object at a
point with respect to some other point
which is not the origin we don't have
that transformation matrix we will
derive it right now what is the first
step okay as we mentioned above
rotations are about the origin
okay look into this slide you will find
out okay it's written that as we
mentioned before rotations are applied
about the origin so to rotate about any
arbitrary point P in space there are two
steps translates so that the P coincides
with the origin okay as you see in this
slide here and you paragraph yes
so to rotate about any arbitrary point P
in space translate so that the P concise
with the origin that is the first step
second step rotate
now rotate because rotate now because
the point has become the origin or the
origin has become the point whatever way
you look at it and then you can
translate it that so these steps to
implement the overall transformation are
translate by minus px krama py which are
nothing but the x coordinates of the
point P and Y coordinates of the point P
so you are basically translating the
point P to the origin or the origin
moves to the point P whatever way you
look at it then you rotate using the
original rotation matrix and then
finally translate it back let us have an
illustration of these steps of the
process first and then what we will do
is look at the overall matrix of the
transformation okay
so rotation about an arbitrary point in
space we are discussing this point may I
take the sample of the house without the
shading and I want to rotate this house
with respect to the point p1 which is at
the bottom left of this screen not with
respect to the origin if I want with
respect to the origin which is here
there is no problem because I have the
rotation matrix for that and the whole
object along with point P one should
rotate
basically when I rotate with respect to
p1 the point p1 will stay everything
else is loaded around that so that is
the difference between rotating about a
point arbitrary point in space in this
case of glittery point is on the object
or with respect to the origin if I
rotate about the origin I repeat the
entire object will rotate along with all
the points but in this case with respect
to point P 1 the point P 1 we will see
it's there the other of the positions of
the other points will be change so what
was the first step if you recollect
translate it to the origin translate P 1
to the origin by a translation vector
minus px minus P a we have seen that now
we can rotate it because P 1 is at the
origin so after rotating by an amount
theta you will find that D and I have
drawn a red line as the x-axis just to
show that D it is equivalent to
transforming the two-dimensional plane
itself but basically you're rotating
only the house not the other axis ok and
finally translate it back to P 1 you
compare the first figure on the top left
and the final figure form on the top
right you can see that the point P 1 has
not change its position the structure
has basically rotated by an amount theta
so this is an example of a sequence of
the stages different stages or steps
involved in rotation about arbitrary
point P in space if you wanted to rotate
this structure about this origin
actually what happened with would have
happened with respect to theta this
structure would have gone and almost
gone past the y axis vertical axis or
would have at least gone near to that so
you can I leave that as an exercise to
for you to rotate this structure about
this origin that will require a simple
rotational matrix to do that so come to
thee let us come back to the equations
we had seen that there are three stages
so we have a translation matrix then a
rotational matrix T 2 and and then a
translation matrix T 3 all these are
general transformation matrices 3 by 3
representing homogeneous Cartesian
coordinates form and the corresponding
matrices are given here ok so that T 1
with minus px py is given on the left
hand side multiplied with the this is
the original rotational matrix with
respect to a point at origin and
cos theta minus sin theta and so on and
all our three by three matrix finally
translating the point back so if you
multiply all these three matrices I
leave it an exercise for you to find out
you can actually multiply by starting
with the last two or the first 2 doesn't
matter
you should always if you are correct you
will get back this final expression so
what you need to almost the system at a
programmer needs to remember these
expressions of the elements of this for
a general purpose rotation pxpy are the
coordinates of the point about you which
one to retain rotate and theta is the
amount of rotation so that is the final
form of the rotation of an object or a
point about any arbitrary point P in
space ok so that is the an example of a
composite transformation that means
several transformations together ok well
you can also talk about scaling about an
arbitrary point in space similarly ok
and since we have transformation
matrices we can scale with respect to
the origin the similar concept as we did
for rotation is applicable here that
means you translate P to the origin
scale translate it back ok so you look
here these are the three stages
translate P to origin scale translate P
back and three transformation matrices
SX XY are the diagonal elements for X
non-uniform scaling could be SX not
equal to sy but these are diagonal terms
