Hi,
I cant understand the concept of explicit representation of curve.I know it means represenation of one variable in terms of another variable using a single valued function.
But i cant understand slide 6 of the lecture at:

www.cs.uml.edu/~hmasterm/Charts/session_2.ppt‎

I cant understand why its saying:

Neither variable is a single-valued function of the other

Can somebody guide me plz?

 

Bad link.

Have you ever seen "linear regression" or y=mx + b ?

 

I cant understand why its saying:

Neither variable is a single-valued function of the other

Can somebody guide me plz?

Well I will accept your statement that you understand what an explicit representation is.

You are quite corect to observe that a function is single valued.
That is an important part of the definition of a function, that for any x there is one and only one value of y.

Unfortunately some engineers regard this as mere mathematics formalism.

But there are expressions such as y = √x, where there are two (or sometimes more) possible values of y that satisfy that equation eg both +2 and -2 satisfy y = 4

In this case we formally choose one and one only of these two (normally the positive root) to define our function.

We do not ignore the other root, just observe that it equals the negative of our chosen root.

You will find loose talk of 'multivalued functions' in some engineering texts and other associated disciplines, but should never find it in a maths book.
Nor should you find it in a tertiary educational establishment, so it is disappointing to see it in your link.

Does this help?

 

Explicit representations in 2 dimensions REQUIRE that y is a single valued function of x in the domain of the function AND x is a single valued function of y in the domain of the function.

The problem with the explicit representation of a circle is that the Cartesian coordinates cannot satisfy this restriction. Therefore we need another method of specifying a circle if we want to render it in a graphics engine.

For Example: Given a constant r, and a parameter θ, whose domain is
0° <= θ < 360°, the locus of points that lie on that circle are given by:
(r cos θ, r sin θ)

 

Hi,
I have got some more questions. Sorry for taking your time. Thanks for elaborating it:

That is an important part of the definition of a function, that for any x there is one and only one value of y.

And for square root example:

But there are expressions such as y = √x, where there are two (or sometimes more) possible values of y that satisfy that equation eg both +2 and -2 satisfy y = 4

This means that y = √x is not an explicit function?? (Plz guide me: yes or no?)

Explicit representations in 2 dimensions REQUIRE that y is a single valued function of x in the domain of the function

i.e y=f(x) (Plz state: yes or no?)

AND x is a single valued function of y in the domain of the function.

i.e x= f(y) (Plz state: yes or no?)

Let's suppose k= f(x,y) &
f(x,y)=x2+y2
f(x,y) is a curve but is it an explicit representation?? A represenatation in terms of theta is a parametric representation??
Plz guide me.

I have also found from web 3 charateristics of Explicit representations (written in bold font)
i) it is impossible to get multiple y values for a given x value, (I am fine with this )
ii) the form is not rotationally invariant and,??

rotationally invariant like symbol ‘o’ & ‘+’. Circle is also rotationally invariant whereas ‘6’ is not rotationally invariant b/c it may become 9. Am I right??


iii) you cannot describe curves with a vertical tangent.??

plz provide some explantion for it. I have searched and found that slope would be infinite but what’s the problem with this?? Does it mean that we cant describe curves having a vertical tangent? How can we describe it using parametric form?

Zulfi.

 

Hi,
I found something related to (iii) at:
http://escience.anu.edu.au/lecture/cg/Spline/printCG.en.html

which says:

If we are asked to represent a straight line in the Cartesian coordinate by an explicit form, we will probably give the following equation:
y = kx + b,

provided that this line is not vertical to the x-axis. Otherwise, we have to represent vertical lines as x = c, where c is some constant value. Such problems inherent in explicit form is easily dealt with when solving a problem by hand. However, it is a nuisance when programming geometrical problems for a computer. Another drawback with respect to the use of explicit form is numerical stability. Referring to the above straight line, we note that the computation of y is numerically unstable if k goes to infinity, indicating the line is nearly vertical. In general, if a curve has nearly vertical tangents, we may expect overflow or rounding error problems when computing the function values. For these reasons, the use of explicit form in computer aided geometric design is very limited. The explicit form is satisfactory when the function is single-valued and the curve has no vertical tangents. However, this precludes many curves of practical importance such as circles, ellipses and the other conic sections.

If possible somebody plz explain it in simple words?
I also found one advantage from web:

Simple to subdivide (consider equal intervals of x)

Is it similar to slide 5 of following link:http://www.cs.sunysb.edu/~qin/courses/graphics/parametric-curves.pdf
Somebody plz guide me.

