Finding Functional Values from the Original Domain Using BSplineFunction

Question

Providing a set of 2D points to BSplineFunction returns a function that requires that the input parameter to the function be between [0,1] no matter what the original domain of the points. In the 2D case that I'm interested in, what is returned from this function are two values: a value in the original units of the domain and its functional value. In other words an x and a y. I am primarily interested in finding the functional value associated with particular domain values. I have seen this before in this forum for example: Does anyone know a better approach than what follows? I have seen essentially a similar question posed before. It seems there is no direct way to provide a domain value and get its functional value. Here's a way to do it but it is slow and computationally expensive:

    (* p = 2D matrix of points*)
    (* f = Function returned by BSplineFunction *)
    (* bsp = A table of 301 returned values providing f with values from \
      [0,1] in increments of 1/300 *)
    (* tstf = A function that returns the first of the two values \
       returned by f (which is a value in the domain of the points p). *)
    (* a= A table of values between [0,1] that minimizes the squared \
       difference between the domain value I want and what is output by f. *)
    (* xy = 2D matrix containing the domain values I desire in the first \
       column and their functional values in the second column. *)

    p = {{400, 0}, {428, 0}, {477, 0}, {545, .5}, {596, 0}, {639,0}, {700,0}};
    f = BSplineFunction[p];
    bsp = Table[f[t], {t, 0, 1, 1/300}];
    tstf[t_?NumericQ] := f[t][[1]];
    Clear[u]
    a = Table[ u /. NMinimize[{(tstf[u] - k)^2, 0 <= u <= 1}, u][[2]], {k,400,700}];
    xy = Table[{k + 399, f[a[[k]]][[2]]}, {k, 1, 301}];

Here's a plot of bsp:

Here's a plot of xy:

They look identical but for bsp the x-values are given by f with values determined by using [0,1] stepping from 0 to 1 in equal increments of 1/300. The returned domain values are not the integer values in xy. Does anyone have a better solution than finding those [0,1] values that produce a specific domain value as I am with NMinimize?

Answer 1

this is essentially the same, but a good bit faster, first tabulate the curve to get good starting values then use FindRoot:

tab = Table[{u, tstf[u]}, {u, 0, 1, .01}];
start = (tab[[# - 1, 1]] &@
      First@FirstPosition[tab, {u_, v_ /; v > #}]) & /@ 
   Range[400, 699];
AppendTo[start, 1];
a = Table[(u /. FindRoot[tstf[u] == k, {u, start[[k - 399]]}]),
       {k, 400,700}];
ListPlot[f /@ a]

verify we found all the integers:

Max[Abs[(f /@ a)[[All, 1]] - Range[400, 700]]]

1.47793*10^-12