3D Viewing - Computer Graphics - ComputerScience and Engineering Video Lecture - Computer Science Engineering (CSE)

we continue the discussion on 3d viewing
pipeline and projection transformations
under the category of 3d weaving and in
the last hour II of lecture now you
remember that we have discussed
basically the weaving pipeline in the
terms of four different stages of
transformations and within those four we
in fact exclusively discussed only one
of these stages the important stage of
normalizing transformation so if we look
back into what we were doing in the last
one our implementation of 3d giving the
input to that is a 3d World coordinate
output primitives and which passes
through four stages of transformations
the first one is to apply the
normalizing transformation the second is
to clip against the canonical view
volume so you know what is a canonical
view volume now and then of course comes
the projective transformation or
projecting to the projection plane which
will actually take the 3d to 2d the
coordinates and then transform into
viewport in 2d device coordinates for
display and so the output of this entire
set of operations are in some sense the
viewing sequence or pipeline is the 2d
device coordinates and if you look out
of these four stages we mainly discuss
the first one which is the applying
normalizing transformations which
consisted of five different steps of of
operations or five different matrices to
be which you are doing these operations
on the 3d World coordinate output
primitives we knew what is a canonical
view volume when we read about and
understand and discuss about clipping we
will see how the clipping operation
could be done because that you always
when you have a graphics display or any
in fact seeing through your eye you have
always a finite amount of display and
you need to clip and throughout the
world which is outside the range of
reading either through the camera or to
the eye and of course in 3d
transformations we have also studied
projection geometry the M generalized
matrix we apply that and the last step
is a two dimensional same
transformation which will also see today
so the first stage if we look which is
the applying the normalizing
transformation and we had seen two
different projection transformations one
under the category of parallel or
orthographic projection orthographic a
non orthographic in case of parallel and
perspective and these would be five
different stages for the normalizing
transformation in terms of parallel and
perspective so just to have a
comparative study you remember we had a
translation matrix first because the
operation is going from left to right so
we are talking about three multiplying
the matrices or coordinate points by
these operations so translation minus VR
P then a rotation then again a
translation or a shear and then a scale
acting only so that that's the
combination the sequencing is very
important you cannot jumble up because
we have studied under transformations
that order is very important sequence of
operations are to be kept unaltered to
have the correct projection or the
transformation done and out of these
five operations the most crucial one was
probably the rotation matrix are where
we first transformed a coordinate system
a view reference coordinate system onto
a world coordinate system we knew how to
align that using the property or
orthogonal property of the rotation
matrix and also we discussed about shear
and scale the form of those mattresses
okay couple of exercises were given to
you during that time to identify the
parameters of shear based on the
direction of projection vector and scale
parameter parameters for the scale
matrix for parallel and perspective as
well both are of similar types the shear
matrix is optional in the case of
parallel projection Infinity spear
orthographic because the direction of
projection in case of orthographic
projection is aligned with the view
plane normal so you do not need a shear
but if it is not you typically will need
it shear okay so this is the first stage
of that pipeline the viewing sequence of
five different steps of the first stage
and then of course after the clipping
has been done you can apply the
projective transformation this also we
seen a few classes back when we were
discussing about three dimensional
transformations and I talked about the
generalized expression of the
transformation
matrix in general and it depends on
basically mini three key or four key
parameters one is the ZP the distance of
the projection plane from the origin and
also the direction vector D X D Y Z DZ
which consists of the direction of
projection vector and the length of the
point Co P or PRP from the point 0 0 ZP
is Q we know that and we had seen the
derivation of this matrix as well few
classes back okay so first second third
stage is clear and the fourth stage of
course is a simple 2d transformation so
we look into the pipeline in a slightly
different framework now we look through
it slightly different framework where we
talk of coordinate systems and matrices
together for this entire 3d viewing
pipeline a sequence of projective
transformations and world coordinate
systems moving together and we see from
how we move from the 3d object
coordinates situated somewhere in space
and then you have a camera looking
through it your world coordinate system
could be some very here a right hand or
left hand system your object points
could be somewhere at this point the
camera looking this way so what will you
see in the graphics screen when a camera
is look at it at an arbitrary point
looking in the direction at some
arbitrary direction you have a set of
objects at this point and the world
coordinate system is somewhere here
vrc's aligned with the camera is looking
down objects or somewhere your objects
which could be outside and away from the
camera will not be in the field of view
because you have a finite zone or a
field of view both for the case of
parallel as well as the perspective
projection geometry and you need to
transform all these points from 3d into
the canonical view volume transform the
VRC into the world coordinate system
then do the projection geometry from 3d
to 2d and get into the viewport ok so
that's a sequence of operations which
are taking place for a generalized
scenery of course you can have a very
simple scenario to start with if you say
my camera is looking in such a manner
that it's pointing towards you or
towards me and the world coordinate
system is also aligned with the camera
it becomes very easy because most of
this transformation matrix will be
unnecessary you will not need them to be
used in your normalizing transformations
and you can simply worry about the
camera parameters
the world coordinate points which are to
