Project Transformations and Viewing Pipeline Video Lecture - Computer Science Engineering (CSE)

hello and welcome everybody to the
lecture series and computer graphics and
today we are going to discuss about
projective transformations and 3d
viewing pipeline so essentially we are
going to discuss about 3d weaving and in
the last few lectures we had basically
discussed 3d transformations and within
transformations the last few part of the
material on 3d transformations was
basically on projective transformations
so as a continuity we continue into with
the 3d viewing pipeline and projection
transformations and that is the main
subject of the lecture today which is
essentially the fifth part of the
lecture series so if you look into the
flow chart look into Road or flow chart
of the implementation of 3d weaving you
essentially give to the 3d viewing
pipeline a set of 3d World coordinate
output primitives on the left hand side
you see the input here 3d World
coordinate output primitives which goes
through a set of normalizing
transformations and that is the
essential part of the discussion today
which you will see and the main focus of
discussion today will be what is it do
you mean by normalizing transformations
in a why it is essential before you
apply what is called as a clipping that
is the second stage clip against the
canonical view volume now these are I
will say larger groups of tasks divided
into various categories first is
applying normalized transformation the
second is clip against canonical view
volume so this is a new terminology
which is we will come across today so we
will define basically immediately next
what is a canonical view volume and how
it helps in the 3d viewing and the
corresponding where does it occur in the
3d viewing pipeline so we need to clip
of course there is a separate section
and discussion on clipping on both 2d
and 3d clipping algorithms and when we
discuss that we it will be clear how to
clip against a 2d enclosed shape or a 3d
volume so for the time being we assume
that there is a clipping algorithm which
klipsch against a canonical view volume
and after that we project onto
projection plane this essentially we
have discussed earlier and the part of
3d transformations I have discussed in
detail about projective transformations
and we also know the general formulation
about projective transformations only so
the third task is known and the last
part is a simple transformation into
viewport we will define what is a
viewport also we have in fact seen this
and we will see again what is a viewport
in a 2d device coordinates for display
and the 2d device coordinates are
generated so input is 3d World
coordinate output primitives in terms of
lines polygons are called text in 3d and
essentially you they those 3d
coordinates are transformed into 2d
device coordinates which must be given
to the electron gun to fire the
corresponding pixels on the screen but
in between there are 4 sub categories
and essentially in this in the 4 sub
categories number one is as you see
again on the screen apply normalized
transformation second is clip against
canonical view volume third is
projection onto the plane and the fourth
of course is the transforming to
viewport out of them we have discussed
the third category that is projection on
the plane of course we will see the
formulation once again
the formula of the general projection
transformation matrix but before that we
need to do clipping so we understand
what is canonical view volume and before
that we will essentially the major part
of the discussion in this particular
lecture is applying normalizing
transformations ok so we will see first
what is a canonical view volume well
there are two types of canonical view
volume which are used in projection
geometry in terms of 3d reading one of
them is the parallel projection
canonical view volume and other of
course should be perspective projection
canonical view volume so the canonical
view volume for parallel projection is
defined by the set of 6 planes it's
essentially you can visualize this to be
a rectangular parallelepiped almost a
square not exactly a rectangular
parallelepiped where along the x and y
dimensions or the x and y axis the
difference between the two binding
planes are at plus and minus 1
respectively so along an x and y the
size is 2 and along z axis the size is
one because it starts from zero equal to
zero and ends at z equal to minus one so
along the varying activities - z axis
the canonical view volume is bound by an
FP or BP which means basically the front
plane and the back plane or the also
called the front clipping plane and the
back clipping plane respectively FP and
BP is what we use and FP is nothing but
the XY plane is there equal to zero and
the back plane is nothing but Z equal to
minus 1 so both planes are parallel to
the XY plane and that is the front plane
in the back plane defined by the set
equal to zero at zero minus one is given
the right-hand side of these equations
and the rest X equal to minus 1 y equals
equal to minus 1 X equal to 1 y equal 1
so basically if you think of a square or
rectangular parallelepiped you need six
surfaces to bind a volume and those is
the corresponding equations of those six
planes are given here by these six very
very simple equations as given on the
top and this is how the side view of the
canonical view volume for parallel
projection is given let us look now into
how this canonical view volume takes a
shape for a perspective projection
correct now you can see there is
relation between the perspective
geometry and perspective projection
canonical view volume as well as the
parallel projection and the parallel
projection canonical view volume in
remember in parallel projection we have
the Rays which are parallel and hitting
the corresponding projection plane and
in the case of perspective the Rays will
focus are meet at a particular point so
that's why you have a tapping effect of
the canonical view volume for
perspective projections we still have
six planes which define them as X equal
to Z x equals minus Z so they're
inclined planes symmetrically located
with respect to the origin or the minus
z axis and y equals z y equals minus ed
and said the front plane the black man
is defined by a Z equals minus 1 which
is the back plane as given in the figure
here you see a minus 1 which is this
point basically the Z equal to minus 1
equation gives this plane and is equal
to minus Zn gives the front plane you
can actually put this dead minik
zero if you like but typically it is a
