slerp breaks flag for quaternion normalization

[hyrodium]

julia> q1 = Quaternion(1.0, 0.0, 0.0, 0.0, true)
Quaternion{Float64}(1.0, 0.0, 0.0, 0.0, true)

julia> q2 = Quaternion(0.0, 1.0, 0.0, 0.0, true)
Quaternion{Float64}(0.0, 1.0, 0.0, 0.0, true)

julia> slerp(q1,q2,0.3)
Quaternion{Float64}(0.8910065241883678, 0.45399049973954675, 0.0, 0.0, false)

The flag false should be true here because slerp keeps norm of quaternion if the input quaternions are unit quaternions.

[sethaxen]

I agree, this would be sensible. The only thing to check is if after repeated application of slerp, the norm of the output drifts away from 1.

[hyrodium]

Thanks for the comment! I will make a PR to resolve this issue after #60 is discussed.

[sethaxen]

Since #60 proposes a breaking change and this issue doesn't, proceeding with a PR anyways would be useful.

[hyrodium]

Oh, I just noticed we also have linpol funciton.

julia> q1 = Quaternion(1.,0.,0.,0.,true)
QuaternionF64(1.0, 0.0, 0.0, 0.0, true)

julia> q2 = Quaternion(0.,1.,0.,0.,true)
QuaternionF64(0.0, 1.0, 0.0, 0.0, true)

julia> linpol(q1, q2, 0.2)
QuaternionF64(0.9510565162951535, 0.3090169943749474, 0.0, 0.0, true)

julia> slerp(q1, q2, 0.2)
QuaternionF64(0.9510565162951535, 0.3090169943749474, 0.0, 0.0, false)

I think slerp is more common word, so it seems better to deprecate linpol function.

[sethaxen]

Are they entirely equivalent, or is there some combination of inputs for which they will produce different results?

[hyrodium]

Note that these functions return different values for non-unit quaternion.

julia> q3 = Quaternion(2.,1.,0.,0.,true)
QuaternionF64(2.0, 1.0, 0.0, 0.0, true)

julia> q4 = Quaternion(3.,0.,0.,0.,true)
QuaternionF64(3.0, 0.0, 0.0, 0.0, true)

julia> slerp(q3, q4, 0.2)
QuaternionF64(2.0, 1.0, 0.0, 0.0, true)

julia> linpol(q3, q4, 0.2)
QuaternionF64(2.2, 0.8, 0.0, 0.0, true)

• slerp returns the first quaternion if the cosine is larger than 1. This seems buggy for me.
  • https://github.com/JuliaGeometry/Quaternions.jl/blob/v0.4.8/src/Quaternion.jl#L285
• linpol normalizes arguments, and the return value is always unit quaternion.
  • https://github.com/JuliaGeometry/Quaternions.jl/blob/v0.4.8/src/Quaternion.jl#L156-L158

My question is, what's the expected behavior of slerp with non-unit quaternion?
I think we have the following choices:

• Normalize all input quaternions, and the return value is always unit quaternion. (like linpol)
• Throws DomainError for non-unit quaternion.
• Return exp(log(q1)*(1-t) + log(q2)*t) for any quaternions.

[hyrodium]

The following are references on slerp, all of which deal with unit quaternions.

Slerp is shorthand for spherical linear interpolation

https://en.wikipedia.org/wiki/Slerp

spherical interpolating between the rotated value ...

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/slerp/

Spherically interpolates between quaternions a and b by ratio t.

https://docs.unity3d.com/ScriptReference/Quaternion.Slerp.html

[sethaxen]

For additional background, linpol was present in the original check-in of the package (351d657), and slerp was added in #6, with no discussion about linpol.

[hyrodium]

@keesvp @SimonDanisch
Do you have any thoughts on this?

[hyrodium]

I tried searching for linpol in GitHub public repositories, but I couldn't find it.
https://github.com/search?q=language%3Ajulia+linpol&type=code

[hyrodium]

On the current master branch, slerp is faster than linpol.

julia> q1 = Quaternion(1.,0,0,0,true)
QuaternionF64(1.0, 0.0, 0.0, 0.0, true)

julia> q2 = Quaternion(0,1.,0,0,true)
QuaternionF64(0.0, 1.0, 0.0, 0.0, true)

julia> slerp(q1,q2,0.2)
QuaternionF64(0.9510565162951535, 0.3090169943749474, 0.0, 0.0, false)

julia> linpol(q1,q2,0.2)
QuaternionF64(0.9510565162951535, 0.3090169943749474, 0.0, 0.0, true)

julia> @benchmark slerp($q1,$q2,0.2)
BenchmarkTools.Trial: 10000 samples with 994 evaluations.
 Range (min … max):  30.258 ns … 47.363 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     30.450 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   30.738 ns ±  1.569 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

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  30.3 ns         Histogram: frequency by time        37.4 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark linpol($q1,$q2,0.2)
BenchmarkTools.Trial: 10000 samples with 991 evaluations.
 Range (min … max):  40.956 ns … 120.753 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     41.835 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   42.864 ns ±   5.312 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

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  41 ns         Histogram: log(frequency) by time      67.6 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.