be displayed on the screen of course we
will see CH as a simplified model which
is only based on a camera but a
generalized scenario is very very
important because in a generalized
scenario most of these graphical
software is based on figs or OpenGL will
ask you to specify if we wing pipeline
to specify the viewing scenario where
you have to specify what is your world
coordinate system it could be somewhere
placed on the earth on the universe on
the Sun wherever it is and then you have
the camera located at some point with
respect to this world coordinate system
the cameras look in a particular
direction you have a sort of object
somewhere positioned here a set of
points line surfaces curve sticks -
whatever the case may be and you would
like to have this generalized scenario
such that you can handle all sorts of
environments all sorts of environments
not only the simplest case the simplest
case cannot handle generalized scenarios
with respect to a camera model which you
will see later on and it's and most
graphical software when you talk of the
programming language layer or the high
level of Figgs
standards it asks you to specify certain
parameters for the viewing pipeline or
the projection geometry so we will come
back of course always keep in mind that
there are two main type of projection
geometries one is the perspective
geometry and the other is the parallel
geometry and again we start with the 3d
modeling object coordinates 3d modeling
object coordinates start they go through
a modeling transformation that is the
first stage that's the first stage you
have the modeling transformation where
basically what you do is it generates
the corresponding 3d world coordinates
for the object model because you may
have 3d modeling object coordinates with
respect an object center and you need to
position them somewhere in the world you
need to vision somewhere in the world
and that is what we call as a 3d World
coordinate generated by the modeling
transformation in this case the modeling
transformation talks of positioning
those objects and the points and lines
curved surfaces of the object at
somewhere in the universe or somewhere
in the world so you get 3d world
coordinates and then you probably start
to give the transformation which we will
call the normalizing transformation we
break it up into two or three different
parts the first part of that cont is
called the view orientation matrix where
you can see the expressions are given on
the left hand side corresponding to
perspective and parallel they are
basically saying in fact they are
nothing but the
first two steps of the first stage of
the normalizing transform that is the
translation vector put the VRP to the
origin T of minus V RP and then the
rotation matrix which will align your
view reference coordinate system with
respect to the world coordinate system
which is our origin of the overall
system so that is the view orientation
matrix which takes the first two steps
of the overall five different steps of
the normalizing transform and the the
the flow chart is not complete here it's
so big that I split into two parts so
after the view orientation matrix we
move on to the next slide we talks of
the remaining stages so the view
orientation matrix has the R and T
mattresses and it will generate what we
call as the view reference coordinates
okay so the dark black boxes talks of
the matrices and the elliptical areas
talks about the coordinates at different
stages okay the outputs at different
stages so if I go back you started with
the modeling object coordinates you move
to world coordinates after that the view
orientation matrix will generate what we
call as the view reference coordinates
and then on the view reference
coordinates you apply what is called the
View mapping matrix you apply the View
mapping matrix which has all the
remaining three steps of those five
different steps of the first stage of
normalizing transform that is the
translation and shear and the scale okay
so you have the T you have the SH and
the s matrices now in terms of
perspective and parallel if you see I
have tried to minimize the difference by
saying that that I will have M matrix
which will probably make my s par is
equal to appearing like s par that the
scaling could be little different in
case of perspective and parallel
transformation because the canonical
view volume is different that's why this
extra M has come to take care of the
same and give you the difference between
perspective and parallel otherwise the
shear and the translation are typically
of the same type okay so after that you
after the View mapping matrix has been
applied from the view reference
coordinates you get the normalized
projection coordinates correct the view
orientation matrix generated the view
reference coordinates and the View
mapping matrix generated the normalizing
projection coordinates in fact the view
orientation matrix along with the view
mapping method
where there's five different steps of
the normalizing transform which we had
discussed in the last hours so those
five steps have been split into two
parts three matrices three steps and one
in two steps and they they are called
here the view orientation matrix and the
View mapping matrix respectively
expressions are given for the view
orientation matrix here in this slide
that is the T and the R and the rest of
the translations shear and scale is a
part of the View mapping matrix and it
has generalized and it has generated the
normalized projection coordinates these
normalized projection coordinates are
still I will say in 3d as you are still
in 3d but now what you have done
basically is if the objects were
somewhere here this was the world
coordinate system the camera was looking
here you have actually done the
following by transforming the camera
getting it aligned with the world
coordinate system so the objects are now
also followed the same path and you have
actually Shrunk the entire view volume
value of reaction geometry into this
small canonical unit view volume okay
which is bounded by this experience we
know the CVV canonical view volume
bonded by six planes top bottom left
right and front and back planes okay so
now the entire world which you are
supposed to see through the camera at
some point is now all aligned and kept
with the world coordinate system and it
is all inside the canonical view volume
or inside the CVV but the points are
still in in 3d they are still in 3d but
in a well bounded or well-defined
normalized their projection geometry and
you have to apply basically two or three
different state the first step is
clipping first you have to clip with
respect to the canonical view volume
then apply a projective transform and
then map do a transformation in 2d that
is all put under one category in the