nonzero value which is less than unity
so that is the front plane and the black
plane being talked about for perspective
projection to look into a comparison I
will roll back this slide and you can
have a look into the two comparative
figures this is the canonical view
volume for parallel projection and this
is the canonical view volume for
perspective projection you see a short
of a rectangular parallelepiped
structure because there is a parallel
here and the Rays tap it off and meet at
a focal point so that's why you have a
short of a pyramidal structure I should
say for a canonical view volume for
prospective position we're only seeing
the side view you have to visualize the
six planes which bind this particular
Rio volume so that's the idea and we
will see later on not in this lecture
when we discuss about clipping how to
clip certain line segments or basically
line segments or even planes with
respect to this canonical volume later
on and we'll discuss in a clipping in a
different section but we move on we move
on because we have to now actually
capture this entire world into a
canonical view volume before we project
it into the projection plane in the
third step so we'll look back into this
flow chart and typically the this is the
diagrams for an arbitrary 3d view I did
cover this under projective 3d
transformations under projective
geometry and I hope you remember all
these terminologies which I will repeat
once again the term VP means the view
plane vu P means the view up Vector
which is given by this yellow line or
arrow representing the view up Vector CW
the center of the window or the view
port which is defined by the coordinates
you mean we mean to a maximum u max we
max respectively this grey color the
light a blue color outside is the view
plane VR P okay is the view reference
point V R P which may not be the center
of the window VR P is the the reference
coordinate on this on the projection
plane with respect to which you have the
U and the V axis the 2d coordinates axis
on the viewport or the projection plane
define and of course the N or the VPN
which is nothing but the surface normal
surface normal to the projection plane
we have a VPN along the normal end to
this
face and so the I hope you remember and
recollect these terminologies based on
which we also define the projection
geometry and the projective equations
for both perspective and and on
orthographic or parallel projections
okay we look into the next diagram this
is also a diagram for perspective
projection geometry and the VP and the
VPN are saying that is a view plane
plane and the view plane normal we have
the centre of window we have a view
reference point and addition we have a
center of projection or perspective
reference projection respectively a
central projection basically or PRP as
it is called projective reference point
PRP is projective reference point or Co
pieces talked about whether arrays meet
and and basically for the parallel
projection the Co P goes to infinity and
that's why the we can have a PRP defined
even for the perspective geometry but
the CEOP basically goes to infinity
because the rays have to be parallel to
the projection plane in this case for
the perspective case they meet at the
center so I hope you remembered this
particular these two diagrams which we
also covered in projective geometry
earlier typically in a general framework
of computer graphics you need to specify
an arbitrary 3d view of course you must
ensure that when we specify an arbitrary
3d view your the camera is looking
towards the objects and and the objects
are within the view zone of the camera
and things like that but let us take a
typical example or a few examples of the
viewing parameters just a few set of
example values to give you a rough idea
about how to set these parameters and
then of course we will see how these
parameters are used for normalizing
transformations and also for projection
geometry the viewing parameters used are
V RP which is V reference point view VPN
V plane normal V of vector V or
projective reference point or Co P or
PRP and the corresponding window
coordinates of the viewport the view
reference coordinates so in bracket
degree these VR P VP and vo p and p RP
are in the first year in world
coordinates WC vrc's view reference
coordinates projection type could be
perspective or parallel
and you can also have the front plane
and the back plane coordinates defined
it is optional with respect to VRC and
the the values are given for the set one
you may not define em since it's
optional set two and say three does not
contain the values for the front
clipping plane and the back keeping
claim with respect to the view reference
coordinate systems so if you look in to
set one these are typical example values
which you can specify for the arbitrary
3d view and make a make a function call
with this set of parameters and it will
set up a specified arbitrary 3d view for
the object for you okay so that's just a
typical example when you use the
graphical software and the corresponding
fixed standards you have to use this
sort of conventions and these are just
example values which you can use yeah
well we go into the steps for
implementing normalizing transformation
matrix in the case of parallel
projection first and then we look into
the perspective projection and the and
the set of states are defined in this
slide the first step is you need to
transform the VRP or the view reference
point to the origin okay translate the
VRP to the origin so you need to have a
translation matrix first and then you
have a rotation matrix which helps you
to rotate the view reference coordinates
such that the N axis or the VPN or the
view plane a normal axis here aligns
with the z axis of the world coordinate
system and in turn also the U with the X
and the V with device so you collides
with X V with Y and the VPN aligns with
the Z I hope you remember the VPN which
we just talked about and what this means
also figuratively will describe as to
what do you mean by translating and not
at least rotating this because the
second step of rotation is the most
crucial step in this process of
implementation and the rest are very
easy steps but I'll complete this
sequence of steps the third step is
basically shear which is not necessary
for pure orthographic projection
generally necessary for parallel we know
orthographic is a special case of
parallel projection and in the case of
non orthographic parallel where the
direction of projection or DOP is not
