- All Implemented Interfaces:
DoubleUnaryOperator
,RealValuedDomainOps
,RealValuedFunctOps
,RealValuedFunctVAOps
,VADomainOps
This class supports both periodic and non-periodic B-splines. Periodic B-splines can be used to approximate period functions. Additional control points and knots are added for periodic B-splines. (see below).
A B-spline consists of
- an increasing finite sequence of real numbers called "knots": u0 ≤ u1 ≤ ... ≤ um
- a set of basis functions.
- a set of control points (just numbers for this class) denoted as P0, P1, … Pn.
- a degree—normally denoted as p. The degree of a B-spline must satisfy p = m - n - 1.
The basis functions are defined as follows: $$\begin{eqnarray} N_{i,0}(u) & = & \left\{\begin{array}{ll} 1 & \mbox {if $u_i \le u < u_{i+1}$} \\ 0 & \mbox{if $u < u_i$ or $u_{i+1} \le u$} \end{array}\right . \\ N_{i,p}(u) & = & \mbox{$N_{i,p-1}(u)\frac{u-u_i}{u_{i+p}-u_i} + N_{i+1,p-1}(u)\frac{u_{i+p+1}-u}{u_{i+p+1}-u_{i+1}}$ for $p > 0$} \end{eqnarray}$$ The value of the function $f$ defined by this spline at a point u is defined by $$f(u) = \sum^n_{i=0} P_iN_{i,p}$$ with a domain [up, um-p] for non-periodic splines and (-∞, ∞) for periodic splines (for a periodic spline, the value of u will be shifted by an appropriate multiple of the period so as to be in the interval [up, um-p], at at the end points of this interval, the values will match, as will their derivatives.
The documentation for the constructor
BSpline(int,double[],double[],boolean)
describes how
periodic and non-periodic B-splines are created. A couple of constructors
use a simplified set of arguments. These use a BSpline.Mode
argument to specify a menu of options for setting up the knots array
and for determining if the spline is periodic.
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Nested Class Summary
Nested ClassesModifier and TypeClassDescriptionstatic enum
Modes for creating a B-spline from a set of data points.Nested classes/interfaces inherited from class org.bzdev.math.RealValuedFunctionVA
RealValuedFunctionVA.Linear
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Field Summary
FieldsModifier and TypeFieldDescriptionstatic final int
Default cutoff for using a straightforward recursive implementation of the method M.Fields inherited from class org.bzdev.math.RealValuedFunction
xFunction
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Constructor Summary
ConstructorsConstructorDescriptionBSpline
(double[] knots, double[] cpoints) Constructor for open B-splines.BSpline
(int degree, double[] knots, boolean periodic, double[] x, double[] y) Constructor specifying a B-spline's degree, knots, and fitting the spline to a set of data points.BSpline
(int degree, double[] knots, boolean periodic, double[] x, double[] y, double sigma) Constructor specifying a B-spline's degree, knots, and fitting the spline to a set of data points, all with the same error.BSpline
(int degree, double[] knots, boolean periodic, double[] x, double[] y, double[] sigma) Constructor specifying a B-spline's degree, knots, and fitting the spline to a set of data points with errors.BSpline
(int degree, double[] knots, double[] cpoints, boolean periodic) Constructor.BSpline
(int degree, int n, BSpline.Mode mode, double[] x, double[] y) Constructor specifying a B-spline's degree, number of control points, mode, and a set of Y values for given X values.BSpline
(int degree, int n, BSpline.Mode mode, double[] x, double[] y, double sigma) Constructor specifying a B-spline's degree, number of control points, mode, and a set of Y values for given X values, with a specified error for the Y values.BSpline
(int degree, int n, BSpline.Mode mode, double[] x, double[] y, double[] sigma) Constructor specifying a B-spline's degree, number of control points, mode, and a set of Y values and their standard deviations for given X values. -
Method Summary
Modifier and TypeMethodDescriptionstatic double[]
createKnots
(int degree, int n, BSpline.Mode mode, double[] x) Create a knot array for a B-Spline.double
derivAt
(double u) Evaluate the function's first derivative.double[]
Get the control points.double[]
getControlPoints
(double[] array) Get the control points, storing them in a pre-allocated array.int
Get the degree if this B-spline.double
Get the maximum value in the domain of the function.double
