Spline and spline fitting module

Classes
class	Eigen::Spline< Scalar_, Dim_, Degree_ >
	A class representing multi-dimensional spline curves. More...
struct	Eigen::SplineFitting< SplineType >
	Spline fitting methods. More...
struct	Eigen::SplineTraits< Spline< Scalar_, Dim_, Degree_ >, _DerivativeOrder >
	Compile-time attributes of the Spline class for fixed degree. More...
struct	Eigen::SplineTraits< Spline< Scalar_, Dim_, Degree_ >, Dynamic >
	Compile-time attributes of the Spline class for Dynamic degree. More...

Functions
template<typename PointArrayType , typename KnotVectorType >
void	Eigen::ChordLengths (const PointArrayType &pts, KnotVectorType &chord_lengths)
	Computes chord length parameters which are required for spline interpolation. More...
template<typename KnotVectorType >
void	Eigen::KnotAveraging (const KnotVectorType &parameters, DenseIndex degree, KnotVectorType &knots)
	Computes knot averages. More...
template<typename KnotVectorType , typename ParameterVectorType , typename IndexArray >
void	Eigen::KnotAveragingWithDerivatives (const ParameterVectorType &parameters, const unsigned int degree, const IndexArray &derivativeIndices, KnotVectorType &knots)
	Computes knot averages when derivative constraints are present. Note that this is a technical interpretation of the referenced article since the algorithm contained therein is incorrect as written. More...

Detailed Description

This module provides a simple multi-dimensional spline class while offering most basic functionality to fit a spline to point sets.

#include <unsupported/Eigen/Splines>

Function Documentation

ChordLengths()

template<typename PointArrayType , typename KnotVectorType >

void Eigen::ChordLengths	(	const PointArrayType &	pts,
		KnotVectorType &	chord_lengths
	)

Computes chord length parameters which are required for spline interpolation.

Parameters

[in]	pts	The data points to which a spline should be fit.
[out]	chord_lengths	The resulting chord length vector.

See also

Les Piegl and Wayne Tiller, The NURBS book (2nd ed.), 1997, 9.2.1 Global Curve Interpolation to Point Data

Definition at line 192 of file SplineFitting.h.

  {
    typedef typename KnotVectorType::Scalar Scalar;
 
    const DenseIndex n = pts.cols();
 
    // 1. compute the column-wise norms
    chord_lengths.resize(pts.cols());
    chord_lengths[0] = 0;
    chord_lengths.rightCols(n-1) = (pts.array().leftCols(n-1) - pts.array().rightCols(n-1)).matrix().colwise().norm();
 
    // 2. compute the partial sums
    std::partial_sum(chord_lengths.data(), chord_lengths.data()+n, chord_lengths.data());
 
    // 3. normalize the data
    chord_lengths /= chord_lengths(n-1);
    chord_lengths(n-1) = Scalar(1);
  }

KnotAveraging()

template<typename KnotVectorType >

void Eigen::KnotAveraging	(	const KnotVectorType &	parameters,
		DenseIndex	degree,
		KnotVectorType &	knots
	)

Computes knot averages.

The knots are computed as

\begin{align*} u_0 & = \hdots = u_p = 0 \\ u_{m-p} & = \hdots = u_{m} = 1 \\ u_{j+p} & = \frac{1}{p}\sum_{i=j}^{j+p-1}\bar{u}_i \quad\quad j=1,\hdots,n-p \end{align*}

where \(p\) is the degree and \(m+1\) the number knots of the desired interpolating spline.

Parameters

[in]	parameters	The input parameters. During interpolation one for each data point.
[in]	degree	The spline degree which is used during the interpolation.
[out]	knots	The output knot vector.

See also

Les Piegl and Wayne Tiller, The NURBS book (2nd ed.), 1997, 9.2.1 Global Curve Interpolation to Point Data

Definition at line 48 of file SplineFitting.h.

