Today's lecture is titled an overview of geometric
modeling. Earlier we have seen in our introductory Meme it

lectures that geometric modeling forms basis
for the integration of many of the cad cam Meme it
activities. So in the first three lectures,
in this particular course we looked at geometric Meme it
modeling applications in design in manufacturing
and that it also forms a basis for integration Meme it
of these activities. So from today onwards
in next few lectures, we look at basics of Meme it
geometric modeling like what you mean by geometric
modeling, what are the various representation Meme it
for representing geometric entities like curves,
surfaces and solids. And how these representations Meme it

are used particularly in design and manufacturing
applications is the subject of this particular Meme it
today's lecture. Meme it
Now geometric modeling can be defined as computer
compatible and mathematical representation Meme it
of the geometry. If you really look at this
particular definition, there are two aspects Meme it
one is computer compatibility this is a must,
second is a mathematical representation of Meme it
geometry. If these two can be fulfilled in
any definition I can call it is a geometric Meme it

modeling. Now one can always give a mathematical
definition which is not computer compatible Meme it
or I can have a representation which is used
purely for visual representation of the geometric Meme it
object. So if I look at these representations
which are like which fulfills only one of Meme it
the requirements then that is not what we
are looking for. We are looking for something Meme it
which can fulfill both. For example if I look
at a mathematical definition to geometry, Meme it
we study in our schools and colleges, a complete
course on solid geometry or coordinate geometry Meme it

which is basically concerned with mathematical
representation of geometric entities like Meme it
line, circle, conics also surface entities
like cylindrical surface, conical surface Meme it
or a spherical surface.
So we study about how these are mathematically Meme it
represented in our coordinate geometric course
or three dimensional solid geometric course. Meme it
So these representations are mathematical
but they are not really computer friendly Meme it
that means what I intend to do with this geometry
cannot be fulfilled by these definitions directly. Meme it

So I am looking for a better definition where
I am able to use the mathematical representation Meme it
also for doing some calculations which are
related to cad cam or also to use this representation Meme it
for visually displaying on let's say a computer
screen. And if I go by other definition that Meme it
is I can always give or I can always visually
represent a three dimensional picture on computer Meme it
screen without going into mathematical definition
also, it's also possible. So, even that is Meme it
not enough, so we are looking for something
which is a combination of this thing. Meme it

So if I just put this picture and say what
does this particular picture represent? Is Meme it
it a two dimensional figure or a three dimensional
figure, what would be your answer? It's somebody, Meme it
some people saying its three dimensional and
some people are saying its two dimensional Meme it
then there are few people who say like it
can be both. Now purely looking at a picture Meme it
you cannot really make out whether it is two
dimensional or three dimensional. For example Meme it
if I look at this particular figure, this
figure can be constructed using a set of lines Meme it

which are drawn in a plane which can give
me an effect of three dimension. For example Meme it
if you look at this particular thing you have
4 plus 3 that is 7 plus 2, 9 lines have been Meme it
drawn to basically give you a feeling of a
three dimensional object. So these 9 lines Meme it
can be drawn in a plane and say that here
is an object which looks like a three dimensional Meme it
object or alternatively I may have really
had a three dimensional representation for Meme it
this that means I have actually constructed
a geometric entity like a cuboid or a box Meme it

and then I have removed the hidden lines and
then are able to show this particular as three Meme it
dimensional representation with hidden lines
removed. Meme it
Now what basically implies here is that what
you see is not what is correct. How it is Meme it
represented internally? Let's say as a file
or as an internal representation is what we Meme it
are concerned. So when we say a mathematical
representation there can be purely visual Meme it
representation without mathematical also or
like you may have seen that many of the pictures Meme it

or sceneries or backgrounds you can always
create three dimensional effect and there Meme it
is no mathematical representations for that.
It's purely a visual kind of effect. We are Meme it
not interested in that either, what we are
looking is something which can give me a geometric Meme it
definition, at the same time the geometric
definition should be computer friendly enough Meme it
to do things which I need to do as part of
cad cam applications. So that's what is basically Meme it
the objective of the studying geometric modeling
as part of this lecture. Geometric modeling Meme it

is a vast subject and in terms of studying
this particular subject, what we basically Meme it
do is to look at how are various geometrical
entities in terms of curves, surfaces and Meme it
solids are represented. Meme it
The representational aspects of curves and
surfaces have some similarities like sometimes Meme it
geometrical representation or geometric modeling
of curves and surfaces together can be classified Meme it
as surface modeling or modeling of you can
say free formed curves and surface entities Meme it