Get the minimum value in the domain of the function.getLSF()
Get the least squares fit, if any, used to initialize this BSpline.int
Get the number of control points.double
Get the period.double
Get the ending value for a period.double
Get the starting value for a period.boolean
Determine if the spline is a periodic spline.final double
N
(int i, int p, double u) Compute the value of a B-Spline's basis function.double
secondDerivAt
(double u) Evaluate the function's second derivative.protected void
setControlPoints
(double[] cpoints) Set the control points for this BSpline.static void
setCutoff
(int value) Set the cutoff for the use of the naive basis-function computation.double
valueAt
(double u) Call the function.Methods inherited from class org.bzdev.math.RealValuedFunction
deriv, deriv, derivAt, domainMaxClosed, domainMaxClosed, domainMinClosed, domainMinClosed, getDomainMax, getDomainMin, isInDomain, isInDomain, secondDeriv, secondDerivAt, valueAt
Methods inherited from class org.bzdev.math.RealValuedFunctionVA
jacobian, jacobian, maxArgLength, minArgLength
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
Methods inherited from interface org.bzdev.math.RealValuedFunctOps
andThen, andThen, applyAsDouble, compose, compose, maxArgLength, minArgLength
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Field Details
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CUTOFF
public static final int CUTOFFDefault cutoff for using a straightforward recursive implementation of the method M. When the argument to N providing the degree of a basis function is below this value, the straightforward implementation is used. Otherwise intermediate results are cached and reused.- See Also:
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Constructor Details
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BSpline
Constructor for open B-splines. A full description is provided by the constructorBSpline(int,double[],double[],boolean)
. The degree of the B-spline is equal to knots.length - cpoints.length - 1.- Parameters:
knots
- the knots for the B-spline to be constructedcpoints
- the control points for the B-spline to be constructed- Throws:
IllegalArgumentException
- See Also:
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BSpline
public BSpline(int degree, double[] knots, double[] cpoints, boolean periodic) throws IllegalArgumentException Constructor. There are two cases depending on whether the B-spline will be periodic or non-periodic. For the non-periodic case, if m is the length of the knots array, n is the number of control points, and p is the degree if the B-spline, then the following constraint must be satisfied:n + p + 1 = m
orm - n - 1 = p.
orm - p - 1 = n.
The degree matches the highest possible degree of the polynomial representation of the basis functions (some authors start the sequence at 1 instead of 0, in which case the degree the smallest integer larger than the degree of the polynomials). The domain of the spline is [us, ue]. where us=knots[degree] and ue=knots[knots.length-p-1].For the periodic case, the values of the spline over a range [us,ue] repeat indefinitely in either direction, and the knots array's initial and ending components will be us and ue respectively. As a result, the period is ue-us. For a periodic spline whose degree is p, the length of the control-points array will either be one less than the length of the knots array, or (p-1) less than the length of the knots array. The constructor will augment these arrays as follows:
- p components will be inserted before the initial component of the knots array, and p components will be inserted after the last component of the knots array. The p components before the last original entry will be copied to the start of the array in the same order as in the original array, with values decreased by the period. The first p components past the initial component of the original array array will be copied to the end of the array, with values increased by the period.
- When the control-points array's length is one less than the length of the knots array, p components will be inserted before the initial component of the control-points array. The last p components of the original array will be copied to the start of the augmented array in the same order as in the original array.
