BSpline Class Reference

Represents a B-spline with control points at \((X_i, Y_i)\). More...

Inheritance diagram for BSpline:

Collaboration diagram for BSpline:

Public Member Functions
	BSpline (final double[] x, final double[] y, final int degree)
	Constructs a new uniform B-spline of degree degree with control points at (x[i], y[i]). More...
	BSpline (final double[] x, final double[] y, final double[] knots)
	Constructs a new uniform B-spline with control points at (x[i], y[i]), and knot vector given by the array knots. More...
double []	getX ()
	Returns the \(X_i\) coordinates for this spline. More...
double []	getY ()
	Returns the \(Y_i\) coordinates for this spline. More...
double	getMaxKnot ()
	Returns the knot maximal value. More...
double	getMinKnot ()
	Returns the knot minimal value. More...
double []	getKnots ()
	Returns an array containing the knot vector \((t_0, t_{m-1})\). More...
BSpline	derivativeBSpline ()
	Returns the derivative B-spline object of the current variable. More...
BSpline	derivativeBSpline (int i)
	Returns the \(i\)th derivative B-spline object of the current variable; \(i\) must be less than the degree of the original B-spline. More...
double	evaluate (final double u)
	Returns the value of the function evaluated at \(x\). More...
double	integral (double a, double b)
	Computes (or estimates) the integral of the function over the interval \([a, b]\). More...
double	derivative (double u)
	Computes (or estimates) the first derivative of the function at point x. More...
double	derivative (double u, int n)
	Computes (or estimates) the \(n\)th derivative of the function at point x. More...
double	evalX (double u)
double	evalY (double u)

Static Public Member Functions
static BSpline	createInterpBSpline (double[] x, double[] y, int degree)
	Returns a B-spline curve of degree degree interpolating the \((x_i, y_i)\) points [42] . More...
static BSpline	createApproxBSpline (double[] x, double[] y, int degree, int hp1)
	Returns a B-spline curve of degree degree smoothing \((x_i, y_i)\), for \(i=0,…,n\) points. More...

Detailed Description

Represents a B-spline with control points at \((X_i, Y_i)\).

Let \(\mathbf{P_i}=(X_i, Y_i)\), for \(i=0,…,n-1\), be a control point and let \(t_j\), for \(j=0,…,m-1\) be a knot. A B-spline [42]  of degree \(p=m-n-1\) is a parametric curve defined as

\[ \mathbf{P(t)} = \sum_{i=0}^{n-1} N_{i, p}(t) \mathbf{P_i},\mbox{ for }t_p\le t\le t_{m-p-1}. \]

Here,

\begin{align*} N_{i, p}(t) & = \frac{t-t_i}{t_{i+p} - t_i}N_{i, p-1}(t) + \frac{t_{i+p+1} - t}{t_{i+p+1} - t_{i+1}}N_{i+1, p-1}(t) \\ N_{i, 0}(t) & = \left\{ \begin{array}{ll} 1 & \mbox{ for }t_i\le t\le t_{i+1}, \\ 0 \mbox{ elsewhere}. \end{array} \right. \end{align*}

This class provides methods to evaluate \(\mathbf{P(t)}=(X(t), Y(t))\) at any value of \(t\), for a B-spline of any degree \(p\ge1\). Note that the evaluate(double) 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 & Destructor Documentation

BSpline() [1/2]

BSpline	(	final double []	x,
		final double []	y,
		final 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() [2/2]

BSpline	(	final double []	x,
		final double []	y,
		final 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.

Member Function Documentation

createApproxBSpline()

static BSpline createApproxBSpline ( double []  x, double []  y, int  degree, int  hp1  )

static

Returns a B-spline curve of degree degree smoothing \((x_i, y_i)\), for \(i=0,…,n\) points.

The precision depends on the parameter \(hp1\): \(1 \le\mathtt{degree} \le hp1<n\), which represents the number of control points used by the new B-spline curve, minimizing the quadratic error

\[ L = \sum_{i=0}^n\left( \frac{Y_i - S_i(X_i)}{W_i}\right)^2. \]

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.
hp1	the desired number of control points.

Returns

the B-spline curve.

createInterpBSpline()

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

static

Returns a B-spline curve of degree degree interpolating the \((x_i, y_i)\) points [42] .

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.

derivative() [1/2]

double derivative

(

double

x

)

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

Parameters

x	the point to evaluate the derivative to.

Returns

the value of the derivative.

Implements MathFunctionWithFirstDerivative.

derivative() [2/2]

double derivative	(	double	x,
		int	n
	)

Computes (or estimates) the \(n\)th derivative of the function at point x.

For \(n=0\), this returns the result of umontreal.ssj.functions.MathFunction.evaluate(double).

Parameters

x	the point to evaluate the derivate to.
n	the order of the derivative.

Returns

the resulting derivative.

Exceptions

IllegalArgumentException

if n is negative or 0.

Implements MathFunctionWithDerivative.

derivativeBSpline() [1/2]

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() [2/2]

BSpline derivativeBSpline

(

int

i

)

Returns the \(i\)th 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()

double evaluate

(

final double

x

)

Returns the value of the function evaluated at \(x\).

Parameters

x	value at which the function is evaluated

Returns

function evaluated at x

Implements MathFunction.

getKnots()

double [] getKnots

(

)

Returns an array containing the knot vector \((t_0, t_{m-1})\).

Returns

the knot vector.

getMaxKnot()

double getMaxKnot

(

)

Returns the knot maximal value.

Returns

the \(Y_i\) coordinates.

getMinKnot()

double getMinKnot

(

)

Returns the knot minimal value.

Returns

the \(Y_i\) coordinates.

getX()

double [] getX

(

)

Returns the \(X_i\) coordinates for this spline.

Returns

the \(X_i\) coordinates.

getY()

double [] getY

(

)

Returns the \(Y_i\) coordinates for this spline.

Returns

the \(Y_i\) coordinates.

integral()

double integral	(	double	a,
		double	b
	)

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

Parameters

a	the starting point of the interval.
b	the ending point of the interval.

Returns

the value of the integral.

Implements MathFunctionWithIntegral.

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The documentation for this class was generated from the following file:

• BSpline.java