etc whereas representation for solids is very
different from that of curves and surfaces Meme it
in cad cam applications. Suppose if I am giving
a definition for a curve, I can always extend Meme it
it to surface one more one dimension higher
and I can also extend it to solids but it Meme it
is usually not done for certain basic reasons.
So we will look at those aspects, why representation Meme it
for curves and surfaces is slightly different
or why there is a different representation Meme it
which is used, which is not used for solids.
So we will be studying about this. First we Meme it

will take up study of curves and surfaces
and then we will go for representational aspects Meme it
of solid objects. Meme it
Now in terms of this study, you will come
across for example if I am looking at curves Meme it
and surfaces particularly, we may be looking
at some of the geometric entities with which Meme it
we are already aware of like starting from
our school etc we study about many geometric Meme it
entities like a line or circle or conic or
if it is surface like it can be plane or a Meme it

cylinder or sphere or a cone, so we are already
familiar with some aspects of these geometries Meme it
which are usually called as a standard geometries
or more so what we call as a known forms. Meme it
But cad cam also deals or when you have to
use a geometric modeling for product design Meme it
and manufacturing applications, one may have
to deal with those surface features or those Meme it
surface entities which are not one of the
standard forms. Usually they are called as Meme it
free formed curves and surfaces. Meme it

So what we would be looking as part of geometric
modeling is start with known forms just quickly Meme it
revise those forms and then go to what we
call as free formed curves and surfaces. And Meme it
then once we have complete representation
for all types of entities then we study about Meme it
how these representations can be used to automate
certain design and manufacturing applications Meme it
so that will be the subject of this thing.
And as I said curves and surfaces will be Meme it
treated almost as one entity and solids will
be treated as a separate entity in terms of Meme it

this particular study. Now study of curves
surfaces and solids is not necessarily like Meme it
domain of a cad cam courses. This is also
a subject of courses and computer graphics Meme it
where one would be interested how to visually
represent an object and also use a mathematical Meme it
definition but not necessarily for design
and manufacturing applications. Meme it
So the subject of geometric modeling has evolved
along with computer graphics, so we will also Meme it
see lot of graphic related terminology as
well as concepts which are coming when we Meme it

study about geometric modeling in this particular
course. Now when it comes to representing Meme it
curves and surfaces, we know that like we
are aware of different types of representations Meme it
which one can use. Meme it
For example we have what is called as a implicit
representation which we study in our geometry Meme it
courses. We also study about explicit representation
which is also a subject of courses on geometry Meme it
and also parametric representation. We are
familiar with parametric presentations for Meme it

some of the geometric entities. Now one has
to basically choose one of these three or Meme it
let's say combination of these representations
for cad cam applications. Now for example Meme it
if I take an example of this, so what you
see here is an entity like a circle which Meme it
is represented in all three representations. Meme it
So if it is an implicit representation I can
have an equation like x square plus y square Meme it
minus r square is equal to zero where a curve
is represented as a function of like f of Meme it

xy is equal to zero is what is the representation
or I can have an explicit representation where Meme it
you try to model a curve as y as y equal to
a function of an x. So the same circle can Meme it
also be represented as an explicit representation
which is shown here. Then you are also familiar Meme it
with parametric representation of circle in
terms of let's say r cos theta or r cos t Meme it
or r sin t where t is the parameter which
is varying for the variable for a curve. Meme it
Now among these three representation, now
each one may have an advantage or disadvantage Meme it

and may be one more suitable for cad cam application
than the other. So we will look into that Meme it
aspect as we go along. Similarly you have
another example here like you have an equation Meme it
of a plane which is given here. so it's a
surface entity either i can represent it in Meme it
a implicit form as ax plus by plus cz plus
d equal to zero where you are using the function Meme it
like f of xyz equal to zero or I can also
say z as function of x and y which is shown Meme it
here as explicit representation or I can also
have a parametric representation where every Meme it

point on a plane like every coordinate point
on a plane xyz can be represented as a function Meme it
of two variables which is u and w. Meme it
It is true that like whenever we represent
a curve or a surface in a parametric fashion Meme it
like whenever we have let's say x y or z which
is basically has one parameter, it can be Meme it
classified as a curve. Whenever you have two
parameters like u and w as it is shown here, Meme it
so it becomes a surface and we can extend
it to higher dimension. I can have xyz which Meme it

are functions of three parameters which can
be used to represent a solid object. So we Meme it
would be looking at these three representations
and to look at which is more appropriate for Meme it
cad cam application. Meme it
Now it is almost universally believed that
among the three representations, parametric Meme it
is the one which is most widely used for certain
strong reasons which we will look into is Meme it
why parametric representation has an edge
over implicit and explicit representation Meme it

for cad cam applications. But when we are
studying about parametric representation and Meme it
its advantages, we should be aware that implicit
and explicit representations to have certain Meme it
advantages. So it is not necessary to say
that parametric representation is the only Meme it
representation. I may use a combination of
these for a given application also but more Meme it
so, a parametric representation for most of
the applications. Meme it
So let's look at why parametric representation,
what are the advantages over let's say other Meme it