- Parameters:
degree
- a non-negative integer giving the degree of the B-spline to be constructedknots
- the knots for the B-spline to be constructedcpoints
- the control points for the B-spline to be constructedperiodic
- true if the spline is periodic; false if it is open.- Throws:
IllegalArgumentException
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BSpline
public BSpline(int degree, double[] knots, boolean periodic, double[] x, double[] y) Constructor specifying a B-spline's degree, knots, and fitting the spline to a set of data points. The use of knots is the same as that for the constructorBSpline(int,double[],double[],boolean)
.- Parameters:
degree
- the degree of the B-spline that will be createdknots
- the knots the B-spline usesperiodic
- true if the B-spline is periodic, false otherwisex
- values in the domain of the spliney
- values in the range of the spline
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BSpline
public BSpline(int degree, double[] knots, boolean periodic, double[] x, double[] y, double sigma) Constructor specifying a B-spline's degree, knots, and fitting the spline to a set of data points, all with the same error. The use of knots is the same as that for the constructorBSpline(int,double[],double[],boolean)
.- Parameters:
degree
- the degree of the B-spline that will be createdknots
- the knots the B-spline usesperiodic
- true if the B-spline is periodic, false otherwisex
- values in the domain of the spliney
- values in the range of the splinesigma
- the standard deviation of the 'y' values for the least-squares fit.
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BSpline
public BSpline(int degree, double[] knots, boolean periodic, double[] x, double[] y, double[] sigma) Constructor specifying a B-spline's degree, knots, and fitting the spline to a set of data points with errors. The use of knots is the same as that for the constructorBSpline(int,double[],double[],boolean)
.- Parameters:
degree
- the degree of the B-spline that will be createdknots
- the knots the B-spline usesperiodic
- true if the B-spline is periodic, false otherwisex
- values in the domain of the spliney
- values in the range of the splinesigma
- the standard deviations for the values provided by the argument y
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BSpline
Constructor specifying a B-spline's degree, number of control points, mode, and a set of Y values for given X values. The number of X,Y data points must be larger than the number of control points.- Parameters:
degree
- the degree of the B-spline that will be createdn
- the number of control pointsmode
- the mode for the B-spline (either BSpline.Mode.UNCLAMPED, BSpline.Mode.CLAMPED, BSpline.Mode.CLAMPED_LEFT, BSpline.Mode.CLAMPED_RIGHT, or BSpline.Mode.PERIODIC); null for the default (BSplineMode.UNCLAMPED).x
- values in the domain of the spliney
- values in the range of the spline
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BSpline
Constructor specifying a B-spline's degree, number of control points, mode, and a set of Y values for given X values, with a specified error for the Y values. The number of X,Y data points must be larger than the number of control points.- Parameters:
degree
- the degree of the B-spline that will be createdn
- the number of control pointsmode
- the mode for the B-spline (either BSpline.Mode.UNCLAMPED, BSpline.Mode.CLAMPED, BSpline.Mode.CLAMPED_LEFT, BSpline.Mode.CLAMPED_RIGHT, or BSpline.Mode.PERIODIC); null for the default (BSplineMode.UNCLAMPED).x
- values in the domain of the spliney
- values in the range of the splinesigma
- the standard deviation for the Y values
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BSpline
Constructor specifying a B-spline's degree, number of control points, mode, and a set of Y values and their standard deviations for given X values. The number of X,Y data points must be larger than the number of control points.- Parameters:
degree
- the degree of the B-spline that will be createdn
- the number of control pointsmode
- the mode for the B-spline (either BSpline.Mode.UNCLAMPED, BSpline.Mode.CLAMPED, BSpline.Mode.CLAMPED_LEFT, BSpline.Mode.CLAMPED_RIGHT, or BSpline.Mode.PERIODIC); null for the default (BSplineMode.UNCLAMPED)x
- values in the domain of the spliney
- values in the range of the splinesigma
- the standard deviations for the values provided by the argument y
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Method Details
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setControlPoints
Set the control points for this BSpline.- Parameters:
cpoints
- an array containing the control points,- Throws:
IllegalArgumentException
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getControlPoints
public double[] getControlPoints()Get the control points.- Returns:
- the control-point array
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getControlPoints
public double[] getControlPoints(double[] array) Get the control points, storing them in a pre-allocated array.- Parameters:
array
- the array that will contain the control poions.- Returns:
- the control-point array (the first argument)
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getNumberOfControlPoints
public int getNumberOfControlPoints()Get the number of control points.- Returns:
- the number of control points
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isPeriodic
public boolean isPeriodic()Determine if the spline is a periodic spline.- Returns:
- true if the spline is periodic; false otherwise.