parallel to the z axis we need to make
it well
taxes before he actually projected so
you may need a shear in case of parallel
projection which is non orthographic
images so that is the third step but
remember it is non optional but will
have a shear matrix at the third step in
the fourth step you have the translate
and scale into parallel projection
canonical view volume we know the planes
which define the canonical view volume
and you need to provide a translation
and scale to adjust your view volume to
the canonical view volume this is also
necessary before in the view pipeline
before you actually use the projection
matrix in fact before you use the Clippy
itself andy this is the form so these
are the sequence of steps and you see if
there are five different steps it
results in a composite matrix with five
different matte matrix multiplication in
the particular order and seen the first
step is the translate to V output the
origin so that is the T with a minus V
RP parameter then you have the R which
we are going to discuss in great detail
today and then s H matrix parallel means
the shear in case of parallel projection
and then P and s successively for pal
projection respectively this s and T
correspond to the last step fourth and
fifth respectively which says translate
and scale into parallel projection
canonical view volume so this is the
sequence of operations one is to do of
the first stage itself of the 3d viewing
pipeline you remember if you remember
the flow diagram which we talked about
at the beginning of the lecture today
there were four stages the first stage
was the normalizing transformation
second was clipping third was projection
and fourth was a simple 2d in the
viewport okay so the first one is what
we are discussing now and these are the
five steps of the first stage itself the
normalizing transformation okay so you
need to normalize such that the entire
world what this stage does is the entire
world is brought into the canonical view
volume so that it can apply clipping
easily and you can also apply the
perspective projection transformation
matrix so people can transfer matrix
although we know the generalized form
before you can apply there in fact two
sub stages
okay prior to that one is clipping which
we will of course discuss later on so
don't worry about it for the time being
I keep repeating don't worry about
leaving you we will do that with respect
to the canonical view volume when we
discuss exclusively clipping into
NTD so we'll discuss merely the first
stage which is the normalizing transform
and that has five stages given here and
the composition matrix in power and part
is nothing but the overall normalizing
transformation matrix for parallel
projections and I repeat it as why it is
composed of sequence of five different
operations or five transformations first
translate T then rotate R then C R SH
then a translation T and again as
scaling as far out of this the most
significant translations will be very
easy as you can visualize we will first
discuss R and then of course many little
bit of time about this shear so rotation
is the one which is very crucial and we
will see why okay this is the step two
of normalizing transformation I repeat
that state again is you have to rotate
the view reference coordinate system the
view reference coordinate system which
is defined with respect to the viewport
and the camera and things like that so
the camera will have its own UV and the
N axis you have to rotate that connected
of course the VRP to the origin and then
rotate it in such a manner said that it
gets aligned with the world coordinates
which has the does that the VPN or the N
axis the view plane normal aligns with
the z axis of the world system and U and
the V also aligns with the X and y axis
respectively so this is the most
difficult step to the most difficult
task in normalized transformations and
just to give an example what I mean by
these let us see this XYZ or XYZ is the
world coordinate system that means you
have defined your originally the world
code system with respect to this XYZ as
origin and however the camera is in such
a position that it cannot be at the
origin it may not be at the origin of
this system it could be at arbitrary
some position looking down some other
direction and there is where you have
the end U and the V axis which define
the 3d coordinate system for the camera
let us say because of the camera you are
basically looking into the visual the
virtual world which gives the
visualization okay and the camera will
have its own view plane normal or the in
axis it will have its own X&Y which we
call it as a u and V and this diagram we
already know where is the view plane
what is the view up Vector what is the
VPN centre of the window CW we were
reference point and all that so the
first stage is to basically take this
diagram or D with respect
to the camera or the view system and
first viewer translation so that the VRP
goes to the origin and then give a a
composite rotation such that the VPN has
to align with Z you with X and V with y
I repeat again first the VRP has to be
taken to the origin here it will go
I repeat VRP is here it has to go to the
origin at this point then the VPN or the
N axis should align with the z axis here
okay I hope you can see the cursor then
the U axis should align it X and the V
with Y okay so these three must
simultaneously happen together with a
composite rotation or three dimensional
rotation matrix three dimensional
composite rotation must take place to
have this alignment done so this is the
most crucial stage or step in
normalizing transformations and this
requires a sequence of operations now
how to get disarmed how to get this are
yes you will probably start thinking
about the various types of 3d rotations
and transformations which take place but
we dig into a little bit of depth of
mathematics here to usually understand
about 3d rotations get some principles
out of the matrix property of rotation
and see how easily we can formulate this
R okay and then we go to the next slide
so we are essentially now under the
first category or first stage of
normalizing transformation within the
five steps of the first stage we are in
the second step okay first stage of the
four stages is normalizing
transformation in that first stage we
have five steps and in that five step
first is translation second is rotation
so you are talking about is given the
slide so we are looking for expressions
for step two which must be derived that
is what we are looking at and we