two representations which are shown for curves
and surfaces in our next part of the lecture. Meme it
The first advantage of a parametric representation
is that it completely separate the roles of Meme it
dependent and independent variables like when
you, for example when I write implicit or Meme it
explicit representation which involves x y
or z, you have one which is like dependent Meme it
on the other, whereas when I am giving let's
an parametric representation like xyz which Meme it
are functions you are basically looking at
them independently. So how x varies as parameter Meme it

of t is of not much concern to you like how
y varies with let's say a parameter which Meme it
is there. So the three coordinates can be
independently written and they can be changed Meme it
if I want to change let's say a curve or a
surface. So this is one of the major advantages Meme it
you can say in terms of parametric. So, instead
of looking at equation as a whole, you are Meme it
looking at the independent aspects of xyz,
how they vary as a part of this particular Meme it
thing. Meme it

And another of course a very interesting aspect
of this kind of independent variables is the Meme it
equation which can be given for let's say
geometric entity can be extended to higher Meme it
dimensions very easily like if I say like
given let's say x equal to r cos t, y equal Meme it
r sin t, z equal to h what do they represent?
What geometric entity? Somebody may say it's Meme it
a cylinder, no, it's a circle or no, no it's
a solid cylinder, I would say it can be anything. Meme it
For example if you look at the values which
are given at the bottom for r t and h, r is Meme it

fixed which is 5 units, t is equal to pi and
z is equal to 20. If I substitute these in Meme it
the equation, I get a unique value for xyz
so it's a point, it's a point, it's not a Meme it
curve, it's not a surface or it's not a solid.
So you have three parameters but how you vary, Meme it
which are the parameters which actually vary
and which are the ones which are actually Meme it
fixed will decide whether these equations,
the three equations for xyz which are given Meme it
here will represent whether it's a curve,
surface or let's say a solid entity. Meme it

Now I can always vary one of the parameters
keeping the other two parameters constant. Meme it
Now if I do that then I am going for one dimension
higher than let's say a single point, a specific Meme it
point which is mentioned. So you will be representing
let's say a curve. Now if I go to that, I Meme it
have the same equations but if you look at
the parameters r is fixed which is till 5 Meme it
units, z is fixed which is 20 units whereas
t is a variable, t varies from let's say minus Meme it
pi to pi. So you have three equations which
are functions of a single variable. We know Meme it

what is the entity which it represents. It's
a circle, so because t is varying from minus Meme it
pi to pi, so I can call it as a circle. Meme it
Now I can extend these two higher dimensions
too and you get different representations Meme it
by fixing a different entities. For example
in these equations instead of fixing let's Meme it
say a variable t, instead of varying let's
say variable t, I may have varied let's say Meme it
for example either r or h. If I do that what
will I get? I will get a straight line instead Meme it

of a circle. So parametric directions like
what basically it's says that you have one Meme it
parametric direction which is more like a
circle, other two parametric direction which Meme it
is like straight lines either by varying r
alone or h alone, I will get a straight line Meme it
which is basically a straight line entity.
So you can like it's basically a three parametric Meme it
directions which are a combination of two
lines and one circle which is it. Meme it
I can go to one more dimension higher, what
you see here is r equal to 5 which is fixed Meme it

whereas two parametric variables that is t
and z are varying within certain range, t Meme it
varies from minus pi to pi and z varies from
0 to 20. So what does this represent? It's Meme it
a cylindrical surface, so it's not a solid
cylinder. It's a cylindrical surface where Meme it
the radius of this cylinder is fixed as 5
units, instead of these two I would have chosen Meme it
some other combination also like what happens
if I vary let's say other two variable, instead Meme it
of t and z suppose if I vary let's say t and
r, I keep z as a fixed. So you get a circular Meme it

disk where the z is constant, so it's again
like a planar entity, it's a surface entity Meme it
with this or I may have gone for varying r
and z keeping the t as a constant. It is a Meme it
rectangle, so that's again a plane entity.
So you are basically choosing a combination Meme it
of these two. Meme it
We have two parametric directions which are
lines, one is the circle. So if I take two Meme it
lines, I will get a rectangle one line and
a circle either I will get a cylindrical surface Meme it

or let's say circular disc. So one can have
a combination of any of these two entities Meme it
and in our each case you get a surface as
a result of this. So it's very clear that Meme it
whenever we have a parametric representation
if xyz are functions of a single variable Meme it
as we have seen in our earlier representation.
If I vary any one of them, I get always a Meme it
curve entity like either a straight line or
a circle has we have seen earlier. Meme it
Now if I vary any two of those, I always get
a surface entity which may be a rectangle Meme it

