Slightly confused about the notation used in this package

[jishnub]

The order k in this package appears to differ from the corresponding usage in other packages such as Dierckx or in scipy.interpolate (which also uses Dierckx I believe). It appears that k here is somewhat equivalent to k+1 in these packages. For example, in scipy:

In [1]: from scipy import interpolate

In [2]: knots = [0,0,1,2,3,4,4]; k = 2; c = [1,0,0,0,0];

In [3]: interpolate.BSpline(knots, c, k)(1)
Out[3]: array(0.5)

In [4]: c = [0,1,0,0,0];

In [5]: interpolate.BSpline(knots, c, k)(1)
Out[5]: array(0.5)

In Dierckx,

julia> k = 2; breakpoints = 0:4;

julia> t = vcat([breakpoints[1] for i in 1:k], breakpoints[2:end-1], [breakpoints[end] for i = 1:k])
7-element Array{Int64,1}:
 0
 0
 1
 2
 3
 4
 4

julia> c = [1,0,0,0,0];

julia> Dierckx.Spline1D(t, c, k, 0, 0)(1) # matches with scipy
0.5

# comparing with this package
julia> basis = BSplineBasis(k, breakpoints); knots(basis) == t
true

whereas in this package

julia> basis = BSplineBasis(2, 0:4)
5-element BSplineBasis{UnitRange{Int64}}:
 order: 2
 breakpoints: 0:4

julia> knots(basis)
7-element BSplines.KnotVector{Int64,UnitRange{Int64}}:
 0
 0
 1
 2
 3
 4
 4

julia> bsplines(basis, 1)
2-element OffsetArray(::Array{Float64,1}, 2:3) with eltype Float64 with indices 2:3:
 1.0
 0.0

julia> basis = BSplineBasis(3, 0:4)
6-element BSplineBasis{UnitRange{Int64}}:
 order: 3
 breakpoints: 0:4

julia> knots(basis)
9-element BSplines.KnotVector{Int64,UnitRange{Int64}}:
 0
 0
 0
 1
 2
 3
 4
 4
 4

julia> bsplines(basis, 1)
3-element OffsetArray(::Array{Float64,1}, 2:4) with eltype Float64 with indices 2:4:
 0.5
 0.5
 0.0

Using order=3 in this package leads to values that seem to match the results for k=2 in scipy, although the knots differ as expected at the edges. I wonder if these differences could be explained in the documentation?

[sostock]

You are correct: what scipy.interpolate and Dierckx call k is equivalent to k+1 in this package. In Dierckx and scipy.interpolate, k refers to the degree of the B-splines, not the order (as in this package). A B-spline of order k is a piecewise polynomial of degree k-1. It is unfortunate that the letter k is used sometimes for the order and sometimes for the degree.

It would indeed be good to add a note about that to the documentation.

[sostock]

@jishnub Do you think the additions to the documentation in #21 are fine, or would you change something?