BasicBSpline.jl

Basic (mathematical) operations for B-spline functions and related things with julia

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Summary

This package provides basic mathematical operations for B-spline.

  • B-spline basis function
  • Some operations for knot vector
  • Some operations for B-spline space (piecewise polynomial space)
  • B-spline manifold (includes curve, surface and solid)
  • Refinement algorithm for B-spline manifold
  • Fitting control points for given function

Comparison to other julia packages for B-spline

Note that the following comparison might not correct. If you have any thoughts, please help in issue#161.

  • Interpolations.jl
    • Currently this package's support is best for B-splines and also supports irregular grids.

    • But seems like no method for B-spline manifold.
  • ApproXD.jl
    • Its functions are similar to Interpolations.jl.
  • Dierckx.jl
    • Wrapper for the dierckx Fortran library.
    • Only 1-d or 2-d B-spline manifold are supported.
    • 5 or less degree of polynomial are supported.
  • BasicBSpline.jl (this package)
    • Any degree of polynomial are supported.
    • Refinement algorithm for B-spline manifold.
    • Fitting algorithm by least squares.
    • High performance on speed.
    • Mathematically oriented.

Installation

Install this package

] add BasicBSpline

To export graphics, use BasicBSplineExporter.jl.

] add https://github.com/hyrodium/BasicBSplineExporter.jl

Example

B-spline function

using BasicBSpline
using Plots

k = KnotVector([0.0, 1.5, 2.5, 5.5, 8.0, 9.0, 9.5, 10.0])
P0 = BSplineSpace{0}(k) # 0th degree piecewise polynomial space
P1 = BSplineSpace{1}(k) # 1st degree piecewise polynomial space
P2 = BSplineSpace{2}(k) # 2nd degree piecewise polynomial space
P3 = BSplineSpace{3}(k) # 3rd degree piecewise polynomial space
plot(
    plot([t->bsplinebasis(P0,i,t) for i in 1:dim(P0)], 0, 10, ylims=(0,1), legend=false),
    plot([t->bsplinebasis(P1,i,t) for i in 1:dim(P1)], 0, 10, ylims=(0,1), legend=false),
    plot([t->bsplinebasis(P2,i,t) for i in 1:dim(P2)], 0, 10, ylims=(0,1), legend=false),
    plot([t->bsplinebasis(P3,i,t) for i in 1:dim(P3)], 0, 10, ylims=(0,1), legend=false),
    layout=(2,2),
)

Try interactive graph with Desmos graphing calculator!

B-spline manifold

using BasicBSpline
using BasicBSplineExporter
using StaticArrays

p = 2 # degree of polynomial
k1 = KnotVector(1:8)     # knot vector
k2 = KnotVector(rand(7))+(p+1)*KnotVector([1])
P1 = BSplineSpace{p}(k1) # B-spline space
P2 = BSplineSpace{p}(k2)
n1 = dim(P1) # dimension of B-spline space
n2 = dim(P2)
a = [SVector(2i-6.5+rand(),1.5j-6.5+rand()) for i in 1:dim(P1), j in 1:dim(P2)] # random generated control points
M = BSplineManifold(a,(P1,P2)) # Define B-spline manifold
save_png("2dim.png", M) # save image

Refinement

h-refinemnet

k₊=(KnotVector([3.3,4.2]),KnotVector([0.3,0.5])) # additional knotvectors
M_h = refinement(M, k₊) # refinement of B-spline manifold
save_png("2dim_h-refinement.png", M_h) # save image

Note that this shape and the last shape are equivalent.

p-refinemnet

p₊=(Val(1),Val(2)) # additional degrees
M_p = refinement(M, p₊) # refinement of B-spline manifold
save_png("2dim_p-refinement.png", M_p) # save image

Note that this shape and the last shape are equivalent.

Fitting B-spline manifold

Try on Desmos graphing graphing calculator!

p1 = 2
p2 = 2
k1 = KnotVector(-10:10)+p1*KnotVector([-10,10])
k2 = KnotVector(-10:10)+p2*KnotVector([-10,10])
P1 = BSplineSpace{p1}(k1)
P2 = BSplineSpace{p2}(k2)

f(u1, u2) = SVector(2u1 + sin(u1) + cos(u2) + u2 / 2, 3u2 + sin(u2) + sin(u1) / 2 + u1^2 / 6) / 5

a = fittingcontrolpoints(f, (P1, P2))
M = BSplineManifold(a, (P1, P2))
save_png("fitting.png", M, unitlength=50, xlims=(-10,10), ylims=(-10,10))

If the knotvector span is too coarse, the approximation will be coarse.

p1 = 2
p2 = 2
k1 = KnotVector(-10:5:10)+p1*KnotVector([-10,10])
k2 = KnotVector(-10:5:10)+p2*KnotVector([-10,10])
P1 = BSplineSpace{p1}(k1)
P2 = BSplineSpace{p2}(k2)

f(u1, u2) = SVector(2u1 + sin(u1) + cos(u2) + u2 / 2, 3u2 + sin(u2) + sin(u1) / 2 + u1^2 / 6) / 5

a = fittingcontrolpoints(f, (P1, P2))
M = BSplineManifold(a, (P1, P2))
save_png("fitting_coarse.png", M, unitlength=50, xlims=(-10,10), ylims=(-10,10))

Draw smooth vector graphics

p = 3
k = KnotVector(range(-2π,2π,length=8))+p*KnotVector(-2π,2π)
P = BSplineSpace{p}(k)

f(u) = SVector(u,sin(u))

a = fittingcontrolpoints(f, P)
M = BSplineManifold(a, P)
save_svg("sine-curve.svg", M, unitlength=50, xlims=(-2,2), ylims=(-8,8))
save_svg("sine-curve_no-points.svg", M, unitlength=50, xlims=(-2,2), ylims=(-8,8), points=false)

This is useful when you edit graphs (or curves) with your favorite vector graphics editor.

See Plotting smooth graphs with Julia for more tutorials.