and we have also the translational
parameters at the t1 and t3 ok and I
leave it as an exercise for you to check
out that the overall transformation
matrix T after multiplications of these
transformation matrices will give you a
expression which is given in the figure
below I would like you to note down this
figure but I also verify whether you are
getting this particular expression if
you are not getting you're making a
mistake please check your expressions
whether you have taken the correct form
of t1 t2 t3 and multiplied one after
another correctly to get this final
expression of a T which is what else
Kaling about an arbitrary point of in
space and that's also a composite
example of the composite transformation
yeah we talked about reflection through
an arbitrary line Y arbitrary line well
we have reflection matrices which we are
reflecting about certain specific
axis like you can reflect about x-axis
about the y axis or a complete diagonal
line which is equations random in y
equal to XR where I got a minus X as you
can think of now even have an arbitrary
line say y equal MX plus C and I would
write to reflect it about that and I
hope you remember reflection we're
talking about reflection with respect to
your plane or a line and you could've
visualized with the mirror about what do
you mean by reflection in 2d space as
well okay now this composite
transformation you need five different
stages while of course we were talking
about before rotation and scaling to
translate to the origin and back that
must be there number one in addition to
that you may need to rotate that line or
the coordinates such that the arbitrary
line coincides with the x axis or the y
axis or y equal to x axis you need all
that because as like the previous case
of rotation and scaling you had the
original transformation matrices with
respect to the origin in this case for
reflection you have the form of the
reflection matrix helps you to reflects
an object about certain specific lines y
equals x y equals 0 x equal to 0 that
means horizontal vertical axis or
whatever it is so you must make the
arbitrary line coincide with some of
these special lines and then only you
can apply a reflection matrix then of
course you rotate translate back as you
have done before in the case of rotation
and scaling to get back the composite
transformation matrix so look back into
the board there are five different
stages translate the line to the origin
if the line is not passing through the
origin if it is passing through the
origin the first and the last step may
not be necessary ok otherwise generally
translate the line to the origin rotate
the line about the origin reflect use
the reflection matrix to reflect the
object or the point reverse the rotation
then reverse the translation process
back as well well five different stages
so I will talk of a GE general
reflection matrix a general reflection
matrix composite metric transformation
matrix for generalized reflection I will
talk of a composite matrix of a
generalized deflection we talk of
translation then a rotation then a
reflection then
reverse of rotation which you know
because since rotation matrix is
orthogonal so these are inverse is
nothing but these transpose and of
course you also take the inverse of the
translation matrix at the end too I
leave it an exercise to compute the
overall composite matrix in that case
now we discuss about commute ability of
transformation that means if we scale
then translate to origin then translate
back some sequence of transformations
let us say arbitrarily chosen is that
equivalent to translate to origin that
means can this process be interchange in
the sense that can the sequence be
disturbed and can we have the same
result if he scale translate the origin
and then translate back he's dead
equivalent to translate to origin scale
and translate back is the order
important that is the question is the
order of transformation which is
dictated by the order of matrix
multiplication or vice versa the matrix
multiplication dictates the order of
transformations or the order is dictated
by the the sequence of matrix
multiplication is the order important or
unimportant what do you think well I am
talking about scenarios that when does T
1 multiplied by T 2 equals T 2
multiplied by T 1 if you go back to
matrix theory
Mattox multiplication the order is
important you cannot change the order
and get the same result that means you
generally will not have if you have
arbitrary matrices and multiply t 1 and
then with respect to T 2 T one post
multiplied by T 2 is not equivalent T 1
pre multiplied by T 2 so that's what
given in this is it same it is generally
not but in certain specific cases you
may have T 1 multiplied by G 2 equals T
2 multiplied by T 1 that means pre or
post multiplication of matrices does not
make a difference but in certain special
cases in general not so it is
mathematics of matrix multiplication
which is dictating my the order
dictating that the order of the
multiplication as well as the order of
transformation is very very important he
cannot jumble up the order and simply
say that I am expecting the same result
and I should get the same result
you neural II will not get it but in
some cases
the cases were t1 and t2 bring post
multiply dot three multiplied gives you
the same we will see in the slide now in