last stage of the viewing pipeline which
I will call as clip transform into 2d
screen coordinates
transforming into projective from 3d to
2d and then an hour shift it to 2d
screen coordinates and all those put
together except a clipping part I say is
all handled by one particular big matrix
called MC vv3 DVP I will expand what
this means but this matrix after
clipping when it is applied will
basically give you your 2d device
coordinates now this M C
vv3 DVP is basically meaning canonical
view volume 3d viewport okay that mean
it's a transformation matrix M which
transforms the canonical view volume
into your 3d viewport for projection so
this has to be done after a clipping
which will transform and then you can
apply your projective geometry max
matrix M general also to get the 2d
points from the 3d and it will give you
your 2d device coordinates of course you
may need a little bit of scaling and
translation at the end to get the 2d
device coordinates within the viewport
you are talking about okay you need to
map everything into your viewport which
we talked about earlier and this is all
coming under the category of M CVV 3d
viewport that's a terminology used in
one book by Foley random Rajas and
etcetera
where they will say that we have a dis
domotics called MC v v3 DV P it means
canonical view volume to 3d viewport and
the expression of that matrix is given
here and this matrix you should apply
after clipping
after clipping when the entire 3d
canonical view volume has been generated
all points under that you apply
basically keeping then apply this matrix
as you can see from the expression the
MCV v3 d VP canonical view volume to 3d
viewport consists of a translation
scalar and a translation so basically
what it is doing it is trying to adjust
the canonical view volume to your 3d
viewport such that you can actually
apply your M general projection matrix
to get the 2d points out of this
remember these are 3d matrices again the
world is still 3d now after doing all 3d
manipulations necessary you basically
apply the projective transform to create
your 2d coordinate points from 3d so
clipping apply MC v v 3d VP then apply
him general to create your 2d
coordinates and then of course you have
to do a little bit of scaling and
translation to position all whatever you
want to see in a particular report if
this is the screen which you are seeing
right now you will say ok I don't want
to see all the text in the entire screen
I would like to select a small portion
here or maybe some portion here so that
is what is a 2d viewport I will say all
I want my display on this particular
graphics package to be positioned here
or somewhere here that is what
call it a viewport and what you may need
to do is the doodle device coordinates
which will be generated will be
generated with respect to the whole
setup you need to scale it down and then
translate in position in the viewport
which you demand which you demand of the
user asks so you need to map it into a
two dimensional viewport those
expressions are very simple since you
all know how to do three dimensional
transformations 2d transformation should
be very simple a simple transfer a
simple scaling and a translation brings
you from this entire screen down to a
small report because you have a big
screen then you basically do a small
skimming operation to bring it down and
then you can position it anywhere in the
screen basically the viewport through
which you are viewing the world which
you decided and that is a simple
translation operation show you at the
end basically you will require a scaling
and translation in 2d which is not there
in the expression but you can visualize
that it comes at the end after all the
projection geometry is done after all
the operations of normalizing transforms
on the viewing pipeline after the
clipping after the m3v CVV the 3d
viewport and then after you do the
projective geometry M general matrix to
transform from 3d to s screen to D and
then now you want to position this
screen because you may not like the
entire display screen to be covered by
just one window of graphic screen you
may need have a graphic screen here you
may have a multi media operation some
text running here and some other
operations down below so obviously you
will not ask one particular application
to cover one the entire screen so that
entire device coordinates where you want
to display will be generated by a 2d
scaling operation and then position by a
suitable translation so that is the
viewing pipeline which we have talked
now and again I will repeat this in
steps to complete the entire 2d viewing
pipeline we can also take a scenario
where it involves not only these
transformations but of course you need
to generate pictures you need to
generate pictures with the help of
different algorithms which will come in
due course of time as we go along in
terms of shading drawing lines and
things like that and we will discuss in
general now the various steps of what we
have seen to the now we will say a very
elaborate or a very general 3d viewing
pipeline and most of the operations you
know the objects are modeled in object
modeling space that is the first one
first step you know that okay so you
have an object centered coordinate
system and objects are centered or more
are kept somewhere in the object space
and then you have to actually apply a
transformations which will be applied to
the objects to position them with
respect to the world space or the view
reference coordinate system at least
with respect to the camera coordinate
system our world spares space our world
coordinate system WC or VR see that is
view reference coordinate system you
would have to apply transformations to
keep these objects so the second state
says transformations are applied to the
objects to position them somewhere in
the world space okay so that is
important view parameters are specified
we had seen in the earlier class a table
with a few examples of what are the
different view parameters you need a
typical X set of examples you can think
of are yes you need to specify say the
view plane normal which direction you
are looking you need to specify the view
up Vector let us say you will also need
to define other parameters of for the
viewing system but typically at least
you need to have the view up Vector and
and view plane normal view up Vector and
the view plane normal to specify the
world coordinate system and the rather
you can look back at the table which was
given in the previous class to have
examples of different type of
perspective and parallel projections and
view specifications so that is all
necessary now that the third step are we
say the view parameters are specified to
define the view volume of the world I a
projection plane of course you need to