or circular disc or a cylindrical surface.
I can go to one more higher dimension where Meme it
you are varying all three parameters, r is
also is varied between 0 and 5. Then I have Meme it
let's say t which is varied between minus
pi and pi and z which is varied between 0 Meme it
and 20. So what does this represent? It's
solid cylinder. Meme it
Now instead of let's say r which is varied
from 0 to 5, let's say I vary the r value Meme it
which is let's say from 3 to 5. So you still
get a solid object but this time it's not Meme it

a solid cylinder but it's a hollow cylinder.
So since these parametric equations are functions Meme it
of three variables where all the three variables
are like, all the three parametric entities Meme it
are being varied so you get a solid object
which is that. The same concept can be extended Meme it
to higher dimensions too. Now one can also
say that for example I have a parametric representation Meme it
which is a function of 4 variables or more
than 4 variables too that may not have really Meme it
a meaningful solid object as per as let's
say a geometric modeling is concerned but Meme it

in some cases it is used. Meme it
For example I would like to represent let's
say a solid cylinder as it is shown here. Meme it
And this solid is moving in space and when
it is moving in space, I also have a function Meme it
which is like a time, so I can bring one more
variable which is r t z and as well as let's Meme it
say another variable which is time. So I can
have let's say a solid or I can represent Meme it
a motion which are functions of 4 variables.
So one can extend this to higher dimensions Meme it

too but in most of the cases, this kind of
representation are used for study of curves Meme it
and surfaces that's what we are actually looking
at. Meme it
Now in explaining this I have taken a very
simple example of let's say a parametric equation. Meme it
The same thing can be extended to other entities
too. It is not necessarily for let's say a Meme it
circle, cylindrical surface or a solid cylinder
but it's true with any parametric entity where Meme it
if I go for parametric equations which are
functions of three parametric variables it's Meme it

always a solid. It may or may not really give
me a meaningful solid but it is a solid. Meme it
Another major advantage of parametric representation
is that it offers more degrees of freedom Meme it
for controlling shape of curves and surfaces
like I have given two, you can say curves Meme it
here which are y as a function of x and xy
as function of parameter. So what you see Meme it
is two different types of representation,
one which you call as a explicit and second Meme it
I think sorry this is not implicit, what is
shown here is it's a parametric form. It is Meme it

wrongly written as implicit, so this is a
parametric form. Meme it
Now if I look from explicit and parametric
form both these things basically represent Meme it
a polynomial which are cubic in nature. A
cubic equation in explicit form and again Meme it
a cubic equation which is in parametric form
where x and y which is there. Now what is Meme it
the basic advantage in these two representations
is like here you have 4 variables. By giving Meme it
different values to p q r and s, I get different
cubic curves. Same thing is true by giving Meme it

different values to abcd efg and h, I get
different parametric cubic curves but a designer Meme it
has more variables to play with in the case
of parametric than in the case of an explicit Meme it
form. Here you have only a 4, whereas here
you have 8 variables. So usually somebody Meme it
who is in the business or who is concerned
with designing a curve for a specific application Meme it
or designing a surface for a specific application
would be interested to have as many degrees Meme it
of freedom as possible or as many variables
as possible so that one can play around. Meme it

And secondly you know that when you are actually
playing with abcd, you are only concerned Meme it
with x coordinate and when you are doing with
efgh, you are only concerned with y coordinate. Meme it
So I can also have their roles which are independent
in this particular case. So same degree of Meme it
polynomial which is cubic in nature because
we know that cubic is one of the most commonly Meme it
used representation also like one can go one
degree below that is quadratic or I can go Meme it
one level higher than cubic that is quartic
or one level higher which is fifth which we Meme it

call as a quintic but cubic is the most commonly
used representation for representing many Meme it
of the curves and surfaces. The reason is
very clear. Meme it
Usually, whenever we have a curve or a surface
you look at the continuity requirement. that Meme it
means the curve should be like, you should
be able to differentiate it at least two times Meme it
because whenever we have first differentiation
of x and y, we are looking at more like how Meme it
the slope varies. And when I am going for
a second differentiation, I am looking at Meme it

curvatures. So curvatures also, curvature
information is also very very important for Meme it
many of the cad cam applications, so cubic
is one. If I go for one order higher that Meme it
is quartic where you have polynomial where
you have terms like u to the power of 4 or Meme it
x to the power of 4, it really doesn't serve
much purpose because your mathematical calculations Meme it
become more and more complex. At the same
time it also gives you a curvature continuity Meme it
even one level higher but that is not really
needed for most of the applications. So in Meme it

