- All Implemented Interfaces:
RealValuedDomainOps
,VADomainOps
This class supports both periodic and non-periodic NURBS, and is
very similar to the BSplineArray
class. For purposes of
this class, a periodic NURBS array adds additional knots and
control points - copies of the ones explicitly specified, so that
the spline will be smooth at its end points. The parameter u will
be shifted if necessary, always by a multiple of its period, so
that the spline's value will be computed using values of the
parameter within a specific range. Details regarding the
additional control points, weights, and knots for the periodic case
are described below.
A NURBS array 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 NURBS must satisfy p = m - n - 1.
- a set of weights denoted as w0, w1 … wn.
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 defined by this spline at a point u is equal to $$f(u) = \sum^n_{i=0} P_iN_{i,p}$$ 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
NurbsArray(int,int,double[],double[],boolean)
describes how
periodic and non-periodic NURBSs are created.
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Constructor Summary
ConstructorsConstructorDescriptionNurbsArray
(int dim, double[] knots, double[] cpoints) Constructor for open NURBSs.NurbsArray
(int dim, int degree, double[] knots, double[] cpoints, boolean periodic) Constructor. -
Method Summary
Modifier and TypeMethodDescriptionvoid
derivAt
(double[] array, int off, double u) Evaluate the function's first derivative, storing the results.boolean
Determine if the domain maximum is in the domain.boolean
Determine if the domain minimum is in the domain.int
Get the degree if this NURBS.int
Get the number of dimensions for the NURBS.double
Get the maximum value in the domain of the function.double
Get the minimum value in the domain of the function.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 NURBS's basis function.void
secondDerivAt
(double[] array, int off, double u) Evaluate the function's second derivative, storing the results.void
valueAt
(double[] array, int off, double u) Call the function, storing the value in an array.Methods inherited from class org.bzdev.math.VectorValuedFunction
derivAt, derivAt, domainMaxClosed, domainMinClosed, getDomainMax, getDomainMin, isInDomain, isInDomain, secondDerivAt, secondDerivAt, valueAt, valueAt
Methods inherited from class org.bzdev.math.VectorValuedFunctionVA
derivAt, maxArgLength, minArgLength, secondDerivAt, valueAt
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Constructor Details
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NurbsArray
Constructor for open NURBSs. A full description is provided by the constructorNurbsArray(int,int,double[],double[],boolean)
. The degree of the NURBS is equal to knots.length - cpoints.length - 1.- Parameters:
dim
- the length of the array of valuesknots
- the knots for the NURBS to be constructedcpoints
- the control points and weights for the NURBS to be constructed- Throws:
IllegalArgumentException
- See Also:
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NurbsArray
public NurbsArray(int dim, int degree, double[] knots, double[] cpoints, boolean periodic) throws IllegalArgumentException Constructor. There are two cases depending on whether the NURBS 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 NURBS, then the following constraint must be satisfied:n + p + 1 = m
orm - n - 1 = p.
orm - p - 1 = n.
where each control point and weight consists of 1 + <the number of dimension> double-precision numbers. For each control point and weight, the array contains the coordinates of the control point (listed in order), followed by the weight, placed contiguously in the array.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) larger than the length of the knots array, in both cases multiplied by 1 + <the number of dimensions>. 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:
dim
- the dimension of the vectordegree
- a non-negative integer giving the degree of the NURBS to be constructedknots
- the knots for the NURBS to be constructedcpoints
- the control points and weights for the NURBS to be constructedperiodic
- true if the spline is periodic; false if it is open.- Throws:
IllegalArgumentException
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Method Details
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getDimension
public int getDimension()Get the number of dimensions for the NURBS. This is also the number of dimensions for the control points, excluding their weights.- Overrides:
getDimension
in classVectorValuedFunctionVA
- Returns:
- the number of dimensions
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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 NURBS 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 NURBS, 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 NURBS 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 NURBS, 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 NURBS 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 NURBS, 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 NURBS. 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 NURBS' degree.
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N
public final double N(int i, int p, double u) Compute the value of a NURBS'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 NURBS's parameter- Returns:
- the value of the basis function at point u
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getDomainMin
public double getDomainMin()Description copied from interface:RealValuedDomainOps
Get the minimum value in the domain of the function.- Specified by:
getDomainMin
in interfaceRealValuedDomainOps
- Overrides:
getDomainMin
in classVectorValuedFunction
- Returns:
- the minimum value
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domainMinClosed
public boolean domainMinClosed()Description copied from interface:RealValuedDomainOps
Determine if the domain minimum is in the domain.- Specified by:
domainMinClosed
in interfaceRealValuedDomainOps
- Overrides:
domainMinClosed
in classVectorValuedFunction
- Returns:
- true if the domain minimum is in the domain; false if it is the greatest lower bound for the domain
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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 classVectorValuedFunction
- Returns:
- the maximum value
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domainMaxClosed
public boolean domainMaxClosed()Description copied from interface:RealValuedDomainOps
Determine if the domain maximum is in the domain.- Specified by:
domainMaxClosed
in interfaceRealValuedDomainOps
- Overrides:
domainMaxClosed
in classVectorValuedFunction
- Returns:
- true if the domain maximum is in the domain; false if it is the least upper bound for the domain
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valueAt
Description copied from class:VectorValuedFunction
Call the function, storing the value in an array.- Specified by:
valueAt
in classVectorValuedFunction
- Parameters:
array
- the array used to store the results of the function call.off
- the offset into the array at which to store the results of the function call.u
- the function's argument- Throws:
IllegalArgumentException
- the function's argument was out of range
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derivAt
Description copied from class:VectorValuedFunction
Evaluate the function's first derivative, storing the results.- Overrides:
derivAt
in classVectorValuedFunction
- Parameters:
array
- the array used to store the results.off
- the offset into the array at which to store the results.u
- the function's argument- Throws:
IllegalArgumentException
- the function's argument was out of range
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secondDerivAt
public void secondDerivAt(double[] array, int off, double u) Description copied from class:VectorValuedFunction
Evaluate the function's second derivative, storing the results.- Overrides:
secondDerivAt
in classVectorValuedFunction
- Parameters:
array
- the array used to store the results.off
- the offset into the array at which to store the results.u
- the function's argument
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