Zulfi.

 

Hi,
I cant understand the concept of explicit representation of curve.I know it means represenation of one variable in terms of another variable using a single valued function.
But i cant understand slide 6 of the lecture at:

www.cs.uml.edu/~hmasterm/Charts/session_2.ppt‎

I cant understand why its saying:

Neither variable is a single-valued function of the other

Can somebody guide me plz?

If a function is single-values, that means that, within the domain of the function, if I give you the value of the independent variable you can tell me the value of the dependent variable, right?

So think of a circle of radius R centered at the origin.

If I tell that x = R/2, can you give me the single corresponding value of y? If not, the y is not a single valued function of x, is it?

If I tell that y = R/2, can you give me the single corresponding value of x? If not, the x is not a single valued function of y, is it?

So neither variable is a single-valued function of the other.

 

Hi,
I found something related to (iii) at:
http://escience.anu.edu.au/lecture/cg/Spline/printCG.en.html

which says:



If possible somebody plz explain it in simple words?

In a computer program, it gets very complicated to use explicit functions unless we know that the functions are, in fact, functions -- namely single valued with no vertical or nearly vertical segments. If we don't know that, then an alternative is to use parametric representations because they tend to be more general and more stable when representing arbitrary curves.

 

Hi,
Thanks for your response. What about (iii) in post#5?
Does it mean conic section?
http://en.wikipedia.org/wiki/Conic_section
What about "Simple to subdivide (consider equal intervals of x)" as asked in my earlier post? Is it similar to the link provided?
Kindly guide me.

Zulfi.

 

Hi,
Thanks for your response. What about (iii) in post#5?
Does it mean conic section?
http://en.wikipedia.org/wiki/Conic_section
What about "Simple to subdivide (consider equal intervals of x)" as asked in my earlier post? Is it similar to the link provided?
Kindly guide me.

Zulfi.

Parametric may or may not have problems with vertical slopes. It really depends on whether the value of the relation remains finite. The circle is a perfect example. While the slope become vertical at both of the x-axis crossing points, the parametic equations remain usable there.

p = <x,y> = <Rcos(θ), Rsin(θ)>

dp/dθ = <-Rsin(θ), Rcos(θ)>

Now, if you try to take the slope at y=0 (i.e., y/x at θ=±∏), then bad things happen. But as long as you can work with the parametic representation, things are good.

But if x or y itself goes infinite, such as with the tangent function, then you are going to have problems.

Even if that's the case, however, you can usually work closer to the singularity before things go south due to overflows in the computations.

 

Hi,
Thanks for your time. Right now i am just exploring explicit representation. So if you get time plz revise your answer for explicit representation.
Thanks again.

Zulfi.

 

Any statement of the form y = "some expression that does not contain y" is an explicit statement.

y = x, y = 3x, y = 3x+2z, y = 5rst are all explicit statements

y = x/y, y = xyz, y = 3x+2y, (x-y) = o are implicit statements

This statement may be a function (and therefore an explicit function)

In some cases it is possible to convert an implicit statement into an explicit one by rearrangement.

eg in the above

y = 3x + 2y implicit statement

y-2y = 3x

-y = 3x

y = -3x explicit statement

This is also a function since for any one value of x there is only one value of y.

So y = 3x + 2y is an implicit function and

y = -3x is an explicit function.

does this help?

 

Hi,
I think its very much useful. Nobody has confronted my views regarding rotationally invariant, curves with vertical tangent and "simple to sub divide". So i feel these are correct.

Zulfi.

 

You are going too fast.

The next thing you need to know is what a parametric representation is

f(x,y) is a curve but is it an explicit representation?? A represenatation in terms of theta is a parametric representation??

A parametric representation is created when we put all the expression variables in terms of a single auxiliary varaible.

We need a separate equation for each original variable.

So for instance for the expression

\({y^2} = 4ax\) : a is a constant.

if we introduce a single parameter, t then use

\(x = a{t^2}:y = 2at\)

we have both x and y in terms of t.

To obtain a point on the original curve we use the same value for the parameter, t, in both equations.

Thus
at t=1, y= 2a and x = a
at t=2, y= 4a and x = 4a
at t=3, y= 6a and x = 9a
and so on.

It is often easy to develop equations for the tangent and other useful things in terms of this parameter.