most of the cases, we represent cubic form. Meme it
Another reason for using a cubic form for
polynomial more commonly is that up to cubic, Meme it
we can also get closed formed solutions. I
can get analytical solutions. If I want to Meme it
know what are the roots of let's say a cubic
equation, it should be possible in many cases Meme it
to know what are the roots in a purely analytical
form or by giving a formulae whereas once Meme it
I go for a higher order than cubic then you
have to use numerical methods. And we know Meme it

that whenever the numerical methods are used,
you have associated mathematical problems Meme it
and also there are other issues. So cubic
has advantage that it serves the, we can say Meme it
this is the minimum representation which serves
most of the applications and also I have analytical Meme it
solutions which are possible. So this is most
widely used representation for this. So I Meme it
have just chosen cubic as an example to represent
this. And of course you can always, other Meme it
representations likes quadratic and linear
are subset of this by making one or the other Meme it

variable zero, I get the other representations. Meme it
And coming back to the parametric representation,
there is one more advantage which is very Meme it
evident is that transformations are easier
to apply. In a cad cam work or let's say when Meme it
you are dealing with computer graphics, you
are not only dealing with geometric entities Meme it
but also their relative positions. How one
entity is placed with respect to the other Meme it
or how let's say an entity is defined with
reference to some Cartesian or other coordinates Meme it

systems or an object is initially constructed
at some convenient position and then it is Meme it
moved within the Cartesian space to some other
position. So when we move it, its parametric Meme it
and other whatever may be the mathematical
representation it changes. So this is usually Meme it
called as a transformation.
So transformations are usually, the most common Meme it
transformation which we use in a cad cam applications
are translation where you translate an object Meme it
from one place to another place or rotation,
it can be about, rotation about an axis which Meme it

may be parallel to xyz axis or it may an arbitrary
axis which is inclined to xyz axis. You also Meme it
deal with the, in fact translation and rotation
are rigid transformations. They do not change Meme it
the shape of an object but I can also have
transformations where I am going to scale Meme it
an object. There are also transformations
called shear where you are again changing Meme it
the shape of an object. So these are non-rigid
transformations. So these are very commonly Meme it
used in many of the cad cam and graphics applications
and we should also look at that particular Meme it

mathematical representation for geometry which
gives convenience in terms of applying transformations. Meme it
Now this I am basically demonstrating giving
a simple example like let's take a circle Meme it
and this is a circle with radius of 7 units.
Now what is shown here is basically like this Meme it
is a parametric representation which is shown
here and what is shown here is basically an Meme it
implicit representation for the same circle.
Suppose if the circle has the center 0 0, Meme it
I can write down it's parametric representation
as x as 7 cos t and y as 7 sin t where t varies Meme it

from, for the complete 2 pi as the parameter
range. The same equation in an implicit form Meme it
can be written as x square plus y square minus
49 is equal to 0. Meme it
Now this circle is moved such that center
now is at 4 and 3 instead of 0 and 0. Now Meme it
when you do a transformation that means the
circle has been moved to another place where Meme it
the shape is not changed, size is not changed,
shape and size has not changed only thing Meme it
is now it is at a different position with
reference to a Cartesian coordinates system Meme it

which I have chosen. What happens to the equation?
A parametric equation which was this thing, Meme it
so you just simply add the translational variables
that it is moving 4 units in the x direction, Meme it
3 units in the y direction directly to the
parametric representation. So if I look at Meme it
this particular aspect, this right hand side
basically denotes the shape and size aspect Meme it
of the entity which you are representing whereas
4 and 3 or translational variable or transformation Meme it
variables which are used. Meme it

Now let's look at that same for when I am
having an implicit representation. The same Meme it
equation which is same circle with radius
of 7 units with at a center has a different Meme it
equation now. Now if I look at these two equations,
it's very difficult to make out, first thing Meme it
is it's the same entity because the transformation
variables have mixed up with the actual representation Meme it
of shape and size. So it's very difficult
to separate it out, in fact circle is very Meme it
simple entity, so you can still make out by
let's say or one can still dig out what are Meme it

the transformation variables and what the
actual shape and size. But when it comes to Meme it
more complex geometric entities, this becomes
more and more complex. Meme it
Since I am able to do that this kind of transformation
variables are I am able to separate it out Meme it
from shape and size aspects. So this parametric
representation has a great advantage in this Meme it
case. Now what is shown here is only a translation.
Same thing is true for rotation also like Meme it
in fact rotation is you will see slightly
a different formulae but basically what one Meme it

does is that you are actually representing
a point as a vector in a parametric representation. Meme it
Suppose if it is a entity in a plane, you
have a vector has two components which is Meme it
x and y. Suppose if it is in a space then
you are looking at xyz. So you are basically Meme it
transforming a vector in a parametric representation.
So, the vector has three components x component, Meme it
y component and z component and whereas you
do not have any such representation here implicitly. Meme it
And we know that the transformations are also
I think you are going to study later. Transformations Meme it

