# Thread: parameterisation of a curve

1. ## parameterisation of a curve

This is from some online notes. let x(s) be a curve in R^3 parameterised by s. Then f(x(s) represents the values of f along the curve.

My question is this. Is the graph of f in R^4 ? I though it would be as f is a function of 3 variables but in the notes its looks like it defines a surface in R^3

2. ## Re: parameterisation of a curve

Originally Posted by Duke
This is from some online notes. let x(s) be a curve in R^3 parameterised by s. Then f(x(s) represents the values of f along the curve. My question is this. Is the graph of f in R^4 ? I though it would be as f is a function of 3 variables but in the notes its looks like it defines a surface in R^3
We have

$\mathbb{R}\mathop \to \limits^{x } \mathbb{R}^3\mathop \to \limits^{f } \mathbb{R}^m\Rightarrow f\circ x:\mathbb{R}\to \mathbb{R}^m$

so, $f(x(s))$ represents the vectorial equation of a curve in $\mathbb{R}^m$ .

what is m?

4. ## Re: parameterisation of a curve

Originally Posted by Duke
what is m?
You didn't define the final set of $f$ , so any $m$ positive integer will do .

5. ## Re: parameterisation of a curve

if m=1, how would you represent f(x(s)) graphically. It next talks about x(s) lying on a flat surface of f and tangent vectors just to give you some context.

6. ## Re: parameterisation of a curve

Originally Posted by Duke
if m=1, how would you represent f(x(s)) graphically.
In the same way that for $m=2$ , $f(x(s))$ represents a movement in the plane, interpret it for $m=1$ as a movement of a particle in a straight line .

It next talks about x(s) lying on a flat surface of f and tangent vectors just to give you some context.
Could you transcribe the exact formulation of the question?

7. ## Re: parameterisation of a curve

Its the 2 and 3 pages of this : https://vlebb.leeds.ac.uk/bbcswebdav...65/2.3-2.7.pdf. I don't understand what the function f is. Thanks.