implement that using the concept of what
we call as combined transformation or
rotation let us take a rotation matrix
rx this rotation matrix you know what
does it do it takes a point in 3d space
and rotates about the x axis okay so the
x axis coordinates of that point will
not be digital I assume the x axis is
pointing towards you okay you rotate it
about the x axis so the x axis
coordinates of that point does not get
disturbed but what are the
to other coordinates we change y&z y&z
are the two coordinates we change and
that is why you see the parameters cos
alpha sine alpha of course we are
talking about looking into the axis and
counterclockwise rotation positive alpha
the second and third axis will have the
cosine alpha sine alpha terms causing
the change in the y and z-coordinates x
axis will not change okay so let us try
to visualize this from a slightly
different angle this rotation I make
here a statement and will validate that
within a with with with a figure and
illustration later on that if you take
the rows of this matrix rx
as unit vectors that's what's it if rows
are unit vectors they remember it is
very easy to see here that due to the
orthogonal property of this rx matrix
the rows and the columns are unit vector
so take the Roses unit vectors and when
these unit vectors are rotated by these
rotation matrix rx they will align with
the Y and z-axis respectively
well it's difficult to straightaway
visualize when I'm making this statement
but I will explain that soon with the
help of a figure but just note down now
that the rows are unit vectors if you
take the rows I will take these rows of
this matrix each row is a unit vector
you can verify that take the first three
terms of each row don't have to take the
fourth storm because the fourth term on
the four by four matrix is coming due to
the homogeneous coordinate system so
take the just the top left three by
three sub matrix and take the three
elements each row the three dimensional
vector we are talking about so each of
these rows are unit vectors and these
unit vectors we I say that when they are
rotated by our X I did say that the x
coordinate doesn't change so 1 0 0 the
first row vector remains 1 0 0 itself
but the next two row vectors which are
again unit vectors they will get aligned
with y&z axes respective it's a very
nice phenomena which we will discover
now with an example but before that
there is an another complementary or
similar concept where I say what were
the column vectors if I say these rows
are unit vectors and they are rotated
and will align with y&z respectively
then the second point says that if I
take unit vectors along the principal
axis if I take
unit vectors along the principal axis of
the coordinate system and those unit
vectors are rotated by our X they form
the corresponding column vectors as
given here in this matrix so if you take
the column vectors 1 0 0 0 cos alpha
sine alpha 0 minus alpha cos alpha you
can form these vectors by taking unit
vectors along the principal axis and
then rotating by our X take the
principal axis vectors unit vectors
rotate them you get the column vectors
and the other case if you take the row
vectors and then rotate by rx they will
get aligned with the principal axis y&z
very interesting phenomena here and we
will see with an example figure as as to
how this happens again I repeat this is
again the row matrix rx has given the
top and what are those row vectors I
have given here the row vectors on the
first row I get a 1 0 0 that is the
first row vector 1 0 0 as given here the
second row vector at the first three
terms upper left sub matrix 0 cos alpha
minus sine alpha is the second row ok
and the third row is 0 sine alpha cos
alpha 0 sine alpha cos alpha so these
are the three row vectors in 3d how will
these row vectors look like ok let us
take a 3 dimensional coordinate system
is given this figure on the left hand
side of the slide with you and the x
axis is not shown because it is coming
out from this slide and pointing towards
you ok you can take a right hand
coordinate system and say assume X to be
pointing towards you so I have not drawn
that because they are essentially
rotating around the x axis so it will
rotate about this origin in 2d x axis is
pointing towards you to the right you
have Y and the vertical one is it so
look back into the figure you will find
x axis pointing towards you Y to the
right and z axis vertically upwards so
now look back into the row vectors which
I have just written look at the row
vectors and try to map it onto this
system you will find that the 1 0 0
vector is nothing but its along the x
axis pointing towards you anyway since
you are rotating by rx it does not
matter 1 0 0 nothing will happen
what about the second vector 0 cos alpha
minus sine alpha you look at this I have
labeled with the same color as the row
vector in text this is the corresponding
yellow
a vector which is giving the second
vector because if you look this is the
angle alpha then the x-coordinate and if
it is a unit vector then the
y-coordinate will be cos alpha and the z
coordinate of that vector will be sine
alpha in the negative direction hence
you have minus sine alpha correct so
that is the second vector the third
vector also corresponding color I have
used it is also sine alpha along the Y
and Zed alpha along the z axis so these
are the two vectors which are the second
and the third row vectors of the
transformation matrix rx now what will
happen if I take these two vectors and
give it a counter clockwise positive
rotation alpha you can visualize what
will happen if I give it a rotation by
an aqua what will happen the
corresponding row vectors will
automatically get aligned with the Y and
z-axis very interesting isn't it you
have them bent and they will talk and
then they will get aligned with the Y
and Z does his respective this was being
talked about as the point in the
previous side I will show that point
once again where you take the row
vectors as giving the transformation
matrix and those row vectors when given
their corresponding rotation by the same
matrix they will get aligned with y&z
now what with the column vectors I can
use the same diagram and visualize the
column vectors now for the column
vectors what you assume is let us take
you take a column vector you take a unit
vector along y&z respectively and give
it a negative rotation that means now
clockwise looking into this figure
negative rotation rx of minus alpha in
this case what will typically happen is
you will get the corresponding column