this particular table the cases were t1
multiplied by t2 is equal to t2
multiplied by t 1 that means you can
take in any order
PR post multiplied if you take two
translations you take two successive
scaling operations two successive
rotation operations or a special case of
a uniform scale followed by rotation
these are the pairs of combinations of
t1 t2 you check for yourself that the
order is unimportant you can apply it in
any order you can apply the translation
t2 and then apply translation t1 or
scale T to then scale t1 is the same as
the scale t1 and then scale T - okay I'm
talking of any tube should take a pair
of a pair of transformations in any
horizontal row not along the column okay
that means you can take t1 as rotation
then t2 as rotation reverse it you get
the same result okay
I am NOT ask you a command transition
and scale take two translations take to
scale reverse the order you get the same
result that means in these specific
cases of for cases as given here these
are four examples only where the order
is in important otherwise in general the
order is very very important you cannot
strap or jumble up the order and reverse
it and expect the same result
just to give a light highlight the
importance of this order I have taken an
example this example shows the order in
which I am I am scaling and then
translating that means you look at the
colors the original object is marked as
read it is the same house object and
then I give it a scaling which gives you
a greenish house and then I give it a
translation we gives a blue house so the
red house gets translated it gets scaled
first to the green and then translate it
to the blue house and if I reverse the
order what I am doing I first translate
I first translate from red I get the
blue sorry I'm sorry from the red I get
the green and then I scale it down I get
the blue house as you can see the final
result is different remember the order
is red green and blue
so you start from red and in blue
reverse the order as you can see that
the final result is different the final
result is marked in blue this is the
final result of scale and then translate
this is the final result when you
translate and scale we'll take another
example rotate and differential scale
that means you are first rotating the
house marked in red to a greenish
position a great position in mark by the
green house and then give it a
differential scale we are talking of
non-uniform scaling that is meant by
different scale the other term which is
used and that's the final position given
by the house marked in blue but if you
alter it that means given a differential
scale first and then rotation you can
see now that the red house given a
differential scale or a non-uniform
scale gives rise to the green house
first the house marked in green is what
I mean and then you give it a small the
roti equivalent amount of rotation you
will end up with the blue again you see
that the final result is different these
two example show that the order in
general is very very important you
cannot just jumble up the pre or post
multiplication operators and expect the
same result and talk of coordinate
systems many types of coordinate systems
in today exist one is called the scream
coordinate system this is the coordinate
system used to address the screen or
what are called the device coordinate
the pixels actually reside on this
screen that is referred to as screen
coordinates we also have world
coordinates which is the user of the
program mur defines a user-defined
application-specific coordinate system
having its own units of measure axis
origin etc those units may not be in
pixel unit it could be in inches
millimetres kilometers whatever it could
be feet some length units sort of is
necessary and the user in the programmer
knows and you have a window based
coordinate system it is a rectangular
region of the world or this screen that
is visible and here its own units
basically again in pixels of course you
have AV port on the window which is a
rectangular region of the screen space
that is used to display the window so
you have a window of the world I should
say on a viewport on this screen to
understand a different coordinate
systems and their relations you look
into this screen in terms of the world
on the
system and the windowed coordinate
system on the left-hand side and the
screen coordinate system and the
viewport coordinate system on the
right-hand side well you see that the
window is a small part of the world and
coordinate system and a viewport is a
small part of the screen coordinate
system and if you are displaying a small
circular patch or a circle the window
takes a part of that arc of the circle
and you can display that in any of the
viewports the two viewports is playing
the window the window of the world and
you can see to the corresponding arcs of
the circle in two different viewports
you can define that we put anywhere on
this screen okay now but you need a
transformation matrix so we are still in
2d transformations you need a
transformation matrix to transform from
the at least it window to the viewport
of course window is just described with
respect to the world and also a viewport
is respect with respect to the screen so
if this is the world for you know you
could have a small window where you can
see only my face maybe see or you can
some characters written on this screen