can be represented in a matrix form like when
an object is translated from one place to Meme it
another, what is the actual translation can
be represented as a matrix form. Similarly Meme it
a rotation can be represented in a matrix
form and you are representing a point in vector Meme it
form. So, all the transformations can be represented
as some kind of a multiplication of vectors Meme it
and matrices. And we know that when it comes
to particularly programming geometric entities Meme it
or when I have to write a program where transformations
and geometric manipulation of geometric entities Meme it

are involved, vectors and matrices are very
convenient to handle in programming rather Meme it
than like solving with many of the algebraic
entities which are sometimes difficult to Meme it
do. So you can say that parametric representation
gives you a programmer friendly environment Meme it
because you are able to represent the geometric
entity as a vector and transformations as Meme it
in a matrix form. Meme it
The other advantage which you have particularly
with parametric representation is that you Meme it

are able to represent the objects that is
curves and surfaces which are inherently bounded. Meme it
Now whenever we study about let's say a line
or give a representation, we really do not Meme it
look at a line, there is a starting point
and then there is an end point for the line Meme it
because in most of the cad cam applications
one has to deal with let say a line which Meme it
has a fixed starting point and fixed end point.
A line which extends infinitely in both the Meme it
directions has not much meaning, you cannot
have an object where there is a straight line Meme it

which extends infinitely in both the directions.
So I want to put bounds on let's say any entity Meme it
which I choose. Meme it
So since whenever you have a parametric representation,
you also try to represent the bounds which Meme it
is the parameter which is being varied and
what is the range of variation is also specified. Meme it
So you are primarily representing the bound
entities like if I take this particular example, Meme it
for example I have a circle which is r cos
t and r sin t and if I take let's say the Meme it

range as minus pi to pi, I get a complete
circle. Now the same representation can be Meme it
used to represent a circular arc, you don't
have to have a different representation like Meme it
imagine representing a circular arc in implicit
or explicit form. It's very difficult to do Meme it
it because you do not have a representation
or you do not have a provision for representing Meme it
the bound entities whereas by simply changing
the parameter range to a different let's say Meme it
minus pi to 0, now equation has not changed
but I am able to represent let's say a circular Meme it

arc instead of a complete circle and this
is true with many of the curves and situations, Meme it
curves and surface situations where you would
be doing it. Meme it
Now somebody you can say it's also possible
to put bounds in implicit and explicit representations Meme it
by putting either range in x or y but it's
not always true like firstly when I am having Meme it
a range in terms of x and y, I may have a
multi valued functions, so I may not be able Meme it
to represent this uniquely that's also one
of the problem. I think I will refer to that Meme it

sometime later in our lectures. So why we
have a difficulty in representing let's say Meme it
segments in implicit and explicit representation,
whereas it is much convenient to do in terms Meme it
of. Now this kind of representation also helps
you when I am trying to find the intersection Meme it
of curves for example. Meme it
Let's say if I am trying to find out let's
say whether two arcs intersect or not, you Meme it
are actually not trying to find out whether
the two corresponding circles intersect or Meme it

not, you are interested in only the arcs segments.
So I can always make use of this parameter Meme it
range to know because corresponding circles
may be intersecting but the arcs may not be Meme it
intersecting. So such a situation if I want
to distinguish then probably I may have to, Meme it
I have an advantage in using a parametric
representation here. Meme it
One more advantage which you see here which
parameter representation has in terms of calculation Meme it
of slopes are some of the geometric properties.
If I represent let's say a curve in an implicit Meme it

or explicit form and you are trying to find
out what is the slope of a curve at different Meme it
places or suppose if I take a circle and I
am trying to find out what is the slope at Meme it
different points. I may have a situation where
the slope becomes infinity or I will have Meme it
a points where the slope becomes infinity.
Now suppose if I am writing a program, I am Meme it
writing a computer program to calculate slope
at various places, the moment you land up Meme it
at a point where the slope is infinity your
program fails because the number infinity Meme it

is something which cannot be handled by a
programming languages that is something which Meme it
cannot be easily defined. Meme it
So in order to evaluate a slope like if I
use a formula like dy by dx which is a very Meme it
most common form, I have a problem. So I cannot
have a check like even to implement a check Meme it
whether the slope is becoming infinity or
not is also difficult because you cannot say Meme it
that if dy by dx is equal to infinity kind
of logical statements are not possible. If Meme it

I am using a parameter representation, you
are actually trying to take two components Meme it
separately that is whatever is the dy by dx
which you are trying to represent, you are Meme it
splitting as dy by du divided by dx by du.
Whenever the slope becomes infinity, dx by Meme it
du become zero. Zero is a number which can
be handled easily in a programming languages. Meme it
So I can always put a check, if dx by du become
0 or equal to 0 then you have something to Meme it
do that means this is the case of infinite
slope and this case has to be handled separately. Meme it

