Array of functions, evaluation in plot

Question

I would like to create two arrays of functions yfunc[[i]] and tfunc[[i]], with each entry of those functions being a function depending on t, that is to say

yfunc={yfunc[[1]][t],yfunc[[2]][t],...} 

and the same for tfunc. Then I would also like to plot members of these array with ParametricPlot in pairs, e.g.

ParametricPlot[{yfunc[[i]][t],tfunc[[i]][t]},{t,0,1}]

This code works:

yfunc[t_?NumericQ, i_?IntegerQ] := i*t;
tfunc[t_?NumericQ, i_?IntegerQ] := 2 i*t;
comp[i_] := {tfunc[t, i], yfunc[t, i]};
Show[ParametricPlot[comp[1], {t, 0, 1}], AspectRatio -> 1]

But another try of creating this function array fails:

yfunc[t_?NumericQ] := Table[i*t, {i, 1, 100}]
tfunc[t_?NumericQ] := Table[2 i*t, {i, 1, 100}];
comp[i_?IntegerQ] := {tfunc[[i]][t], yfunc[[i]][t]};
Show[ParametricPlot[comp[1], {t, 0, 1}], AspectRatio -> 1]

I guess in this case it is a problem with the evaluation. What would be the correct way to do it in the second way?

Thanks!

Answer 1

The problem with not using tfunc[t][[i]] and trying as you do to call tfunc[[i]][t] is that tfunc is not a list and cannot be indexed. The result of calling tfunc[t] is a list, which can be indexed. If you want to use a list of functions, you can use anonymous functions:

yfunc = Table[With[{i = i}, Function[t, i*t]], {i, 1, 100}];
tfunc = Table[With[{i = i}, Function[t, 2 i*t]], {i, 1, 100}];

comp[i_?IntegerQ, t_] := {tfunc[[i]][t], yfunc[[i]][t]};
Show[ParametricPlot[comp[1, t], {t, 0, 1}], AspectRatio -> 1]

Note that since Function has attribute HoldAll, With[{i=i},...] is used to inject the value of i into the expression. If you don't do this it wont work as: Table[Function[t,i],{i,2}] evalutes to {Function[t,i],Function[t,i]}.