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An accurate way to interpolate between two quaternions is to use Spherical Linear Interpolation (Slerp) because it preserves the unit length, whereas straightforward linear interpolation does not, as shown by this example:

Clear[slerp];
slerp[q1_, q2_, f_] := Module[{omega},
  omega = ArcCos[Dot[Normalize@q1, Normalize@q2]]];
  If[PossibleZeroQ[omega], 
    q1,
    (Sin[(1 - f) omega] q1 + Sin[f omega] q2)/Sin[omega]
  ]
]

q1 = {1,0,0,0};
q2 = {0,1,0,0};
q3 = {0,0,1,0};

(* Linear interpolation using Interpolation *)
qint = Interpolation[{{0,q1},{1,q2}},InterpolationOrder->1];

Plot[{Norm@qint[t], Norm@slerp[{1, 0, 0, 0}, {0, 1, 0, 0}, t]}, {t, 0, 1}, 
  AxesLabel -> {"time", "length"}, PlotRange -> {0.7, 1.01}, 
  PlotStyle -> {Automatic, Dashed}, 
  Epilog -> {Text["Linear Interpolation", {0.5, 0.75}], 
  Text["Spherical Linear Interpolation", {0.5, 0.98}]}
]

Mathematica graphics

What's nice thing about Interpolation, however, is that it is easy to give it a list of vectors to get an interpolating function valid over the specified range, and it works fast:

Interpolation[{{0,q1},{1,q2},{2,q3}},InterpolationOrder->1]

InterpolatingFunction[{{0,2}},<>]

How can I create something like an InterpolatingFunction that preserves the unit-length property?

This is what I came up with, but it seems kludgy and it is very slow (this code does not check input bounds):

Clear[slerpInterpolation]; 
slerpInterpolation[q_List] := Function[{t}, 
  Module[{times, quaternions, pos, u, dt},
    times = q[[All, 1]];
    quaternions = q[[All, 2]];
    pos = Last@Flatten@Position[t - times, x_ /; x >= 0];
    dt = times[[pos + 1]] - times[[pos]];
    u = (t - times[[pos]])/dt;
    slerp[quaternions[[pos]], quaternions[[pos + 1]], u]
  ]
]

It is too slow in practice, giving about 10 evaluations per second on my laptop. Compare that with regular interpolation:

qlist = Transpose[{Range[100000] - 1, Normalize /@ RandomReal[{-1, 1}, {100000, 4}]}];
sint = slerpInterpolation[qlist];
lint = Interpolation[qlist, InterpolationOrder -> 1];

AbsoluteTiming[Do[sint[541.236], {10}]][[1]]/10

0.0967707

AbsoluteTiming[Do[lint[541.236], {10000}]][[1]]/10000

7.2522*10^-6

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1  
Might find something useful here –  Daniel Lichtblau Sep 10 '12 at 15:15

2 Answers 2

up vote 11 down vote accepted

Here's a somewhat complete implementation of Shoemake's spherical linear interpolation that functions completely analogously to Interpolation[] and InterpolatingFunction[]. As already noted, much of the slowness is due to your use of a sequential search. In any event, if you prefer, you could use the built-in interpolation as suggested in the other answer, but I think using a bisection routine is a bit more instructive.

Anyway...

SphericalLinearInterpolation::inddp = "The point `1` is duplicated.";

SphericalLinearInterpolation[data_] := 
 Module[{dtr = Transpose[SortBy[data, Composition[N, First]]], diffs, times},
        SphericalInterpolatingFunction[data[[{1, -1}, 1]], dtr] /; 
        If[MemberQ[diffs = Chop[Differences[times = First[dtr]]], 0], 
           Message[SphericalLinearInterpolation::inddp, 
                   First[Extract[times, Position[diffs, 0]]]]; False, True]]

SphericalInterpolatingFunction::dmval = 
  "Input value `1` lies outside the domain of the interpolating function.";

MakeBoxes[SphericalInterpolatingFunction[range_, rest__], _] ^:= 
 InterpretationBox[RowBox[
    {"SphericalInterpolatingFunction", "[", "{", #1, ",", #2, "}", ",", "\"<>\"", "]"}], 
    SphericalInterpolatingFunction[range, rest]] & @@ Map[ToBoxes, range]

SphericalInterpolatingFunction[stuff__][l_List] :=
 SphericalInterpolatingFunction[stuff] /@ l

SphericalInterpolatingFunction[range_, __]["Domain"] := range

SphericalInterpolatingFunction[{r_, s_}, data_][t_?NumericQ] :=
  (Message[SphericalInterpolatingFunction::dmval, t]; $Failed) /; ! (r <= t <= s)

slerp = Compile[{{q1, _Real, 1}, {q2, _Real, 1}, {f, _Real}}, 
   Module[{n1 = Norm[q1], n2 = Norm[q2], omega, so},
    (* vector angle formula by Velvel Kahan *)
    omega = 2 ArcTan[Norm[q1 n2 + n1 q2], Norm[q1 n2 - n1 q2]];
    If[Chop[so = Sin[omega]] == 0, q1, Sin[{1 - f, f} omega].{q1, q2}/so]]];    

SphericalInterpolatingFunction[range_, data_][t_?NumericQ] := Module[{times, quats, k},
        {times, quats} = data;
        k = GeometricFunctions`BinarySearch[times, t];
        slerp[quats[[k]], quats[[k + 1]], Rescale[t, times[[{k, k + 1}]], {0, 1}]]]

See J. Blow's article on why spherical linear interpolation might not always be the best method for interpolating quaternions.

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I would +1 but it would take me time to process. –  Chris Degnen Sep 9 '12 at 23:18
    
Take your time, then. I'll only be able to do experiments with the compiled version later... –  J. M. Sep 9 '12 at 23:29

The main slow-down in your code is the determination of the value of pos, which currently takes 90% of the time. You don't need to process the whole array to find where your time value is located. J. M.’s proposal is to perform bisection, but I’d suggest to use Mathematica’s own tool for that: Interpolation!

You have a function linking value of time and index, you just need to make a linear interpolation of its inverse:

func = Interpolation[Table[{qlist[[i, 1]], i}, {i, 1, 100000}]]

then you can simply change the code that calculates pos in your slerpInterpolation to:

pos = Floor[func[t]];

On my laptop, this decreases the per-iteration timing from 55.1 ms to 6.5 ms.

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