So the advantage that zero has an advantage
over infinity in terms of handling in programming Meme it
languages that is possible with parameter
representation that is one of the advantage. Meme it
It's not necessarily restricted to slopes,
we may have a many situations where you may Meme it
end up with values like infinity in many cases
whereas those can be very easily handled when Meme it
it comes to a parameter representation. Meme it
Then of course one of the, a very clear advantages,
a major advantage as far as cad cam applications Meme it

is concerned is in terms of a discretizing
an entity. I have again taken an example of Meme it
a circle. In most of these cases I have taken
a circle because this is one entity with which Meme it
we are already familiar and most of the advantages
can be demonstrated using this single example. Meme it
Here is a circle which is shown and you have
an implicit representation for this circle. Meme it
The radius of this circle is given as 8 units,
center of the circle is 0 0 which is shown Meme it
here. Now you have both implicit representation
and explicit representation. Now this circular Meme it

entity has to be approximated with let's say
straight line. This we do quite often in cad Meme it
cam applications. Meme it
For example, I have a curve which may be a
circle or which may be any free form curve Meme it
and in order to display this particular curve
or in order to approximate this particular Meme it
curve, I take number of points on the curve
and join them by straight line and say that Meme it
you have a piecewise linear approximation
per a curved entity. Now whenever for example, Meme it

I have let's say an algorithm for which works
only for polygons not for curved entities. Meme it
I can always take a circle and convert into
a polygon with number of edges and still apply Meme it
the same algorithm. Meme it
So there are many situations where I can basically
go and we need to do this kind of discretization Meme it
of a curved entity in terms of straight line.
Same thing is done for surfaces also. I may Meme it
have a free form surface and this free form
surface has curved geometry. So instead of Meme it

dealing with a curved geometry, I would like
to approximate it as let's say facets which Meme it
are basically plane or a polygonal faces.
So you try to approximate a surface entity Meme it
with a piecewise polygonal approximation.
Now whenever we do that, there are problems Meme it
with implicit and explicit representations
which is clearly shown here as a simple example. Meme it
Now one thing is suppose if I am trying to
take number of, I want to display this particular Meme it
circle. So in order to display this particular
circle, I use a piecewise linear approximation Meme it

by calculating, computing number of points
on this circle and joining them by straight Meme it
lines. If I use a representation like this
how do I compute the, how do I go about let's Meme it
say computing points on a circle, maybe I
will vary one of the variables. Let's say Meme it
I vary the x, I know the range of x for example
in this case that is it goes from minus 8 Meme it
to plus 8, so I can take this particular range
and try to calculate y. So I will get number Meme it
of points and join them by straight lines
to do that, but I have a problem firstly when Meme it

I give a value of x in some places I have
two values for y. So which one to choose? Meme it
That means I have to keep tracking in both
the directions or I may have a cubic curve Meme it
where you have an implicit and explicit equation
which is cubic in nature. Meme it
So if I give x then I have three roots and
I have to evaluate all the three roots. Some Meme it
may have a meaningful roots like that means
I may have a real roots or I may have imaginary Meme it
roots too. So I have to keep track of like
how the curve is moving and if there is a Meme it

multivalued functions, how to track this particular
curve is also a big issue whereas in a parametric Meme it
representation, whenever you have a curve
when I move from let's say one point on the Meme it
curve which I call as a starting point to
end point. One thing which is continuously Meme it
varying is a parameter, so just by varying
one parameter I am able to get as many points Meme it
as possible, you should do. So if I want to
draw let's say do an approximation of this Meme it
circle as a piecewise line, so what I will
do is I will vary the t. Here I know what Meme it

is the range of t, t has a range from let's
say 0 to 2 pi or minus pi to pi depending Meme it
on how you represent. And I take small intervals
of t and compute x and y values between 0 Meme it
and 2 pi and I get number of points and I
can join them to get let's say a linear approximation Meme it
to this particular circle whereas you have
a difficulty when I go for implicit and explicit. Meme it
Moreover you also have some kind of a uniformity
when I am using a parametric representation Meme it
like even if I am able to track let's say
both the directions, for a given value of Meme it

x if I have two values of y, if I take equal
intervals of x values I have highly unequal Meme it
intervals for the y values. That means a small
change in x value here results in a large Meme it
difference in y value whereas here if I take
a x value large difference in x value somewhere Meme it
towards the center results in only a small
difference in y value. So you have a non-uniform Meme it
kind of variation for x and y, if I vary one
variable and try to calculate the other variable. Meme it
When it comes to parameter representation,
you have an advantage. It's not true for all Meme it

the entities. For circle it's okay but for
others it is not necessary that by varying Meme it
a t or by varying a variable I will always
get equal size line segments but it would Meme it
be much more uniform than implicit or explicit
representation in many cases. So by varying Meme it
a t as let's say certain value taking let's
say if either a interval of let's say 5 degrees, Meme it
10 degrees or 1 degree depending on my requirement,
I am able to get the straight line approximation Meme it
which is more uniform and which has a unique
value. Meme it