vectors as 1 0 0 0 cos alpha minus alpha
remember rx of minus alpha so what will
happen is in this matrix the sine alpha
here will change sign to - and in the
third column the minus will become plus
correct so that is what is happening
here that the reverse could also take
place you could have given a positive
alpha and seen what could have been the
result but in this case a negative alpha
with the same figure I can show you that
if you take unit vectors along y&z give
it a clockwise rotation negative alpha
you will get the column vectors exactly
as given by the transformation matrix so
this is an interesting phenomena which
we will use to derive the general
transformation matrix in which you have
an arbitrary coordinate system and you
need to give a composite addition to get
it all aligned with another coordinate
system respectively
ok I'll roll back those two points once
again you see here these are the two
points ok where I say I repeat again if
rows are unit vectors and then they are
rotated by a rotation matrix rx remember
then they will get aligned with the Y
and z-axis respectively we have seen
that these rows are the unit vectors
obtained itself from the Rx and the next
point is when unit vectors along the
principal axis are rotated by an by the
same by a transformation matrix rx they
form the corresponding column vectors of
rx so that's a very interesting
phenomena and we have seen that with the
help of this example you can copy this
figure and the corresponding row vectors
column vectors and keep checking it
yourself with both positive alpha and
minus alpha rotation of course you can
use rx ry r RZ you will also get the
same effect okay so you keeping this
concept in mind that the row vectors can
be rotated and formed and they and they
form the vectors as given in the
rotation matrix and and the column
vectors you take which are in fact the
row vectors get aligned with the Y and Z
and in the case of columns you take unit
vectors along Y and Z and what happens
is they're they form the vectors as
given by the rx so if that is true and
this this is is possible to use this
concept and this idea because of the
property of orthogonal Y of the matrix
because unless the matrices would have
been orthogonal you wouldn't have had
the rows and the columns giving unit
vectors that is number one okay and so
if that is the case I make a step jump
and come to a general scenario of a
combined rotations wherever I use this
property based on the orthogonality of
the r matrix where i say that if this is
a scenario of three different vectors I
will say on a system XYZ I have a vector
the
p1p3 the the whitish vector given by the
white arrow p1 p2 and the dashed vector
in blue color p1 PN so there are three
vectors and in this unit three vectors I
must tell you that the P 1 and P n is
nothing but a vector which is normal to
the surface obtained by P 1 P 2 P 3
so actually what it means is you can
take a cross product of the two vectors
p1p2 and p1p3 that will give you the
normal vector P n that is the relation
so basically PN P 1 P n is normal to
both P 1 P 2 as well as P 1 P 3 whereas
P 1 P 3 and P 1 P 2 may not be
orthogonal to each other or
perpendicular to each other so P n can
be visualized again as I say as a normal
to a surface formed by P 1 P 3 and P 1 P
2 vectors but what I essentially want to
do is I want to get this set of vectors
aligned with the at least two of them I
can get aligned with two of the other
axis in XYZ because I have
perpendicularity with respect to P N and
P 2 P 1 P n is perpendicular to the P 1
P 2 so before transformation if this was
a scenario I would like to give this set
of three vectors a combined
transformation a combined transformation
where these three let us say get aligned
with D homogeneous if this is the XYZ
and I have this P 1 P 3 P 2 P n vectors
and I will give it a combined
transformation to get itself aligned
with XYZ and now we go back into the
slide and I want to give it a combined
transformation after transformation
I want the following to take place the P
1 P 2 actor before transformation was
given in the previous figure here in the
same slide gets aligned with the z axis
and the corresponding normal to that the
P 1 P and the blue color dashed 1 gets
aligned with the x axis and of course P
1 P 3 also takes its own place somewhere
because it has to be perpendicular also
to the P 1 p.m. so basically this P 1 P
2 P 3 vectors will actually these two
vectors will lie in the plane Y Z plane
that is bound to happen with this set of
transformation if I take place so
if you remember the previous two
concepts which we talked about in
Article II matrix in terms of rows and
columns which one was the property which
given a rotation arbitrary vectors are
some vectors get aligned with the
coordinate axis XYZ we have to deal with
rows because if you take the rows as
unit vectors of the transformation
matrix and give it a transformation they
in fact get aligned with the
corresponding principal axis okay so
that relation will keep in mind and the
corresponding transformation rows or
columns of the transform matrix can now
be in fact defined or obtained directly
from these vectors elect if you just
expand this concept from 2d which was a
rotation on the x axis to a general
scenario just a step more of course you
can derive analytical expressions then
check this also but but if we just draw
an analogy to a general
three-dimensional what will happen is
now that you can derive this
transformation matrix from the vectors
directly that's the advantage okay
remember the previous examples we are
taking with rows and columns with the
transformation matrix rx keep that in
mind and will derive now okay so let the
effective rotation matrix be a
combination of three rows as given by
this where I will say the first row
corresponds to something in X as R at R
1 X are ticks are tubes respectively
second with respect to similarly with Y
and similarly the third row is with
respect to this and this is the
effective rotation matrix which will
cause that transformation to take place
and if that is so this rotation matrix
what it is supposed to do before
transformation there's P 1 P 3 P 1 P 2 P
1 in vectors will be transformed three
vectors will be transformed in such a
manner that the P 1 P 2 P the purple
color vector will get aligned with the
z-axis the whitish - vector p1 PN will
align with the x axis and of course the
P 3 vector also lies somewhere in the Y
Z plane to implement this concept we
look back into the equations and here
