 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