So computing becomes much easier like for
example if I have to write a program to display Meme it
a line as a series of circles, naturally I
would prefer this rather than using that as Meme it
a mathematical equation for writing a program.
Now, approximation of a curved entity in terms Meme it
of lines and polygons is not only for the
display purpose. It displays one aspect like Meme it
as I said, circle can be approximated as lines
in fact surface is also can be approximated Meme it
as polygons. This is one of the extensively
used, you can say tool in computer graphics. Meme it

Whenever you see let's say an object which
is represented let's say a good seen a computer Meme it
graphic seen which you see, many a times the
object looks like a curve but actually internally Meme it
it is basically approximated as a series of
triangles or series of polygons because in Meme it
most of the computer graphics algorithms whenever
I do operations like rendering etc, I have Meme it
an advantage in dealing with triangles and
polygons than with curved entities. Meme it
So one of the most common you can say aspects
in computer graphics for display purpose is Meme it

approximate for let's say a curved surface
with a series of triangles or series of polygons Meme it
that's it. Many a times this approximation
is so smooth that you can't even notice when Meme it
you see an object on the computer screen like
for human eye, it almost looks like a curved Meme it
surface but internally it is approximated.
Now this kind of approximation is not necessarily Meme it
for purely visual purpose or displaying or
for computer graphics applications, you also Meme it
need this kind of approximation for applications
like manufacturing. For example if I have Meme it

to generate let's say NC tool path, for example
I have a curved geometry and I have to take Meme it
my tool along a curved geometry let's say
for machining purpose. Meme it
So what you do is in a typical CNC machine,
machining programming that means a part programming Meme it
either you use a linear interpolation or let's
say a circular interpolation like a circular Meme it
interpolation basically uses codes like G02
or G03 whereas a linear uses G0 1. Now, any Meme it
entity which is not line or a circle is again
approximated as piecewise linear approximation. Meme it

So, if I have a free formed curve, I take
number of points on the curved entity and Meme it
then join them by series of straight line
tool paths to get let's say curved path, so Meme it
that's a very common. Meme it
So approximating a curve or a surface as let's
say piecewise linear or polygonal approximation Meme it
is not necessarily for display. It is also
true for an NC tool path calculation. So here Meme it
also we see that a parametric representation
has an advantage. So what I basically try Meme it

to do particularly in this lecture is to give
you enough inputs to convince that parametric Meme it
representation is the way to go for cad cam
applications. But at the same time, we should Meme it
not really like undermine the other representation.
There are situations where implicit or explicit Meme it
representations have an advantage. For example
I can give you taking the same example of Meme it
a circle. Meme it
Suppose if I have a point, I want to know
whether this point is inside the circle or Meme it

outside the circle. How do I do that or I
have a straight line and then there is a point, Meme it
I want to know whether the straight line is
on which side of the straight line whether Meme it
it's on the left or right. So in those situations,
you have other representations which may be Meme it
useful. For example if I want to know whether
a point is inside or outside a circle, I can Meme it
use an explicit representation very conveniently.
Substitute the coordinates of the point in Meme it
the equation of a circle and check whether
the value which you are getting is 0, negative Meme it

or positive. So, depending on that you can
say the point is either on the curve or outside Meme it
the curve or inside the curve, whereas if
I want to use the parametric representation Meme it
for that, I may not be able to use it that
conveniently as I do it. Meme it
So, summary is that parametric representation
has major advantages but a combination may Meme it
be useful in many situations. So I may try
and opt for a combination of these representations Meme it
if the other representation also has an advantage.
So, we basically try to look at the representation Meme it

and their advantages. Now, what will do is
carry forward this like we will take up start Meme it
with representation for curves, how some of
the curves are represented parametrically Meme it
and then we move on to surfaces. Once we have
finished with a curves and surface representation Meme it
both for the known forms as well as free form,
then will take up some applications where Meme it
these curve and surface representations are
used to automate or to take certain decisions Meme it
in a cad cam environment. So that would be
like sequence of lectures which we would following Meme it

starting from today. So if you have any questions
please raise them. So any questions or is Meme it
this clear? Then I will, yeah somebody. Yeah,
any questions? Then I will stop it here and Meme it
we will take up with a parametric representation
of curves in our next lecture. Meme it