the corresponding form of the
patience of this combined or effective
rotation matrix is given and we'll start
with the bottom row last row of that
matrix where I define our Z to be a
vector of the last three coefficients of
the last row elements of the matrix
given by what it is nothing but the unit
vector along p1 p2 so if I put the
elements of the last row equal to the
unit vector along p1 p2 that will help
me to get this vector aligned with the z
axis after transformations similarly
next what I have to do I have to take
this p1pn vector and align with the x
axis so my first row which consists of
the elements are 1 X R 2 X and R 6
respectively which is given by the
vector R X is nothing but the vector P 1
P N and what is P 1 P n letter it is
nothing but a cross product of p1p2 and
p1p3 unit vector along P 1 and P n can
be obtained by the cross product of p1p2
and p1p3 so that so that is the unit
vector which is given by P 1 P 2 cross P
1 P 3 cross product divided by the norm
correspondingly will give the unit
vector along p1 PN and that gives the
elements are 1 X R 2 X R 3 X
respectively so the top row and the last
row elements or coefficients of the
matrix are only obtained you need the
middle row which is nothing but the r 1y
r 2i r 3 y respectively
they are nothing but easily you can
visualize that they have to be nothing
but a cross product of the two vectors
p1p2 and p1p3 that will give you nothing
but the vector which will align
automatically with the y axis because
after you have a line P 1 P 2 with Z P 1
P n with X automatically the cross
product of these two will get aligned
with the Y axis respectively so the
cross product of RZ and rx will be the
elements of the y axis so this is the
philosophy or concept by which you come
up with an effective rotation matrix win
by which any arbitrary 3-dimensional
coordinate system can get aligned with
another coordinates
respectively of course we are talking of
Cartesian coordinate system X Y Z and n
UV let us say and you want to put this
here or do you want to put that here
sort of a thing you can do that very
simply if you know the corresponding
vectors unit vectors of the other axis
with respect to the one which you are
mapping onto or aligning it with and if
you know the these vectors with respect
to this coordinate system let us say the
source and the target then the you just
form the transformation matrix with the
help of these vectors put the
coefficients as these unit vectors apply
that and the system gets aligned so that
is the basic philosophy of the rotation
matrix which is essential for
implementing step 2 of the
transformation or normalized
transformation okay so understand this
theory now and I repeat this figure once
again this is essentially showing the
figure in which the rotation matrix of
steps 2 has to be formulated in a manner
in which we are illustrated with respect
to this figure where I made P 1 P 2
aligned with the z axis and the P 1 P
and aligning the x axis respectively and
if I can do that
similarly I can do it for my normalizing
transformation where I need to take the
VRC or the view reference coordinate
system and align it with the world
coordinate system and I will have the
similar form of the rotation matrix and
and you remember I repeat this step 2
this is what we have been discussing how
to implement this step 2 in normalizing
transformations and it say that you
rotate the VRC says the view plane
normal or the N axis aligns with the z
axis and also you make you align with X
and we align with Y axis respectively
okay so that was your world coordinate
system XYZ and you had to align this
right this figure comes begin I repeat
this because this is essentially what we
were planning to do and we have to do
and we have understood the basic theory
of how to do that how to do that now you
need to just align VPN with the z axis
you with the x axis V with the Y axis
this was our original problem and we
know to solve it we notice solve it
because if you know the vectors VPN U
and V with respect to the world
coordinate system XYZ you can form your
rotation matrix with the help of these
vectors and that is the rotation matrix
which will help you to
the system aligned with the wall
coordinate system so let us look at this
form okay that's simply like the one we
did take this axis Z the last row of
coefficients the our Z which is nothing
but the view plane normal so to take be
unit vector along the view plane normal
that will give the the last row of the
transformation matrix for rotation in
the in the transformation matrix the
last row and is it and the X will also
be obtained by a cross product of the
view vector and our Z let us go back to
the figure to understand because
significance of our X here it is a cross
product of view up Vector and the and
the view plane normal if you take the
cross product of view up Vector view up
Vector is nothing but points which is
the object it's lying in the enve axis
and if the cross product of the we are
vector with the VPN will actually give
you nothing but it will actually give
you the u axis so the specification of
view of vector and VPN itself these two
vectors sufficient to tell you where is
your you access and a combination of U
and V of vector will tell you are the
VPN and you will tell you where is your
V but to get you given vup and given VPN
I repeat given V u P or the view of
vector are given view plane normal or
the VPN the cross product of these two
will give you the V axis so the that is
the concept of taking the cross product
of U plane normal which is nothing but
our Z and Y gives the elements of X and
of course the elements of Y this is the
second row can be obtained easily by the
cross product of RZ generic so these are
the expressions these say the equations
which will help you to get the
expressions for the combined
transformation or the rotation matrix
okay so that is the step two so the step
two coefficients can be obtained by rx
ry RZ respective individual rows and
they are computed nothing is required
except you require the view plane normal
and view of vector once VPN and vup are
given you can actually obtain the rx ry
RZ
elements required for the rotation
matrix which is this step two and that
is the step to the remember there were
five different transformation matrix
required for the overall combined
normalized transformation matrix for a
parallel projection I have put in
bracket dub
blue CSV be to be projected to PPC CV I
explained what this terminologies mean
well WCS VB you can expand to world
