> From: John Randall <[EMAIL PROTECTED]>
>
> Raul Miller wrote:
> > In other words, I am trying to find the inverse of a moderately
> > complex set of functions describing curves.
>
> My understanding of the example is this. You have a function f and
> its graph G={(x,f(x))}. You rotate this through an angle t to get a
> graph G'. Now (where possible) for each u in R, you look at the
> vertical line through (u,0) and see where it intersects G'. Choose
> the maximum of these values, say v. Then v=g(u) gives a new function.
>
> If f is a polynomial, then solving the intersection problem for a
> given u also involves polynomials. In general this will be hard.
>
> Are you looking for the implicit function theorem? This gives a local
> inverse on the graph of an implicit function whenever the tangent line
> is not horizontal.
In parametric form, some function f:R->R, parameter t, angle a.
G(t) = (x(t),y(t))
x(t) = t
y(t) = f(t)
R[a] is rotation by a
G' = R[a] mp G
G'(t) = (x'(t),y'(t))
x'(t) = cos(a) * t - sin(a)*f(t)
y'(t) = sin(a) * t + cos(a)*f(t)
Now we need a way to find explicit f'(x).
Which as was discussed earlier, piece-wisely defined
using a max (or min). So that
G'max = {x'(t),max{y'(t)}} = {x,f'(x)}, x in range of x'(t)
Something along these lines?
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