tell where is a projection plane and
also the viewport the two-dimensional
viewport on the screen so these are some
of the parameters could be optional but
you need to specify at least minimum
some of these parameters to specify
where you are looking correct you can
have a world you can have a virtual
reference coordinate system but you can
have a camera looking from this angle
onto the object or from this angle or
from so you need to specify your where
is your camera and what direction it is
looking of course if the object is here
and the camera could be looking
somewhere else he will not be able to
see anything unless there are objects
somewhere there but typically you need
to position it so that you want to see
what sort of view you would like to have
we really like to have the top view or
the bottom view of an object if this is
an object let us say or the side view or
the front view so you need to position
your camera you need to specify the
viewing reference coordinate system with
respect to the world coordinate system
here you know where the object is placed
so all these being parameters are
very very important which talks of a
general three-dimensional viewing
pipeline so I repeat again the step the
first step was the objects are model
into in the object space or object
modeling space second you apply a
transformations which are applied to the
objects to position them somewhere in
the world space and then you specify the
view parameters which are specified to
define the view volume of the world and
the projection plane and the viewport on
the screen so these are the first three
stages of the viewing pipeline then once
the transformations are applied you of
the projection geometry and all
normalizing transforms are are available
the five different steps which we talked
about rotation translations sheer scale
and all that in theory you need to clip
this object with respect to the U volume
now actually sometimes the objects are
often clipped with respect to the
canonical view volume because the
expressions for clipping actually
becomes very simple when we use with
respect to the canonical view volume
they remember the expressions of the six
planes which bind the canonical view
volume both in the case of perspective
and parallel there were six planes and
there were six different simple
expressions you can almost visualize it
X equal to plus 1 X minus 1 X equal to Y
X for a minus y 0 equal to minus 8 means
Z equal to 1 and so on so if the
expressions of those planes are so easy
it helps you to make the clipping
algorithms work very fast almost you can
very good it grows from very nice
expression to obtain the clipping
clipping means what you have a line and
you have a bounded volume within the
through which you are seeing this line
let us say so of course what it means is
if this is your view volume and the line
is somewhere inside you will not be
allowed to see outside the wave column
whatever the line exists remember it's a
virtual world a line or a rod can
actually go through any particular
material so if you have a if you have a
virtual volume you will be allowed to
see only inside of it and that is where
you require to clip the line and come
out with only that section of the line
which the graphics screen it should be
displayed on the graphics screen not the
outside and the expressions for this
clipping becomes very simple when you
operate with the canonical view volume
because the canonical view volume has
very simple expressions for the planes
which bind the canonical view volume but
of course generalized expressions are
available for clipping in which you can
keep actually in any 3d space with any
arbitrary volume that's also possible so
you either clip with respect to the v1
in prescribed by the viewing geometry or
is or you clip with
to canonical view volume okay so after
the clipping is done then the results
are projected onto the projection plane
window so this is from 3d to 2d and it's
finally mapped into a sorry this is
still in 3d it is not into the 3d
viewport so this is something like the
canonical view volume comes into the
picture and you have if it defines a
small 3d viewport through which you are
able to see the entire world interworld
is squeezed into a canonical view volume
and 3d viewport okay in this 3d viewport
you are still in 3d remember before
going to 2d fin a simple case then you
slightly apply the projection geometry
and go from 3d to 2d otherwise what is
done is in a case of complicated
structure like a house or any complex
building or environment where there are
lots of objects and each objects has lot
of surfaces in the front back side and
all that you need to remove certain
hidden objects now this is a special
there are specified are very good
algorithms to do this particular task of
removing hidden objects hidden surface
removal you see so we will see what is
this hidden remove objects to view well
it is not for the timing you can assume
that if you are looking to the camera is
looking towards me let us say in this
specific example you are not able to see
your back even when I am looking at you
I am able to see your frontal face and
maybe this side but may not be the back
side so this back side of yours and my
back side are all objects on the onto
the back of mine or they are all hidden
they are all hidden the notionally you
can visualize that and there is no point
drawing those on the screen because
anyway I will be overlapping this
objects and you will be overlapping
whatever is behind you so you need to
eliminate those or remove those hidden
surface cell from the scene in terms of
drawing it on the screen and because
otherwise even spend a lot of time which
wastes lot of time in drawing those so
hidden surface removal algorithms are
very very very essential very important
to reduce the compression complexity of
drawing objects in fact typically what
is found that less almost in most of the
cases 20 or 30 percent or at the maximum
on average 50 percent of the surface is
what you need to draw for a general
three-dimensional scenario and that's
why and it is made possible with the
help of hidden surface removal
algorithms but all the hidden surfaces
are
so you to have that in the graphics
viewing pipeline as well and so the
hidden syrup dips are then removed and
the then when the hidden objects and
hidden surfaces lines etcetera all are
removed you only are left with the
objects which are now visible to the
camera visible to the user
and you want to show those objects
rendered them so the objects are what
are called scan converted and then
shaded if necessary
okay now that means we are talking of
two more types of sections methods or
algorithms which are dealing with scan
conversion and shading so in along with
