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

Example

use bsplines::{Curve, DataPoints};
use nalgebra::dmatrix;

// Five 2D data points, one column per point.
let data = DataPoints::new(dmatrix![
    -2.0,-1.0, 0.0, 1.0, 2.0; // x
     0.5,-0.5, 1.5,-1.5, 0.5; // y
]);

// Interpolate the data with a cubic curve and evaluate it.
let curve = Curve::interpolate(&data, 3)?;
let point = curve.evaluate(0.5)?;
let velocity = curve.evaluate_derivative(0.5, 1)?;

// Or approximate it with a penalized least-squares fit.
let fitted = Curve::fit(&data, 3).loose_ends().penalized(0.5, 2).build()?;

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.

Structs

Constraints

The parameters at which the points of two merged curves stay fixed.

ControlPoints

The control points P of a curve and of its derivatives; together they form the control polygon.

Curve

A B-spline curve: a parametric function that maps the domain [0, 1] into N-dimensional space.

DataPoints

Input points that a curve is interpolated through or fitted to. Data points are consumed by curve generation; they are not part of the resulting curve.

FitBuilder

Builds a least-squares fit of data points, with fixed or loose ends and an optional penalization; created by Curve::fit.

Knots

The knot vector U of a curve and the knot vectors of its derivatives.

Parameters

The parameter values ū assigned to the data points for interpolation and fitting, one per data point.

Enums

Error

The error type for all fallible B-spline operations.

KnotMethod

The method generating a clamped knot vector for a curve of degree p with n polygon segments.

ParameterMethod

The method assigning a parameter value ū to every data point.

Traits

Points

Common accessors for point sets stored as one column per point.

Type Aliases

Result

Convenience alias for a core::result::Result with the crate-wide Error.