coordinate system view volume world
coordinate system view volume has to be
transformed to a parallel projection
canonical view volume PPC VV world
coordinate system view volume to a
parallel projection canonical view
volume yeah that is what we have been
discussing as the overall combined
transformation matrix and consists of
five different transformation matrices
first you remember the steps translation
rotation shear translation scaling and
after the translation with respect to
minus V RP so far in the most part of
the talk in the last half an hour or so
I have discussed how to get this R and
the equations to obtain the elements of
our are given the same slide in the top
the third the last row of R is given by
the VPN view P multiplied by VP and
cross product gives you the first row of
the elements of X and our Y can be
obtained by the cross product so once
you get R the first two stages are over
you need to only find out what is how to
get the shear transformation matrix and
what should be translation and scale so
we will discuss in the remaining time
these three transfer matrix and complete
the first stage of the normalizing of
the viewing pipeline of the normalizing
transform okay so this was the equation
the combined transformation matrix for
parallel projection which transforms a
world coordinate system via volume to a
parallel projection canonical view
volume WCS VB to a PPC VV is given by
these five transformation matrices and
we know that how to get R and T what is
the shear matrix this here is necessary
if you remember carefully this demand
while discussing these steps the shear
matrix is necessary for a parallel
projection which is non orthographic
calibrations could be two types it could
be autographed non orthographic as well
as extra material symmetric and all that
what is the difference between
orthographic and non orthographic among
the class of parallel projections well
in case of parallel projections there is
no doubt that the Rays will be all
parallel
and they will be going towards the
direction of projection somewhere
towards infinity see your peak is
interposition it goes to infinity that's
why the Rays are all parallel but there
is a directional projection that
direction of projection may not be
normal to the view plane if it is normal
it gives orthographic if it is not
normal to the view plane if this is the
view plane then if they are all
perpendicular to the view plane you get
perfect orthographic projection but you
can have non orthographic parallel where
the direction the parallel rays are now
not normal to the surface they just come
and hit it in inclined manner so that is
a case where you need to provide a
shield to the view volume now otherwise
if it is pure orthographic razor in your
parallel or I'm sorry normal to the
projection plane then you don't need to
provide this third step or the middle
step of shear and to explain this shear
step I have two figures where before
this is a side view of shearing of the
view volume what actually may happen is
as you can see in the left hand side
bottom of the frame you have a view
plane normal and the direction of
projection is somewhere different it is
not parallel to the view plane normal
this can happen in a parallel projection
which is non orthographic in nature in
case of an orthographic automatically
direction of projection and view plane
normal will actually be parallel you
don't have to might do not need to
provide this shear but here the
direction of projection is not aligned
with the view plane normal and you need
to provide shear to this view volume to
make that direction of projection
aligned with the view plane normal and
that is given in the side with a side
view of the shearing of the view volume
if we can provide it shear the right
hand side figure you know this what you
can say is this this rhombus type of
figure basically becomes a perfect
rectangle or a square and and that helps
to make your directional projection and
if you play normal the same they are now
aligned and that can be provided by two
types of shear along the x and y axis
you need to give this year and the shear
parameters in the matrix is given here
as H bar and it has just two elements
here s H X bar and SH Y bar and I leave
it as an exercise for you the first
exercise in these today's lecture
to obtain this X power and SH Y power
elements of this year you have discussed
a virtual matrix in theatres summation
you should be able to write that but the
answer is given to you here where the
elements are nothing but the ratio of X
to Z component of the direction of
projection vector d o P vector is a
directional projection vector so X
component of that divided by Z give you
the SH X paw and the ratio of Y to Z
gives you the SH Y part ok the SH Y part
is given by dop y by D of Z respect ok
well so we are discussing this stage of
overall combined transformation and when
there are two stages left one is scale
and other is translation and I leave it
an exercise for you this this last two
stages of translation and scale are
required because the canonical view
volume has to be the view volume has
been transformed after rotation
translation shear but it has to be
shrunk to the canonical view volume
defined by please explains in parallel
projection and to do that before itself
scaling and you might need to translate
it says that the front plane gets
aligned with the corresponding XY plane
and the corresponding u max you mean V
Max remain act together and so you need
a little bit of translation in 3d to
align it perfectly and then give it an
overall scale factor you see that the T
is it is a vector the T matrix whereas
the S is given the functional form these
are the three diagonal elements of the
shear matrix which are given in the last
equation and you provide that scale you
provide that scale but to actually fit
it and form the canonical view volume in
parallel in the time remaining we'll
quickly go through the perspective
transformation matrix for normalizing
transformation and these steps are
almost similar to case of parallel
projection only now the canonical view
volume will have a different shape is
like a pyramid unless in the case of a
rectangular parallelepiped in the case
of a parallel projection okay so these
are the equations six planes we talked
about in case of perspective projection
so it is forming a pyramid we are having
a side view in the X Z or Y Z plane only
we are looking if you look into either X
0 revise it you will have this figure
it's like basically
pyramidal structure ok the steps I will
