Hello and welcome to this module number 8

on Manufacturing Systems Technology. A quick recap of the last module, we actually discussed

in the last module how to perform a complex transformation process on a 3D object with

respect to an orthogonal coordinate system using a concatenation of two translations

and one rotation matrix. The other important aspect that I would like to cover today is

that you know geometries can be plotted using simple lines and arcs and curves and regular

shapes, but they cannot be really estimated when the topology is that you are born mapping

is very complex in nature. So, in this particular module I would like

to look at how to represent such curves and what are the flexibilities which are available

when you try to force fit a polynomial function to a certain region of the curve, so that

on a very local basis I could have a proper fit of a particular region. The essence of

all this is that in a computer aided design, it is always important sometimes to be able

to estimate the exact topology even if it is very complex. So, instead of having one

whole regular geometric feature to be able to represent the whole topology, it is a very

good idea to split up the whole geometry into small synthetic curves which are again joined

end to end in a manner, and so that the whole topology can be mapped in a very accurate

manner, ok. . So, we now discuss the representation schemes

of the curves that we have been talking about and obviously in CAD/CAM systems, usually

thousands of such curves or lines are stored and manipulated, so that you can have various

objects of all different complexities topologically being mapped as in this particular case we

are studying. So, when we talk about the mathematical representation

of a curve, a very simple representation can be just in terms of how the coordinates are

related to each other. For example, let us say if you have a straight line. We are talking

about a straight line y equal to x plus 1. So, obviously there is a relationship governed

between x coordinate and y coordinate of all points which will lie on that particular straight

line and that governing relationship would be y equal to x plus 1. So, it is a very very

simple way of looking at just by looking at a relationship between different coordinates.

. This representation in mathematics is known

as the non-parametric representation of describing a curve in a similar ground. We can actually

talk a little differently. We can say that instead of having a xy description; let us

say that there is a parameter, an extra parameter t which is somehow function of which actually

becomes the x coordinate. So, it may be simply a relationship like x equal to t, but supposing

the curve is a non-linear curve, when we are talking about let us say a parabola or some

other non-linear curve, where it can be a square of t or it can be a cube of t. There

would be slight differences in case of straight line. It may be appearing to be one and the

same thing, but for different set of curves when it comes from a linear to a non-linear

mode, it may appear to be different if you are just changing these parameters order from

1 to 2. So, what you assume is that let us say in

the same straight line case where we talked about a non-parametric equation y equal to

x plus 1, we assume x to have a parameter equal to t. Obviously, y would also have a

parameter t plus 1 and so, if you vary this t from some limits, let us say limit 0 to

5 or 0 to 1 or 0 to 3. Simultaneously, y would also be limited within that domain. So, what

I am trying to say is that instead of looking at the whole global picture of simple x and

y relationship, parametrically we are trying to associate an extra parameter which we are

trying to vary locally, so that we can have an idea of the local variation of the straight

line at some level. Obviously, in case of a straight line, the geometry is regular,

it is linear. Both those variations would be one and the same. So, the non-parametric

equations can be further divided into two different cases. One is a clear cut case just

called the explicit non-parametric form of the equation; the other is a hidden form which

is also known as implicit. I will just explain what this implicit and explicit representation

is in the non-parametric domain are really. . So, let us look at an explicit case first

of the non-parametric representation. So, let us say for a two-dimensional general and

you know curve that we are talking about there may be a representation v1 in a manner that

you know the y coordinate is varying as a function of x, and you are representing this

by looking at v1 as xfx as the fundamental way that you know the coordinate frames are

varying of a certain curve, ok. So, therefore, there is always a relationship between y and

x in the manner given here y equal to fx. So, it is a very clear cut relationship which

is available. Simultaneously for a three-dimensional curve

also, you may assume a third dimension z and say that in one case, it is related with respect

to a function f of x. So, y varies as fx and in another coordinate z, it varies another

function g of x. Again there is a clear cut relationship between the x and the y and x

and the z as given in this particular example. So, this is actually called an explicit or

a clear definition in which you can define a particular non-parametric equation for a

curve, ok. . So, in the implicit case for example, in this

particular case as you can see, there are let us say you know curve defined by all sort

of variables between x 1 to x n and let us say you know only on a three-dimensional case,

we have two functions f xyz is equal to 0 and g xyz equal to 0. So, there is not enough

clarity that how x varies with respect to y or how x varies with respect to z, or as

a matter of fact how y varies with respect to z. The clarity is not available, although

all the information which is necessary for assuming a sort of functional relationship

is available within these two equations 1 and 2. Such a state of description of a curve

would be known as an implicit representation of the non-parametric form of the curve.