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getPeriodStart
public double getPeriodStart()Get the starting value for a period. A periodic B-spline explicitly handles a range of values of the parameters. For other values of the parameter, the parameter is shifted by the period until it falls within this range. For a non-periodic B-spline, the domain minimum is returned.- Returns:
- the starting value for a period if the spline is periodic; otherwise the domain minimum
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getPeriodEnd
public double getPeriodEnd()Get the ending value for a period. A periodic B-spline explicitly handles a range of values of the parameters. For other values of the parameter, the parameter is shifted by the period until it falls within this range. For a non-periodic B-spline, the domain maximum is returned.- Returns:
- the ending value for a period if the spline is periodic; otherwise the domain maximum
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getPeriod
public double getPeriod()Get the period. A periodic B-spline explicitly handles a range of values of the parameters. For other values of the parameter, the parameter is shifted by the period until it falls within this range. For a periodic B-spline, the period is numerically equal to getPeriodEnd()-getPeriodStart().- Returns:
- the period for a periodic spline; 0.0 if the spline is not periodic
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getDegree
public int getDegree()Get the degree if this B-spline. The smallest degree is zero (some authors use 1, in which case the degree will be larger by 1 than the value returned by this method).- Returns:
- this B-spline's degree.
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getLSF
Get the least squares fit, if any, used to initialize this BSpline. This method will always return null if it is called two or more times. It is intended for use by subclasses of LeastSquaresFit or other classes that need information about a least squares fit when the fit is created.- Returns:
- the least squares fit; null if there is none.
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createKnots
Create a knot array for a B-Spline.- Parameters:
degree
- the degree of the B-Splinen
- the number of control points for the B-Splinemode
- the mode for the B-spline (either BSpline.Mode.UNCLAMPED, BSpline.Mode.CLAMPED, BSpline.Mode.CLAMPED_LEFT, BSpline.Mode.CLAMPED_RIGHT, or BSpline.Mode.PERIODIC); null for the default (BSpline.Mode.UNCLAMPED)x
- the x values used to create the B-Spline- Returns:
- the knot array
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setCutoff
public static void setCutoff(int value) Set the cutoff for the use of the naive basis-function computation. When the argument to N providing the degree of a basis function is below this value, the straightforward implementation is used. Otherwise intermediate results are cached and reused.- Parameters:
value
- the cutoff, which must not be negative
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N
public final double N(int i, int p, double u) Compute the value of a B-Spline's basis function.- Parameters:
i
- a integer representing a knot span with a value equal to the index of the knot at the start of a spanp
- the degree of the basis functionu
- the value of the B-spline's parameter- Returns:
- the value of the basis function at point u
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getDomainMin
public double getDomainMin()Description copied from class:RealValuedFunction
Get the minimum value in the domain of the function.- Specified by:
getDomainMin
in interfaceRealValuedDomainOps
- Overrides:
getDomainMin
in classRealValuedFunction
- Returns:
- the minimum value
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getDomainMax
public double getDomainMax()Description copied from interface:RealValuedDomainOps
Get the maximum value in the domain of the function.- Specified by:
getDomainMax
in interfaceRealValuedDomainOps
- Overrides:
getDomainMax
in classRealValuedFunction
- Returns:
- the maximum value
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valueAt
Description copied from class:RealValuedFunction
Call the function.- Specified by:
valueAt
in interfaceRealValuedFunctOps
- Overrides:
valueAt
in classRealValuedFunction
- Parameters:
u
- the function's argument- Returns:
- the value of the function for the given argument
- Throws:
IllegalArgumentException
- the function's argument was out of range
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derivAt
Description copied from class:RealValuedFunction
Evaluate the function's first derivative.- Overrides:
derivAt
in classRealValuedFunction
- Parameters:
u
- the function's argument- Returns:
- the value of the function for the given argument
- Throws:
IllegalArgumentException
- the function's argument was out of range
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secondDerivAt
public double secondDerivAt(double u) Description copied from class:RealValuedFunction
Evaluate the function's second derivative.- Overrides:
secondDerivAt
in classRealValuedFunction
- Parameters:
u
- the function's argument- Returns:
- the value of the function for the given argument
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