Does this help?

 

Hi,
Thanks for your continuous guiding. I am not in a position to start parametric form. It would be too fast. However i have checked implicit and i found with due regards that your definition

y = 3x + 2y implicit statement

y-2y = 3x

-y = 3x

y = -3x explicit statement

This is also a function since for any one value of x there is only one value of y.

So y = 3x + 2y is an implicit function and

y = -3x is an explicit function.

does this help?

is not correct.
In case of implicit form the RHS is set to 0. I have checked this on several web sites.

Zulfi.

 

is not correct.
In case of implicit form the RHS is set to 0. I have checked this on several web sites.

Actually it is strictly correct.

We divide all possible statements, about our variables, into those which are explicit by defining an explicit statement and those which do not follow this definition, as I did in post#12.
Those which do not follow this definition we call or define as implicit.

Any statement of the form y = "some expression that does not contain y" is an explicit statement.

All other statements are defined as implicit.

So my statement y = 3x + 2y is indeed an implicit statement since it does not conform to the definition of explicit.

Yes I chose a particularly simple one for example purposes to develop the idea. We would normally convert it to an explicit statement to work with. We could even convert it to one containing zero on the RHS if we wished.

However we can still work with it as an implicit statement and follow the rules for handling implicit statements if we want to.

 

Hi,
Thanks for your reply. I highly appreciate your efforts. I depend upon you and other people like you for my knowledge. I found following stuff while searching web:

Slide2 where it has shown this: (out of quote)
http://graphics.cs.cmu.edu/nsp/course/15-462/Spring04/slides/10-curves.pdf
Implliiciitt Reprresenttattiion
• Curve in 2D: f(x,y) = 0
– Line: ax + by + c = 0
– Circle: x2 + y2 – r2 = 0
• Surface in 3d: f(x,y,z) = 0
– Plane: ax + by + cz + d = 0
– Sphere: x2 + y2 + z2 – r2 = 0

https://www.cse.msu.edu/~cse472/lectures/14.pdf
And Slide 3 (out of quote)
Y=x2 (explicit)
And x2-y=0 implicit
See definition on slide 5:
Implicit representation –
represents a curve as a function
of all variables equal zero:

 f(x, y) = 0
 ax +by + c = 0 Line
 x2 + y2 – r2 = 0 Circle
http://escience.anu.edu.au/lecture/cg/Spline/printCG.en.html

An implicit equation of the form f(x,y) = 0 can avoid the difficulties of multiple values and vertical tangents inherent in the explicit form. For example, a unit circle with its centre at the origin is given by x2+y2-1 = 0.

I dont want to hurt you because you are my source of knowledge.

Zulfi.

 

OK so you have been looking around the net,

How does it describe my statement y = 3x + 2y?

Edit: As a matter of interest if you rewrite it in the form 3x + y = 0 is it explicit or implicit?

Have you done any calculus?

 

Hi,
I dont know. I have surely done calculus. I think you need to solve this.

I found following from wiki:
http://en.wikipedia.org/wiki/Implicit_function

In mathematics, an implicit equation is a relation of the form R(x1,..., xn) = 0, where R is a function of several variables (often a polynomial). The set of the values of the variables that satisfy this relation is a curve if n = 2 and a surface if n=3. The terms implicit curve and implicit surface are usual to denote curves and surfaces defined in this way. The implicit equations are the basis of algebraic geometry, whose basic subjects of study are the simultaneous solutions of several implicit equations whose left-hand sides are polynomials. These sets of simultaneous solutions are called affine algebraic sets.
For example, the implicit equation of the unit circle is x^2 +y^2-1 = 0.

My friend, if you tell me the answer plz tell me the source also.

Zulfi.

 

You get into all sorts of difficulties with this definition. Not least the conflict about R being a function or not with multivalues. We have already discussed that.

Since you have done some calculus you may have heard of 'implicit differentiation'.
Introducing derivatives, integrals, moduli and many more into the definition complicates matters further.

I am only prepared to discuss further if you take note of what I have said about starting with a definition of an explicit statement and regarding everything that does not satisfy this as implicit, not the other way round.

If you want discuss horses, you define what a horse is and regard everthing else as not-horse. You do not define a bear and then get into trouble when you meet a tiger.

Note also, very very carefully,

I have consistently said explicit statement.

You wiki definition requires an equation.

Not all statements are equations. What about an inequality or a locus?