Parametric equation of a surface-eliminating the parameters

[kingwinner]

Homework Statement

The parametric equation of a surface is given by:
x = s + t, y = (s²/2) + s f(t) + t, u = s + f(t) where f is some given function (e.g. f(t)=t/2).
I would like to eliminate s and t from these equations and describe the surface in terms of x, y and u only.

Homework Equations

N/A

The Attempt at a Solution

The first equation implies that s=x-t
Put it into the third equation, I get
u = x - t + f(t)
=> u - x = f(t) - t
Now if I can solve for t in terms of u and x, then I can put it into the second equation and I'm done.
But how can I solve for t in the equation u - x = f(t) - t ?
I think this may be related in some way to the idea of "inverse function", but I don't know how to handle it in this case...

Can someone please help me out?
Thank you! :)

 

[tiny-tim]

x = s + t, y = (s²/2) + s f(t) + t, u = s + f(t) where f is some given function (e.g. f(t)=t/2).
…
=> u - x = f(t) - t
Now if I can solve for t in terms of u and x, then I can put it into the second equation and I'm done.
But how can I solve for t in the equation u - x = f(t) - t ?
I think this may be related in some way to the idea of "inverse function", but I don't know how to handle it in this case...

Can someone please help me out?
Thank you! :)

Hi kingwinner!

If f(t) = t/2, it's easy!

If f(t) = t², you need to solve a quadratic equation.

If f(t) is more complicated, there may be no non-computer solution.

 

[kingwinner]

Hi tiny-tim,

If we assume that the general function f is invertible, can we eliminate s and t from these equations and describe the surface in terms of x, y and u only (perhaps with f and f^-1 in the expression) ?

 

[tiny-tim]

If we assume that the general function f is invertible …

All functions are invertible (at least locally) …

no, there's no general solution, you have to work it out for each one.