hidden surface removal after that we'll
also have to study about scan conversion
and also shading algorithms to find out
how to effectively not only shared
objects but if possible and necessary
create shadows that's very very
important so we listed here but we can
say that this is under the general
category of 3d viewing pipeline which
you need to rendering algorithms are
necessary under the category of general
viewing pipeline as well okay so well
look at again from a flowchart angle
when you have the flowchart of the 3d
viewing pipeline you have the object
model which creates the object space
then under the object space you scale
rotate and translate the objects
generalize transformations you go to the
world space in the world space you
specify the view calculate the surface
normals calculate illumination remove
the back faces we had seen a few of
these terminologies now we will look
into all those algorithms as we go along
in this sequence of lectures and you're
still in the word space after we specify
the view calculate the surface number
calculate the elimination that is
etcetera and then you apply the
normalized transformation and clipping
to create go into the what we call is
the eye space going to add a canonical
view volume 3d viewport from the world
space okay and this again the flowchart
is not complete we are in giving the
general steps of the 3d viewing pipeline
we give it in steps now I give it in a
flow chart again and in the indy and
where you know where the matrices to be
put and after each matrix operation with
space you are talking about so we have
moved from worlds of depth space to
world space word space to eye space and
from eye space is where you apply your
projective transformation or projection
transformation matrix or parallel
projection transformation as the case
may be and there is where you move from
three
Rudy you move from 3d to 2d and that
makes you move from the AI spacing
coordinates to the image space in 2d and
there is where you can apply the hidden
surface removal algorithms in the image
space or you can apply it in also in 3d
space as well and there you provide the
shading the texture all that is
necessary for drawing the objects then
you map into the viewport you map it in
the viewport and draw this screen we
talked about this viewport and the
screen because you may not be it may not
be necessary and you may not want to
draw the entire picture on the entire
screen if it covers intestine you may
not be a today we talked about a dynamic
environment with several user several
tasks in a multitasking multi
programming environment where you will
not have be having graphics programs but
other programs also running in a system
and you may have also multiple graphics
program one of them could be just
rendering a scene with the house or some
building structures and all that you can
have another graphic screen displaying a
movie or a multimedia animation and
stuff like that another could be doing
if is giving a visualization of a
multimedia office a molecular biology or
some other visualization of
computational fluid dynamics as some
other engineering CAD cam applications
let us say so you need of course not
always the case that you complete the
the entire screen should be complete
with one graphic screen you define your
viewport and display the size of the
viewport depending on the demand of your
application you can always scale the
viewport if necessary and put it up to
cover the entire screen and then bring
it back that's alright but you always
need to put it on it before so you map
to viewport and draw the raster on the
screen and that generates the device
coordinates as you see on this screen
and you see in the last step of the
flowchart when you map into the viewport
and draw on the screen the corresponding
device coordinates are generated the
corresponding device coordinates are
generated and they come through starting
from the object space so I go again
object space world space I space image
space and ultimately the device
coordinates where are it in the viewport
you want to draw your graphic objects is
now specified starting from an arbitrary
three dimensional world where the
objects were given objects were here
camera has its own position world
coordinates somewhere it has gone
through all transformations clipping 3
to 2 D idn't surface knew about shading
all the tile rhythms and
you have it in a viewport you have in a
viewport the objects which are drawn
which the camera is supposed to see on
the objects you change the objects the
position of the objects or II change the
camera position well by altering the 3d
view specifications whatever is being
drawn the viewport will also change okay
if you change any of the parameters in
the corresponding leaving pipeline which
we have seen today so this is the
overall 3d viewing pipeline which talks
of a very generalized scenario of
viewing and we have discussed the first
four stages in a very gross framework in
approximate framework then we looked
into different steps of the 3d viewing
pipeline and finally also we came back
and in a very exhaustive flowchart about
3d viewing pipeline now what happens let
us say if if the you don't want to use
let us say in a very simplified scenario
you want to create a very simplified
scenario where you know that your camera
is aligned with respect to the world
coordinate system you have the objects
in front of it and you want to just
display the system okay you are not let
us say but of course if you want if you
are involved in the using standard
graphic software's
standard graphical packages like figs
and OpenGL you have to specify go to
this 3d viewing pipeline to display in
most cases they'll ask you to specify
those parameters but let us say if you
want to write a small programming
ourselves in OpenGL where you will
simplify this entity DVD pipeline and
actually the first few stages of
normalization and projection geometry of
course and then you will say if I can
specify what are my camera parameters in
terms of what the view I am looking at
what is the focal length or some
parameter for the camera in terms of its
lens and screen width and things like
that and what is the view volume I am
looking - can I not transform a point in
3d into the 2d screen yes it's possible
you may not have to go to the entire
viewing pipeline and do that you can use
the 3d viewing pipeline there is
absolutely no problem because it is so
general that it can take care of all
sorts of transformations all sorts of
scenarios which is possible but let us
say you want to create a simple formula
one or two steps let us say where you
know that I have a camera in the world
in a virtual
coordinate system or VRC okay view
reference coordinate system situated
aligned already with the world and I