some part of it or you can actually this
to be also your screen and you would
like to have a small viewport and
nothing else outside the viewport you
want to make it visible it says
something like looking through a a
window in your house and you see a small
part of the world to the window that is
the example of a window or a viewport
the port through which you are able to
see that is what is being talked about
and you need a transformation to
transform from window to viewport so the
purpose is to find the transformation
matrix that map's the window in world
coordinates to the viewport in screen
coordinate system and we talked about a
windowed XY space system denoted by X
min Y min xmax ymax the corresponding
minimum and maximum coordinates in x and
y direction respectively and similarly
also you have a viewport instead of X Y
you use typically and other coordinates
UV sometimes X Y and UV are used
interchangeably within the device
coordinates or window or you put we will
have to get used to this UV and
similarly in the UV space you have a
minimum or a maximum coordinates defined
as human u Max or women V Max
respectively okay so the overall
transformation is translate the window
to the origin and so you have to scale
it to the size of the viewport and then
translate it to the viewport location so
these are the two transformations
involved in the overall transformation
matrix very 1d we
to be transformed to the viewport
because you want to display section of
the world in a particular section of the
screen which is the viewport you want to
do not display you don't in general want
the entire world world means they enter
all set of pictures which are drawing on
this screen or which you have modeled
basically you want to draw all those
pictures on the screen you want a
certain section of your world which you
have defined yourself the world in this
case talks of the pictures and the
objects which you want to draw your
model basically in mathematical using
mathematical tools or mathematical
functions and you want a part of that to
be displayed not maybe on the entire
screen part of the screen so that part
of the screen is the viewport the part
of that world which you want to display
is the window you want to take this
window and put it in the viewport so
that's the transformation we are talking
about and let us look at this
transformation matrix as given in this
slide now we are talking of the overall
transformation that takes mwv that is
the notation which I have used if you
look back into the slide
it consists of three transformations a
translation scale and translation again
we are not already used to this type of
transformations so T with human women
multiplying by a scaling matrix and then
a transformation matrix again and this
are the scale factors for the scaling
matrix is given by the ratio of the
maximum minimum coordinates along X and
u respectively and along V and y
respectively and if you plug this SX and
XY into the scale matrix and then
multiply T s and T the final form is
given at the bottom of your screen M is
the overall matrix for transformation WV
subscript indicates world to viewport
sorry window to viewport so we are
talking about about overall
transformation matrix to transform from
window to viewport M window to viewport
is given by this SX sy are the
corresponding scale factors for the
scaling matrix explain why I mean human
women are also known and you max V Max
is also taken care of by the SX X Y so
this is an example of a transformation
in the device world two device
coordinates oh yeah so after we had seen
this nature of the transformation matrix
which could be used to provide the
transformation
from the world coordinate systems or the
programmers coordinate system or the
users coordinate system to a viewport
within the device or screen coordinates
we will take an illustration now of an
object which will do this task okay we
will take an example illustrative
example of what do you mean by this
transformation okay as we go along so
window to viewport transformation we
have seen the matrix representation it's
a combination of a translation a scale
and a translation back again and we take
an object here a simple square sheet
object is taken as you see in this slide
here it's a small square shaped object
which represents the window in world
coordinate it typically means that means
a programmer or a user may have all
these pictures and objects drawn all
over in this world over all x and y
pause values it could be positive or
negative but at the end what he would
have wanted is or he would have liked or
he desires now at this at any given
point of time let us say he wants his
objects drawn from X min Y mean to xmax
ymax all these points to be all the
objects to be drawn here to be put in a
particular position in the viewport
somewhere in the viewport it must be put
in viewport belongs to a small part of
the screen okay and the first action is
to translate it to the origin translate
it to the origin and that is so explain
why I mean becomes zero zero and xmax
ymax which is the top right corner of
the box at the window will become now
what x max minus X min Y max minus y min
that is the top left corner coordinates
after translation you want to scale it
back because the size of the viewport
may not actually match with the size or
shape of the window size is more
