Re: Unusual Linear Mapping??

In article <ef3382e.1@xxxxxxxxxxxxxxxx>, david <david@xxxxxxxxxx> wrote:

Mmmh, to be honest I'll try anything that can get me from [0, 0.5, 1]
-> [0, 0.95, 1]. Though I know it is a difficult task! My first
post was probably misleading by using the word linear, I'll try any
suggested method which is uniquely defined to perform the mapping. Do
you think it would be possible by performing some kind of parabolic
curve interpolation? Any further hints or suggestions most welcome.

Thanks for your help,
David
----------------------
When I said the parabolic curve might not be what you want, I referred
to its possibly not being a one-to-one mapping. Sure enough, if you let

k = 0.94117647058823529;
a = 2 - 4*k;
b = 4*k - 1;
c = 0;

then the parabolic mapping y = a*x^2 + b*x + c satisfies your conditions
but it is not one-to-one. It reaches a peak of y = 1.083 at somewhere in
the neighborhood of x = .78 and descends after that down to y = 1 at x =
1.

A possible mapping that is one-to-one is y = a + b*exp(-c*x) for
appropriate values of a, b, and c. To satisfy your requirements we need:

a + b = 0,
a + b*exp(-c/2) = k, and
a + b*exp(-c) = 1,

which leads to the problem of solving

(exp(-c/2)-1)/(exp(-c)-1) = k

The values I get for c, b, and a are

c = 5.5451774444795694,
b = 1/(exp(-c)-1), and
a = -b

which yields the mapping

y = (exp(-c*x)-1)/(exp(-c)-1);

Try it and see if it suits your needs.

(Remove "xyzzy" and ".invalid" to send me email.)
Roger Stafford
.

Relevant Pages

  • Re: Unusual Linear Mapping??
    ... When I said the parabolic curve might not be what you want, ... to its possibly not being a one-to-one mapping. ... Try it and see if it suits your needs. ... Roger Stafford ...
    (comp.soft-sys.matlab)
  • Re: Unusual Linear Mapping??
    ... Roger Stafford wrote: ... In article, david ... When I said the parabolic curve might not be what you want, ... to its possibly not being a one-to-one mapping. ...
    (comp.soft-sys.matlab)
  • Re: Unusual Linear Mapping??
    ... The mapping you specify cannot be linear. ...
    (comp.soft-sys.matlab)