have the object in front can I pick up
the 3d points on the objects and
transform it to 2d on this screen okay
so we look now in the last few minutes
of this lecture today a camera model
which gives a very simple formula to
transform a 3d point or 2d point this is
something similar to the projection
geometry combined with the
transformations okay so what are this so
I'm just taking a simple example as a
case study where we specify our initial
camera model and by identifying the
following parameters so you know what's
the scene which consists of polygonal
elements each represented by their
vertices so basically you assume for the
time being we are not talked of solid
modeling so far we assume that there are
3d vert is is specified in the world and
it is in front of the camera and the
camera is also and defined by the
following parameters the camera position
may be it could be the origin of the
world coordinate systems but you can
still have CX CY cz as the position of
the camera you have a point that
represents the center of attention of
the camera or a co a center of attention
sometimes also called the focus of
attention fo a in fact it is true with
our eye that we always have a focus the
center of attention to the subject or
object we are looking to and you have a
center of attention or a focus of
attention for a point for this camera
which is specified also by a point in 3d
as AXA why is it the components for the
the coordinates for the point a and the
last important point is the field of
view angle alpha this is this is
something which gives you the canonical
view volume type of a scenario
now you specify that not with planes but
a field of view Akal alpha I am give you
the diagram by meaning what is this see
where is this a what is this alpha which
represents the angle subtended at the
apex of the viewing pyramid
okay the last fourth point I repeat
again is very important the field of
view angle alpha which represents the
angle subtended at the apex of the
viewing pyramid so you need to know si
you need to know a unit you know alpha
and these parameters can I take points
in 3d which are the vertices of polygon
elements and shipped it to 2d just a
simple projection using these parameters
which are there in points 2 3 & 4 see a
and alpha points 2 3 & 4 the parameters
C a and alpha if they are known see na
on our scalar quantities they are 3d
vectors and if C a and alpha alpha is a
scalar quantities and angle in degrees
or radians if these three are not can we
can we come up with a simple projection
model okay so let us look at this
scenario specification of C and a is
given in this figure already you have an
idea what is this alpha it is the angle
subtended at that vertex C angle
subtended at that vertex C of that
pyramidal structure now this is similar
to a canonical view volume for
perspective and that is the most common
and natural model for what is called as
a pinhole camera which used for image
processing tasks as well when a camera
is model of course there are other
parameters linear nonlinear parameters
to model a camera but this is quite
simplified and okay for a computer
graphics point of view where I have C a
and alpha 3 parameters and if you look
into the figure again in addition of
course you will have two more surfaces
one is called a near plane and a far
plane okay or the is also called the
front clipping plane and the back
clipping plane we've already known about
FP + BP front clipping plane and back
clipping plane you know with respect to
those six surfaces where you are
clipping your 3d points and surfaces and
lines so now this pyramid is was
infinite I have made it finite by make
putting an airplane and a far plane on
the front and the back side with respect
to the camera which is looking through
the center of the pyramid you have a
central line which passes from C to a a
being the focus of center of attention
of the camera and there are other
binding for surfaces of the pyramid
which define dizzy volumes so now what I
say is all objects which are inside this
finite pyramid I hope you can visualize
this theorem it is infinite pyramid but
putting the near plane or the front
clipping plane or far plane is the back
tipping plane if we put these two then
you have come up with what we will call
as a finite pyramid okay and so the
specification of near and far bonding
planes must be given and these planes
are considered perpendicular direction
of view vector the direction of V vector
is nothing but the vector going from C
to it okay a is marked here C is here
okay a is here okay see you today is the
direction of projection let us say or it
could be unity to see does it matter is
the direction is more important and
these planes near and far planes are
considered for
integral to the direction of view vector
or direction of projection DOP remember
this and are at a distance N and F from
the camera respectively
so these near planar distance n from C
so they're perpendicular distance along
the central line which passes from C to
a or the direction of view vector or
direction of projection vector along
that line the distance from C to the
near plane is N and from C to the far
plane is okay so n minus F is the
distance along that center of line
within the fine and view pyramid within
within which all of these objects must
lie to be to be viewed you know clipping
will do the operation where any objects
which are outside or in front of the
near plane between the camera and an
airplane will not be viewed anything
which is beyond the far plane also will
not be under the gate okay it came the
moving pyramid the 3d viewing space
again I have just added the coordinate
system UV and W of course you can align
the W with the center of line which is
piercing to the center of the finite
pyramid see 2/8 you can align the W with
that but it could be very general you
have an airplane far plane and the mid
space view volume is UV W which I will
say although it's 3d ultimately when you
say a three-dimensional image space with
UV W what you are talking basically is a
unit cube at UV W ok you need cube or a
unit pyramid and it's very simply switch
off W you will get u n V that is another
way in the case of perspective
projection you switch off W you get the
device coordinates or viewport
coordinates or 2d coordinates U and V
but of course in the case of perspective
projection in which the image space
volume is also a pyramidal structure you
need to have a projection matrix which
so this is the scenario and let us see
how we can generate the projection
matrix for mapping points in 3d on to 2d
this is the Society of the viewing space
I was talking of these distances in an F
from the center C to the near plane and
the centre is at this vertex bearing an
angle alpha and from the center to the
