umontreal.iro.lecuyer.functionfit
Class BSpline

java.lang.Object
  umontreal.iro.lecuyer.functionfit.BSpline

All Implemented Interfaces:

MathFunction, MathFunctionWithDerivative, MathFunctionWithFirstDerivative, MathFunctionWithIntegral

public class BSpline
extends Object
implements MathFunction, MathFunctionWithIntegral, MathFunctionWithDerivative, MathFunctionWithFirstDerivative

Represents a B-spline with control points at (X_i, Y_i). Let P_i = (X_i, Y_i), for i = 0,…, n - 1, be a control point and let t_j, for j = 0,…, m - 1 be a knot. A B-spline of degree p = m - n - 1 is a parametric curve defined as

P(t) = ∑_i=0^n-1N_{i, p}(t)P_i, for t_p <= t <= t_m-p-1.

Here,

Ni, p(t)	=	((t - ti)/(ti+p - ti))Ni, p-1(t) + ((ti+p+1 - t)/(ti+p+1 - ti+1))Ni+1, p-1(t)
Ni, 0(t)	=	{

This class provides methods to evaluate P(t) = (X(t), Y(t)) at any value of t, for a B-spline of any degree p >= 1. Note that the evaluate method of this class can be slow, since it uses a root finder to determine the value of t^* for which X(t^*) = x before it computes Y(t^*).

Constructor Summary

BSpline(double[] x, double[] y, double[] knots)
Constructs a new uniform B-spline with control points at (x[i], y[i]), and knot vector given by the array knots.

BSpline(double[] x, double[] y, int degree)
Constructs a new uniform B-spline of degree degree with control points at (x[i], y[i]).

Method Summary

static BSpline	createApproxBSpline(double[] x, double[] y, int degree, int h) Returns a B-spline curve of degree degree smoothing (xi, yi), for i = 0,…, n points.
static BSpline	createInterpBSpline(double[] x, double[] y, int degree) Returns a B-spline curve of degree degree interpolating the (xi, yi) points.
double	derivative(double u) Computes (or estimates) the first derivative of the function at point x.
double	derivative(double u, int n) Computes (or estimates) the nth derivative of the function at point x.
BSpline	derivativeBSpline() Returns the derivative B-spline object of the current variable.
BSpline	derivativeBSpline(int i) Returns the ith derivative B-spline object of the current variable; i must be less than the degree of the original B-spline.
double	evaluate(double u) Returns the value of the function evaluated at x.
double	evalX(double u)
double	evalY(double u)
double[]	getKnots() Returns an array containing the knot vector (t0, tm-1).
double	getMaxKnot() Returns the knot maximal value.
double	getMinKnot() Returns the knot minimal value.
double[]	getX() Returns the Xi coordinates for this spline.
double[]	getY() Returns the Yi coordinates for this spline.
double	integral(double a, double b) Computes (or estimates) the integral of the function over the interval [a, b].

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

BSpline

public BSpline(double[] x,
               double[] y,
               int degree)

Constructs a new uniform B-spline of degree degree with control points at (x[i], y[i]). The knots of the resulting B-spline are set uniformly from x[0] to x[n-1].

Parameters:

x - the values of X.

y - the values of Y.

degree - the degree of the B-spline.

BSpline

public BSpline(double[] x,
               double[] y,
               double[] knots)

Constructs a new uniform B-spline with control points at (x[i], y[i]), and knot vector given by the array knots.

Parameters:

x - the values of X.

y - the values of Y.

knots - the knots of the B-spline.

Method Detail

getX

public double[] getX()

Returns the X_i coordinates for this spline.

Returns:

the X_i coordinates.

getY

public double[] getY()

Returns the Y_i coordinates for this spline.

Returns:

the Y_i coordinates.

getMaxKnot

public double getMaxKnot()

Returns the knot maximal value.

Returns:

the Y_i coordinates.

getMinKnot

public double getMinKnot()

Returns the knot minimal value.

Returns:

the Y_i coordinates.

getKnots

public double[] getKnots()

Returns an array containing the knot vector (t₀, t_m-1).

Returns:

the knot vector.

createInterpBSpline

public static BSpline createInterpBSpline(double[] x,
                                          double[] y,
                                          int degree)

Returns a B-spline curve of degree degree interpolating the (x_i, y_i) points. This method uses the uniformly spaced method for interpolating points with a B-spline curve, and a uniformed clamped knot vector, as described in http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/.

Parameters:

x - the values of X.

y - the values of Y.

degree - the degree of the B-spline.

Returns:

the B-spline curve.

createApproxBSpline

public static BSpline createApproxBSpline(double[] x,
                                          double[] y,
                                          int degree,
                                          int h)

Returns a B-spline curve of degree degree smoothing (x_i, y_i), for i = 0,…, n points. The precision depends on the parameter h: 1 <= degree <= h < n, which represents the number of control points used by the new B-spline curve, minimizing the quadratic error

L = ∑_i=0ⁿ()².

This method uses the uniformly spaced method for interpolating points with a B-spline curve and a uniformed clamped knot vector, as described in http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/.

Parameters:

x - the values of X.

y - the values of Y.

degree - the degree of the B-spline.

h - the desired number of control points.

Returns:

the B-spline curve.

derivativeBSpline

public BSpline derivativeBSpline()

Returns the derivative B-spline object of the current variable. Using this function and the returned object, instead of the derivative method, is strongly recommended if one wants to compute many derivative values.

Returns:

the derivative B-spline of the current variable.

derivativeBSpline

public BSpline derivativeBSpline(int i)

Returns the ith derivative B-spline object of the current variable; i must be less than the degree of the original B-spline. Using this function and the returned object, instead of the derivative method, is strongly recommended if one wants to compute many derivative values.

Parameters:

i - the degree of the derivative.

Returns:

the ith derivative.

evaluate

public double evaluate(double u)

Description copied from interface: MathFunction

Returns the value of the function evaluated at x.

Specified by:

evaluate in interface MathFunction

Parameters:

u - value at which the function is evaluated

Returns:

function evaluated at x

integral

public double integral(double a,
                       double b)

Description copied from interface: MathFunctionWithIntegral

Computes (or estimates) the integral of the function over the interval [a, b].

Specified by:

integral in interface MathFunctionWithIntegral

Parameters:

a - the starting point of the interval.

b - the ending point of the interval.

Returns:

the value of the integral.

derivative

public double derivative(double u)

Description copied from interface: MathFunctionWithFirstDerivative

Computes (or estimates) the first derivative of the function at point x.

Specified by:

derivative in interface MathFunctionWithFirstDerivative

Parameters:

u - the point to evaluate the derivative to.

Returns:

the value of the derivative.

derivative

public double derivative(double u,
                         int n)

Description copied from interface: MathFunctionWithDerivative

Computes (or estimates) the nth derivative of the function at point x. For n = 0, this returns the result of evaluate.

Specified by:

derivative in interface MathFunctionWithDerivative

Parameters:

u - the point to evaluate the derivate to.

n - the order of the derivative.

Returns:

the resulting derivative.

evalX

public double evalX(double u)

evalY

public double evalY(double u)