Explicit functions for curve representation

Discussion in 'Homework Help' started by zulfi100, Dec 7, 2013.

Jun 7, 2012
320
0
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?

Last edited by a moderator: Dec 7, 2013
2. MikeML AAC Fanatic!

Oct 2, 2009
5,450
1,066

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

3. studiot AAC Fanatic!

Nov 9, 2007
5,005
513
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?

4. Papabravo Expert

Feb 24, 2006
10,163
1,797
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)

Jun 7, 2012
320
0
Hi,
I have got some more questions. Sorry for taking your time. Thanks for elaborating it:
And for square root example:
This means that y = √x is not an explicit function?? (Plz guide me: yes or no?)
i.e y=f(x) (Plz state: yes or no?)
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.

Last edited: Dec 8, 2013

Jun 7, 2012
320
0
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?
I also found one advantage from web:
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.

Last edited: Dec 8, 2013
7. WBahn Moderator

Mar 31, 2012
17,777
4,805
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.

8. WBahn Moderator

Mar 31, 2012
17,777
4,805
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.

Jun 7, 2012
320
0
Hi,
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.

10. WBahn Moderator

Mar 31, 2012
17,777
4,805
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.

Jun 7, 2012
320
0
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.

12. studiot AAC Fanatic!

Nov 9, 2007
5,005
513
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?

Jun 7, 2012
320
0
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.

14. studiot AAC Fanatic!

Nov 9, 2007
5,005
513
You are going too fast.

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

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?

Jun 7, 2012
320
0
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

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

Zulfi.

16. studiot AAC Fanatic!

Nov 9, 2007
5,005
513
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.

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.

Jun 7, 2012
320
0
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:

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

Zulfi.

18. studiot AAC Fanatic!

Nov 9, 2007
5,005
513
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?

Last edited: Dec 10, 2013

Jun 7, 2012
320
0
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
My friend, if you tell me the answer plz tell me the source also.

Zulfi.

20. studiot AAC Fanatic!

Nov 9, 2007
5,005
513
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?