far plane the distance is F so this is
under side view I hope the picture of
the pyramid is getting clear this was a
3d picture of the finite pyramid
obtained from the invite structure we
defined the finite view volume and the
image space is also defined when the
coordinates on the right hand side here
in terms of
UVW each along each axis you have 2
minus 1 to plus 1 and this is a
two-dimensional view where I will cut is
it could be a side view it could be a
front top view through which you can get
this type of triangular structure for
the near pin flower plane and the two
sides binding along the UV axis okay let
us go ahead and derive the waving
transformation matrix in terms of the
camera parameters and as we go ahead you
note down the equations and the matrices
which come along let us look at this
equation in the figure as given here you
find a point P with coordinates UV and W
which is projected onto the projection
plane at a distance D from the co P the
projection plane is lying in the UV
plane and the projection point is u
prime V prime W Prime and it can easily
derive u prime V prime W prime using
symmetry of triangles you have done this
a little bit earlier as well and they
are nothing but D u by minus w DV by
minus W and minus D respectively minus D
can easily be found out because the
minus W coordinate is minus D okay you
can rewrite these three u prime V prime
W Rama is given here equal to do by WT v
by W a - just the multiplying and
dividing by W so basically these three -
2 D is nothing but representing in
homogeneous Cartesian coordinate form
uvw and one is transformed to D u DV TW
and minus W because if you divide the
first three coordinates by the fourth
one you are going to get your 2d
coordinates and which are nothing but
your u prime V Prime and W prime is
anyway you know it is equal to minus D
so once you know I can write this in
form of a matrix express this is the
transformation matrix PD I can easy any
one of these forms you know dragon LED
and then minus one in the third element
of the fourth column or you can write
this as the entity matrix three ones and
the one replaced by the zero in the
fourth row fourth column but the third
element in the fourth column will be
minus 1 by T either of these matrices
could be used is absolutely no problem
because you are working in the
homogeneous Cartesian coordinate
representation
we look back into the equation again you
will find that this is a post
multiplication the coordinates in 3d are
post multiplied with one of these matrix
to get the corresponding u prime V prime
W prime is given by do DVD W and minus W
respectively so if you look back into
the equations please note down this
matrix form and try both these matrices
first multiply a point uvw in 3d space
post multiplied by PD any one of them
will do and you will get this okay so
this projective transformation is fine
but you have to work with the view
volume that is new to transform the
finite truncated viewer view pyramid to
the canonical view volume of size uvw
ranging from minus 1 to plus 1 and we
will do that with one axis of tramadol
let us first analyze the W axis only ok
and let us say if you use a
transformation matrix basically it is
some sort of scaling and translation
which you have to do and if you use such
a transformation matrix where the
unknown elements let us say are a and B
why these two because his a will handle
is scaling along the z axis or w axis a
we handle that and B we randomly
transformation projection also minus 1
you need to provide that basically this
helps you to transform a point 0 0 minus
n multiplied by P to a point 0 0 1 that
is the near point will be transformed to
the front clipping plane in canonical
view volume which is plus 1 and the far
clipping plane 0 0 minus s in 3d will be
transformed to 0 0 minus 1 that is the
backup engine so if you know that P is
able to give these two transformations
from 3d to 2d basically or in fact in 3d
itself then you basically solve for a
and B using the above equation that
means you substitute P in these two and
you get basically expressions like this
these are the two constraints you get
for a and B remember N and F are known
to you and you since you have two
equations with two unknowns a and B you
can solve for a and B straight away
using these two equations and you get
very simple nice forms as a solution for
a and B based on F and n remember F and
n are also camera parameters in some
sense because they are the distance to
the nearly and near front clipping plane
and back clipping plane respectively the
F okay so let us look back into the
market
this is the transformation matrix now
these transformation matrix will handle
the Z transformation from the finite
view parameter to a canonical view
volume what about U and V with respect
to x and y well we have to also go to
any one of these axis and the UV axis
transformations of the pyramid are
understood by taking the side view again
if I take the pyramidal seidel view you
can see these two white lines are of
length if this is 0 to n and then 0 to F
with an ml alpha by 2 because their
entire angle is alpha we are taking half
of this pyramid which is alpha by 2 the
angle at the vertex so these two
vertical lines will be of length n 10
alpha by 2 and if it has to be like that
and what basically happens is now these
the points are these two at the edge of
the pyramids will be having these 3d
coordinates 0 and 10 alpha by 2 and
minus n and this point will also have 0
either you are V we are talking about
and F tan alpha by 2 which is the
vertical along U or V and digit Z part
is minus F so these have to be
transformed to correspondingly the
coordinates 0 1 minus 1 and 1 that is
the desired normalized 3d coordinates
for both this points Sophia and am
implies the matrix P which you have seen
then you get this transformation you get
this transformation from this point into
alpha by 2 minus n2 a transformation
matrix like this and similarly for the F
tan alpha by 2 if please note down these
equations and try to yourself multiplied
by P you get this point now you
basically need if you see here you need
to have a transfer matrix P in such a
manner that this becomes 0 1 minus 1 and
plus 1 and similarly this also becomes 0
1 minus 1 and that can be done by
modifying P this can be done by
modifying P because the P which we
derived with only the z axis had these
two terms f plus there is the third
column it has these 2 terms now if you
introduce in the diagonal element these
two term cot alpha by 2 it will help you
to basically get these two terms
vanished and this will become 0 F minus
F + 0 n minus 1 and the transformation
will take place
basically okay and of course so you can
note down this new form of P using the