quickly go through their steps it's
almost the same translate the VRP to the
origin rotate the VRC the second step is
also the same you don't need a she and
now prior to that you needed translation
because there is a center of projection
or a perspective reference point or a
PRP or Seop which comes you need to
provide a translation first then you
need to provide a shear so that the
centerline of the view volume becomes
the z-axis like the direction of
projection which you had in the case of
parallel projection and then you of
course do a scale so the sequence of
tips are almost similar especially the
first two and then of course the last
three a little bit jumbled up in a sense
that the translation first then this
year and then this scale okay the this
is the the scenario of the cross-section
of the view volume after first three
transformations what are the first three
transformation translate with respect to
- VRP rotate which we have discussed
first the rotation of the one system to
the other and then of course you
translate it to the - with respect to -
COP or PRP then you need to give it a
shear because what may happen after the
first three stages of translation
rotation translation is the view volume
a before giving it a scale it may appear
something like this it may not be
perfectly aligned with the z-axis you
need to do this otherwise the canonical
view volume will not come in shaping it
cannot apply the projective
transformation later on so the view
volume centerline VV CL as given here
passes the center of the window from the
origin and this has to be aligned with
respect to the minus eight axis and that
is what is required and that will be
provided to by the SH power matrix which
will give you that shear okay so this is
the overall and just as a comparative
study of the overall combined
transformation matrices for parallel and
perspective as you can see here the
nature is almost same as specifically
the first two transformations my T of
minus V RP n are are same then we have a
translation and shear in the case of
perspective in the case of a parallel on
their top you have first year and then
translation so that is the reverse and
the end anyway we have to have the
scaling to scale it to the corresponding
canonical view volume so that's the
comparison between the two projection
geometries of the transformation
matrices and now if you go look back
we'll stop with this today that we come
back to this original viewing pipeline
the implementation of the 3d viewing
pipeline which we started to talk about
today and you remember I repeat again we
start with 3d World coordinate output
primitives apply a normalized
transformation then clip then you have
to projecting the projection plane and
ultimately the 2d transformation to get
the 2d device coordinates so far what we
have discussed in the entire lecture
today is the first stage only the
applying normalizing transformation and
that to two parts of it one is the
parallel geometry parallel projection
another is the perspective geometry and
and among those there are five steps of
transformations in the first stage
itself of applying the normalized
transformation and among those five
steps the second step rotation was
probably the most crucial one of course
you have to apply this shear in the
scale as well so applying the normalized
transformation consists of five steps
both in the case of perspective and
parallel projection geometries then you
of court need to provide a clip in the
canonical volume and then we project
into projection plane now the third
stage is the projection plane this we
have studied earlier in the case of
three-dimensional transformation itself
I just flashed this slide so that you
can recollect the general transformation
matrix for projection and once that is
done you need just a 2d scaling in terms
of the transform into viewport in 2d
coordinates for display and the next
slide is the third stage that is because
the second stage of clipping we will
discuss when we talk of clipping
entirely in 2d and 3d so we know now the
first stage the second stage of clipping
will come in due course of time when you
talk of clipping third stage in fact we
have already discussed under three
dimensional transformations and this was
the generalized formula if you remember
the generalized formula for perspective
projection matrix was given as this this
we derived in the previous lecture under
three dimensional transformations under
the Tridentine transformations we
discussed a lot about perspect about
projection geometry parallel axonometric
to going to metric one point two point
three point perspective geometry and
also transformation matrix in the case
of perspective geometry and also this is
generalized projection matrix both
applicable for perspective and parallel
and if you recollect back into the nodes
which you have taken from the slide no
you can put special cases of the values
of DX dy DZ direction of projection and
which will take this GOP to either
infinity or in a finite distance we will
give you projections of perspective or
perspective or parallel orthographic
okay so with this we stop today and we
continue with the 3d viewing pipeline in
the last lecture this is the perspective
projection matrix which we use as the
third stage of the viewing pipeline I go
back and we stop with this same slide
which we were talking and we started
with this slide where we have discussed
extensively about applying normalized
transformation and we have discussed
also seen the expression of the third
stage which is projected on the
projection plane we will probably not
discuss canonical view volume clipping
now a little bit of transform into
viewport we will discuss but in the next
lecture we will start from this and we
will see the viewing pipeline broken up
into several stages where we start from
object model coordinate system pass
through world coordinate system go into
the eye space or image coordinates and
then to device with its entirely a
sequence of transformations which take
your object somewhere in space the
camera is somewhere here the object is
somewhere here model and the sequence of
transformations which will slowly take
all these into the device coordinates in
the screen on the TV or the projection
system which you see okay so we stop
here with the first lecture on 3d
viewing and we'll continue this with the
3d viewing parlin with the different
matrices revisited again in the
different stages and on how to get the
2d device coordinates and important
thank you
you