Question about parameterizing curve of intersection.

[ozone]

I couldn't find any resources in my book or online dedicated to this subject. I honestly don't even know where to begin for this problem.

Homework Statement

Let $f(x,y) = 4 / (1+ x^2 + y^2)$ and let S be the surface given by the graph of f(x,y)

b) Let C2 denote the curve in the xy-plane given by $r(t)= t, 3/2 − t^2$ and let C denote the curve on the surface S which has C2 as its shadow in the xy-plane. Find the parametric equations r = r(t) for C

Homework Equations

The Attempt at a Solution

 

[algebrat]

how about the "graph" over the curve,

$$(r(t),f(r(t))=(x(t),y(t),z(x(t),y(t)))$$

Then it is a curve, on the surface, and it's shadow is r(t), correct?

 

[ozone]

Hrmm.. I'm not too sure honestly. I think this all has to do with arc-length/curvature. If that is correct then I think I will go study more about that and see if I can't figure it out

 

[algebrat]

I would not discourage you from exploring the concepts, and so develop your understanding of the subject as a whole. However, I think you'll find, while my hint is somewhat abstract, it is more or less correct, and that curvature and arc-length do not apply here. But again, please do investigate and compare the concepts!

Also, my use of the word graph is not a bad definition for you to understand, Stewart uses it in his textbook on calculus; here is the definition of graph (there are other definitions) on wikipeida:

http://en.wikipedia.org/wiki/Graph_of_a_function

In other words, what is the graph associated with the function (x,y) --> z=f(x,y)

 

[ozone]

I don't deal well with this sort of abstraction.

In my mind what you are saying is we can come up with a new function which is simply our old function r(t) plus a new variable which is the sum of the variables of our original function.

Correct me if I am wrong.

But it would appear to me that we need our original function to come up with the parameters, since S is the measure of the surface of $f(x,y)=4/(1+x2+y2)$.

Oh and one last thing. A "shadow" is simply a projection correct?

 

[HallsofIvy]

There is no abstraction here! You are given x and y in terms of t and told how to calculate z in terms of x and y. So what is z in terms of t? It is just basic algebra.