understand explicit form of curves

[zak100]

Hi,
I want to understand the concept of explicit form of curves. I want to know whether the x2+y2=k is an explicit form or not.
I also want to understand the following two characteristics of explicit curves which i got from a slide on web:
ii) the form is not rotationally invariant and,??

In my view rotationally invariant is 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.??

Please provide some explanation for it. I have searched and found that slope would be infinite, but what’s the problem with this?? Does it mean that we can't describe curves having a vertical tangent? Is it related to conic section?

Hope this would clear my question. Kindly guide me.
Zulfi.

 

[stapel]

I want to understand the concept of explicit form of curves. I want to know whether the x2+y2=k is an explicit form or not.

What is your book's definition of the "explicit form" of a curve?

 

[zak100]

Hi,
Thanks for your response. I am not using any book. I am just exploring web. I have found that its a single valued function of other variable.
www.cs.uml.edu/~hmasterm/Charts/session_2.ppt‎

I got following :

the form is not rotationally invariant and,
you cannot describe curves with a vertical tangent

​

from:
http://www-users.aston.ac.uk/~cornford/cs2150/pdf/curves_lec_8up.pdf
I cant understand the terms rotationally invariant & curves with vertical tangent in the context of explicit representation.

Also I got following:

An explicit representation of curve enables us to directly compute y at any value of x. 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.


​

from:
http://escience.anu.edu.au/lecture/cg/Spline/printCG.en.html

This is what i want to understand. Somebody plz guide me.

Zulfi.

zak100Newcomer

Posts: 4Joined: Mon Dec 09, 2013 11:40 am

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[zak100]

Hi,
Thanks for your reply. Actually I am not using any book. I am studying through web material. According to this: explicit functions of one variable is a single valued function interms of other variables. Plz correct me if i am wrong.
I got following :

the form is not rotationally invariant and,
you cannot describe curves with a vertical tangent

​

from:
http://www-users.aston.ac.uk/~cornford/cs2150/pdf/curves_lec_8up.pdf
I cant understand the terms rotationally invariant & curves with vertical tangent in the context of explicit representation.

Also I got following:

An explicit representation of curve enables us to directly compute y at any value of x. 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.


​

from:
http://escience.anu.edu.au/lecture/cg/Spline/printCG.en.html

This is what i want to understand. Somebody plz guide me.

Zulfi.