Crate bsplines

bsplines is a library for vectorized, N-dimensional B-spline curves and their derivatives based on nalgebra.

Features

• Create N-dimensional (N = 1, 2, 3,...) curves of arbitrary polynomial degree p.
• Efficient curve evaluation for all available derivatives k = 0, 1,... , p.
• Built with nalgebra to store point data in contiguous arrays
• Multiple methods for
  • curve generation
  • curve parametrization
  • knot generation
  • curve manipulation
    • knot insertion
    • reversing
    • splitting
    • merging

What are B-Splines?

B-splines are parametric functions composed of piecewise, polynomial basis functions of degree p > 0. These piecewise polynomials are joined so that the parametric function is p-1 times continuously differentiable. The overall functions are parametrized over finite domains with a so-called knot vector with the co-domain being an N-dimensional vector space, that is defined by control points. They can describe curves, but also surfaces. These characteristics lead to many desirable properties. The piecewise definition makes B-spline functions versatile allowing to interpolate or approximate complex-shaped and high-dimensional data, while maintaining a low polynomial degree. Because of the polynomial nature, all possible derivatives are accessible.

Still, evaluations or spatial manipulations can be executed fast because only local polynomial segments must be considered and the associated numerical procedures are stable. Lastly, polynomials represent a memory-efficient way of storing spatial information as few polynomial coefficients suffice to describe complex shapes.

Literature:

Piegl1997	Piegl, L., Tiller, W. The NURBS Book. Monographs in Visual Communication. Springer, Berlin, Heidelberg, 2nd ed., 1997.
Eilers1996	Eilers, P. H. C., Marx, B. D., Flexible smoothing with B -splines and penalties, Stat. Sci., 11(2) (1996) 89–121.
Tai2003	Tai, C.-L., Hu, S.-M., Huang, Q.-X., Approximate merging of B-spline curves via knot adjustment and constrained optimization, Comput. Des., 35(10) (2003) 893–899.

Modules

• curve
  Implements the B-spline curve.
• manipulation
  Implements different curve manipulation methods.
• types