matrix and I hope you are able to note
it down the inverse of P I leave it as
an exercise for you to find out a little
bit of matrix manipulation try out the
inverse of P which should be this you
can note it down for the time being and
practice at home four by five Matt is
not too very difficult one this is the
matrix P and this is the inverse of the
matrix it is given as this is the cot
alpha has become tan alpha by two
diagonal terms are very straightforward
and but the other practices also will
change places
so the overall viewing transformation
matrix the overall viewing
transformation matrix is a concatenation
I'm sorry the product of two matrices P
D and P and remember the PD you can take
any one of these two forms D DD minus 1
or 1 1 1 minus D and multiples
multiplied by the P matrix which we have
just seen remember P is this matrix and
the PD hat diagonal elements D and the
first three diagonals
and it had a minus 1 so if you want
apply PD with that P you will the
product of the matrix is what you will
get as this one of the forms of PD gives
you this the other form of the PD is
what you also basically get in this
particular form and please note down
these forms and displaying it for you
this is the form of the unit
transformation matrix and if you look
into the parameters all the parameters
all the parameters are based on the
camera model all the parameters are
based on the camera model if you take D
what is a D D is something like the you
can visualize it to be the focal length
of the lens or a camera parameter where
this is basically the distance of the
projection plane from the origin that is
the D what is the alpha that is the
angle subtended at the vertex of the
pyramid okay which defines your view
volume how much you are looking both
sideways and top-to-bottom and what on
the far end how much you are looking you
can't look beyond a certain point and
something very close also you cannot
visualize properly which is very
realistic so you have a front clipping
plane distance F and a far or a back
clipping plane distance and F and n you
have the D where you have the projection
plane you have the alpha so all these
parameters are there in this
equation as you see this is this is one
form of the projection transformation
matrix P f which is VD multiplied by P
if you take the other form of the matrix
P D which is has it is almost singular
in I am sorry identity matrix type 1 1 1
and minus 1 by T you will typically have
the nature of the expression of P D in
this form so can you note down this row
matrix is and I hope if you can start
from scratch and derive these matrices
expressions yourself I hope you are able
to copy down the expression yourself as
given in the slide please copy down
these expressions and try to derive
these from the figures which are given
have probably jumped one or two stages
here step slate basically so that I
leave with something in the exercise for
you to write down matrices from the
expressions directly because you have
gone through 2d transformations gone to
3d transformations gone through
projective transformations gone through
viewing pipeline and now give any
equations you should be able to write in
matrix or matrix back the equations do
the manipulation and again the most
important thing is to visualize what
these transformations mean because these
transformations are nothing but
basically implementing a graphical
scenario from you for the user for the
viewer and you must know why these
transformations are very important you
will have algorithms in graphics all
right but the mathematics part is also
very very important which you cannot and
along with that the geometrical
interpretation which you must have in
mind so we wind up this lecture today on
3d viewing pipeline with this set of we
are in fact discussed about general
transformations of viewing pipeline
which consists of four broad stages
applying normalized transformation
clipping to canonical view volume use
the projection matrix and then actually
scale and translate into viewport these
were the four broad stages we discussed
the second normalized in the last
lecture the normalizing transformation
which had five different steps correct
rotation translation sheer scale and
finally a rotation sorry a scale again
at the end and those were the five
different steps for the normalizing
transformation which helped you to bring
the word view reference coordinate and
the entire objects align them with
respect to the world coordinate system
for proper viewing then of course you
need to scale it down to the canonical
view
volume or three we put clip you have the
clipping done which we'll discuss later
on and then of course do the perspective
projection from D to D that was the
general viewing pipeline and then of
course we went to an example case study
where what about for a simplified
situation if let us say only camera
model parameters are given is it
possible to generate a view yes it's
possible you can bypass if not necessary
the entire generalized 3d viewing
pipeline and if the camera parameters
are given what are the camera parameters
given the D which is the distance of the
projection plane from the center of the
camera the angle subtended by the the at
the vertex alpha which defines the
pyramidal view volume and N and F the
distance of the front and black keeping
plane if these are given these
parameters are given you can define also
come up with your projection
transformation matrix come up with the
projection transformation matrix which
will take you from 3d to 2d we have seen
the last two expressions in two
different forms almost similar
expressions which take you from 3d to 2d
and without in fact going to the
extensive pipeline but you must also
know the extended viewing pipeline which
helps you to understand the generalized
scenario which all graphic standards and
packages mostly follow so in your using
standard graphics packages and
software's and graphical standards let
us say the fix is an example which we
have been following in this course with
extensively we have not come up with
specific examples but will devote one on
one hour at least for certain examples
of figs and OpenGL together we and we
only need to follow such graphic
standards you have to follow a 3d
viewing pipeline you cannot but of
course you want to light a small
software error interface which is built
on simple camera parameters the camera
models yes what you can do basically is
use such a transformation matrix and get
the transformation done from T to 2d
which includes the economic concept of
even canonical view volumes and it helps
you to even click if necessary
ok so that is the last slide and we wind
up this lecture end of the lectures on
3d viewing projective transforms and
paving pipeline thank you very much