important whether than the shape we
typically take rectangular shapes so
size is a important factor so we scale
so scaling matrix will come into play
and it will implement a scaling it will
scale the translated window to the
origin to another shape or size at the
origin itself and then we need to
translate it back to the
point where the viewport is in these
screen coordinates so you are already
into U and V after the transfer the
scaling and we now translate in the UV
space to a point you mean we mean given
by the lower lower left bottom most
coordinate of the viewport that the
viewport bottom must coordinate in the
UV space where we wanted the viewport to
be okay so that this is an example of
the window to viewport transformation as
it happens using the transformation
matrix the composite transformation
matrix so we have seen lots of examples
of composite transformation matrices now
we will take a small exercise about
transformations of at least one case of
more than one line say let us say we
take transformation of parallel lines I
will probably encourage you to take
transformations of intersecting lines
also but let us say you have two
parallel lines so of course anywhere
which is parallel let us say let's say
there are two parallel lines and I want
to give it a transformation that means I
want to rotate both of these lines
simultaneously I want to translate them
I want to maybe scale them do certain
transformations we are talking of a
general transformation matrix so let us
consider two parallel lines as given in
this screen one runs from A to B with
coordinates x1 y1 to x2 y2 as given in
this screen here and the second line
runs from coordinate C to D x3 y3 to a
point x4 y4 and let us say that means
there are two lines a B and C D ok and I
want to transform these lines by your
general transformation matrix ABCD
now of course I have not taken
homogeneous coordinates here does not
matter you can work with the general
transformation matrices in 2d without
deep translation let us say a
translation also we know what to do
there is actually no problem with
translation when you are translating
parallel lines they will remain parallel
anyway but what about other if you scale
if you rotate okay if we reflect if we
give it a shear transformation parallel
lines if we do any sort of
transformation what happens well I live
it is I will not solve this problem
entirely for you but he gave you hints
and direct you to solve it the slope of
the lines can be computed by this
expression M is the slope of the line we
given two end points of a line can
always compute the slope I hope that you
know this from fundamental
mathematics geometry is given by y2
minus y1 divided by x2 minus x1 or y4
minus watch x4 minus T so both are same
that is a slope of both the lines a B or
C D okay you solve the problem by
transforming this lines by a
transformation matrix T given by ABCD so
how you solve it what I would request
you to do is you transform a by T B by T
C by T also and also D all the four
coordinates that means the two pair of
endpoints not the corresponding pair for
each line transform it by T and then
compute the new slope compute the new
slopes and you say that is M Prime from
the transformed endpoints and try to
compute the slope of the transformed
lines M prime as a function of these
elements ABCD and M that is the original
slope of the line and you will find that
the slope of the transformed lines both
the lines get an equivalent
transformation their slope also changes
in the same way that means from M you
reach another M Prime given by this
expression this is the exercise which I
am leaving it to you at the end of this
talk almost that the M prime can be
written as an expression given by B plus
DM a plus C M respectively as you see if
we look back into this expression here
the functional form for M prime e
depends on the original value of M that
is the slope of the parallel lines
before transformation and the elements
of the transformation matrix it doesn't
depend on what it doesn't depend upon
the coordinates that means you can take
any points a and B on the line any point
C on those two parallel lines the the
transformed lines which had the two
parallel lines which had a slope M after
transformation will also have the same
slope M prime that means parallel lines
after rigid body transformation will
remain parallel they will not change
rigid body transformation has this
property you take two parallel lines of
an object or any object shape it could
be a house or any other structure give
it a rotation give it a shear give it a
reflection give it any track or even
translation for that matter and the the
two parallel lines remain parallel yes
the slope of that line
that is given by the expression here
which you must work out solve it to
yourself and see if you cannot solve it
you can contact me by email or sending
by post for me to correct it or you can
contact your teachers but what the
lesson we learn is parallel lines
transformed remains parallel but their
slope changes that means they had a
slope M and it has now become M Prime
which is a different slope but they gain
momentum that is the interesting I would
also ask you to take another exercise to
find out if you have two intersecting
lines and their intersection point is
computed after transformation you get a