So, what we learnt so far is how do you non-parametrically define a curve with respect to the x and y

coordinates. You have a clear cut case, where the y coordinate or the z coordinate as in

a three-dimensional curve is completely a function of the x coordinate and another case

of information is around, but it is in a hidden manner you have two different functions, where

there is a relationship between x, y and z being indicated and you will have to infer

from that how x varies with respect to y or how x varies with respect to z, or even as

a matter of fact y and z what are the variations with respect to each others. So, that is the

implicit way of describing the non-parametric representation of the curves.

. So, having said that let us look at the corresponding

parametric representation. I think I had already mentioned that if I involve another extra

parameter t to define all the points, for example as you can see here the x coordinate

is varying as a function of t, right. So, x is varying as a function of t some function

capital X of t. Similarly, y is varying again as a function of t and so is z varying as

a function of t and then, we say that we associate a range for this t value. Let us say the t

varies between some t minimum and t maximum. So, what we can do is that instead of moving

over the whole domain of xyz, you can actually now limit the value of t on the parameter

in the manner, so that on a very local basis you can describe what is going on as a relationship

between xyz between that parametric domain varying between t minimum and t maximum. I

will just come to an example problem where we show how this parametric equation can be

developed. . Let us say we are looking at a straight line

here for example, in the xy plane and it is given by two points v1 x1 y1 and v2 x2 y2

as the two ends of a straight line. Further we are also having a situation where we are

describing a third point here vxy which is a variable point between v1 and v2. If we

want to develop the non-parametric equations and the parametric equations of this straight

line, how do we actually approach the problem? So, a familiar non-parametric representation

of the line can be let us say x2 minus x1 times of y minus y1 equals y2 minus y1 times

of x minus x1. This is completely true and valid because if we look at the slope of the

line from this variable point to one of the end points, it would be defined as y minus

y1 by x minus x1. So, the slope does not change if you go from the local domain to the global

domain of variation where you are having both the end points. So, this can be represented

as y2 minus y1 by x2 minus x1. So, it is completely justified in writing in this manner now if

I were to. So, this is a simple case of parametric form of representation, where I can say that

y is varying with respect to x in a manner, so that y2 minus y1 by x2 minus x1 times of

x minus x1 plus y1. So, there is a variation, there is a completely valid relationship between

y and x which is governing the whole equation or your straight line and this is a non-parametric

representation. Can I now represent this in a parametric manner is the question.

So, let us say we want to just slightly change the way to represent this whole thing by adding

a parameter here. So, parametric representation, so we add a parameter t and define t in a

manner, so that t is equal to vv1 by v1 v2. In other words, t is described by the fraction

of x length that is x minus x1 as a part of the overall length in the x direction in this

2 minus x1 of the particular straight line, and there is a equivalence between this fraction

and the way that y minus y1 would be defined with respect to y2 minus y1. So, that is the

fraction of y from y1 with respect to the overall length y2 minus y1. So, obviously

these two fractions are going to be similar if we assume a straight line like relationship.

So, the xy would vary linearly between the points x1 y1 and x2 y2, so that always the

fraction of the x length with respect to the overall x length should be equal to the fraction

of the y length with respect to the overall y length if this point varies between x1 y1

and x2 y2, ok. . So, if we just look at

the way that we can reassemble this equation, we can have the first equation just out of

this relationship t which is the x fraction equal to

the y fraction. The first relationship x can come out to be 1 minus tx1 plus tx2 and the

second relationship y can come out to be 1 minus ty1 plus ty2 from these equations. So,

there is a flexibility that we have now because now if we vary let us say put t equal to 0

here, x becomes equal to x1 and y becomes equal to y1, right which means that at the

fraction of x or y length equal to 0, the points xy are supposed to be at x1 y1. Similarly, if t equal to 1, x becomes

equal to x2 and y becomes equal to y2 from these relationships which means that at fraction

of x or y length equal to 1, the vxy is supposed to be at v2 x2 y2, ok.

So, the points really move between the initial point and the final point corresponding to t

equal to 0 and t equal to 1. What is important for me to say is that just by merely varying

this parameter between a local domain, let us say varying between 0.5 to 0.7, I can really

zoom down the parametric equation to a point which is corresponding to 50 percent of the

fraction to 70 percent of the fraction. So, the parameterization of the equation enables

me in a way to look at a geometric object locally, provided there is a global description

given by a non-parametric form of equation of

the particular curve. So, this is the power of such non-parametric representation.

Now, I am going to go to the next level and tell you that how to represent synthetic curves in a parametric

and non-parametric manner, and there you will have a very good feel that a very

complex topology constituted of many small synthetic curves with some relationships of

interconnects between each other, how they can be traced on a profile topologically,

so that they can match the exact profile into question and that will be in the next module.

Thank you. 1