new point of intersection and the entire
object which is transformed by an
equivalent transformation matrix the
corresponding intersection point is also
transformed by the same T this is
derived from the law or the concept that
all points of rigid object are
transformed equivalently by the same
transformation matrix by the same
transformation matrix which is the
transformation matrix for the entire
object or for all points or for all
lines or for all parts of the object as
long as the body D is rigid doesn't
change its shape or size we did talk
about deformable structures which we are
not discussing here we are talking of
rigid structures which we see in front
of us not like the cloth on the paper
which can be deformed we are talking of
rigid objects and when they are
transform rigidity is maintain internal
structure details are maintained and by
the transformation takes the points or
the object to some other point all right
but that internal structure details are
decency and their distances between the
any two points the slope corresponding
slope relations of two lines any parts
of the object remain the same so that is
the lesson which we learn from rigid
object transformations we will wind up
with simple examples of transformation
once again just to enlighten you if you
have done with a lot of mathematics you
must trust me you have to believe me
that this is the beginning of
mathematics of this course as we go
along with more and more algorithms we
will have more and more of this
mathematics more to do with in algebra
in some cases algorithms differential
and mostly vector calculus and vector
geometry and concepts like that as we
move along and we will go ahead with
some simple nice illustrations very
simple all right but nine
since that will the examples illustrate
the effect of transformation which is
fundamental to animation when you are
talking about animation which we'll
discuss at the end of this course
animation inverse motion or in some sort
of movement and movement along a path so
you must define the path refine the
pattern of movement and it is that the
you need the transformation matrices
tree in fact move any parts of the
object or the object by itself so we
will see how transformations are used to
create animation with the help of simple
structures okay let's take an
illustration here I take the same house
structure and I wanted to rotate it
about the origin it will create an
animation type of motion but you have to
assume here that I am doing it
incrementally the amount of rotation I
am doing in by small theta as step by
step and until if I do it very fast this
is the effect which I will get so it
rotates by not a smooth motion all right
and I am trying I was trying to create
an effect of an animation I go back and
give you the motion again in the example
of rotation or the sequence of a small
rotation steps giving it a rotation so
if in effect it has rotated and it has
created animation you see another effect
here where the object is scaled and
rotated simultaneously rotated about its
centroid I have at some point within the
object and it is also scaled it was a
small object and it is enlarging out
okay I will repeat the process again
simultaneously it is enlarged zoomed out
and as well as rotated okay these are
two objects one coming through a spiral
path another coming through a scale and
and and zooming scale or zooming effect
along with rotation which we have seen
earlier I play it again two objects
simultaneously joining but following two
different paths of transformations okay
and the very last one is a simple case
of rotation I think I showed this
structure earlier you see a structure of
a house which rotates about the origin
and then of course also at after
rotating about to 90 degrees it also
change its position at that end this
means you rotate about itself I'll leave
it that process once again rotation by
origin rotate about itself rotate about
origin 90 degrees rotate it
rotate about origin rotate itself rotate
about origin by anti degrees so these
are the sequence of an examples of
transformations rotations we have seen
scaling and/or a combination of rotation
scaling and then of course if you have
multiple objects as like in the previous
example multiple object paths can be
created and you can have multiple
objects animation okay
so we stop the lecture on two
dimensional transformations right now
here and the next lecture will start
with three dimensional transformations
and in and when we discuss objects which
will be transformed in 3d and in testing
we will find out along with
transformations in 3d which involves all
these transformations like translation
rotation scale sheer reflexion you have
a special case of transformation in 3d
which doesn't exist in 2d you can almost
guess it now it is what we call as
projective transformations perspective
orthographic isometric various types of
projective transformation exist only in
3d there is no scope of that or no
application of that in two dimensional
transformations two dimensional
transformations we have five different
categories translation rotation scale
sheer reflection and in 3d we have all
these five plus projective
transformations we will see when we go
to the next lecture please solve these
examples of 2d transformation problems
which are given inherent within the
lectures itself thank you very much
you