  {
    knots.resize(parameters.size()+degree+1);      
 
    for (DenseIndex j=1; j<parameters.size()-degree; ++j)
      knots(j+degree) = parameters.segment(j,degree).mean();
 
    knots.segment(0,degree+1) = KnotVectorType::Zero(degree+1);
    knots.segment(knots.size()-degree-1,degree+1) = KnotVectorType::Ones(degree+1);
  }

KnotAveragingWithDerivatives()

template<typename KnotVectorType , typename ParameterVectorType , typename IndexArray >

void Eigen::KnotAveragingWithDerivatives	(	const ParameterVectorType &	parameters,
		const unsigned int	degree,
		const IndexArray &	derivativeIndices,
		KnotVectorType &	knots
	)

Computes knot averages when derivative constraints are present. Note that this is a technical interpretation of the referenced article since the algorithm contained therein is incorrect as written.

Parameters

[in]	parameters	The parameters at which the interpolation B-Spline will intersect the given interpolation points. The parameters are assumed to be a non-decreasing sequence.
[in]	degree	The degree of the interpolating B-Spline. This must be greater than zero.
[in]	derivativeIndices	The indices corresponding to parameters at which there are derivative constraints. The indices are assumed to be a non-decreasing sequence.
[out]	knots	The calculated knot vector. These will be returned as a non-decreasing sequence

See also

Les A. Piegl, Khairan Rajab, Volha Smarodzinana. 2008. Curve interpolation with directional constraints for engineering design. Engineering with Computers

Definition at line 81 of file SplineFitting.h.

  {
    typedef typename ParameterVectorType::Scalar Scalar;
 
    DenseIndex numParameters = parameters.size();
    DenseIndex numDerivatives = derivativeIndices.size();
 
    if (numDerivatives < 1)
    {
      KnotAveraging(parameters, degree, knots);
      return;
    }
 
    DenseIndex startIndex;
    DenseIndex endIndex;
  
    DenseIndex numInternalDerivatives = numDerivatives;
    
    if (derivativeIndices[0] == 0)
    {
      startIndex = 0;
      --numInternalDerivatives;
    }
    else
    {
      startIndex = 1;
    }
    if (derivativeIndices[numDerivatives - 1] == numParameters - 1)
    {
      endIndex = numParameters - degree;
      --numInternalDerivatives;
    }
    else
    {
      endIndex = numParameters - degree - 1;
    }
 
    // There are (endIndex - startIndex + 1) knots obtained from the averaging
    // and 2 for the first and last parameters.
    DenseIndex numAverageKnots = endIndex - startIndex + 3;
    KnotVectorType averageKnots(numAverageKnots);
    averageKnots[0] = parameters[0];
 
    int newKnotIndex = 0;
    for (DenseIndex i = startIndex; i <= endIndex; ++i)
      averageKnots[++newKnotIndex] = parameters.segment(i, degree).mean();
    averageKnots[++newKnotIndex] = parameters[numParameters - 1];
 
    newKnotIndex = -1;
  
    ParameterVectorType temporaryParameters(numParameters + 1);
    KnotVectorType derivativeKnots(numInternalDerivatives);
    for (DenseIndex i = 0; i < numAverageKnots - 1; ++i)
    {
      temporaryParameters[0] = averageKnots[i];
      ParameterVectorType parameterIndices(numParameters);
      int temporaryParameterIndex = 1;
      for (DenseIndex j = 0; j < numParameters; ++j)
      {
        Scalar parameter = parameters[j];
        if (parameter >= averageKnots[i] && parameter < averageKnots[i + 1])
        {
          parameterIndices[temporaryParameterIndex] = j;
          temporaryParameters[temporaryParameterIndex++] = parameter;
        }
      }
      temporaryParameters[temporaryParameterIndex] = averageKnots[i + 1];
 
      for (int j = 0; j <= temporaryParameterIndex - 2; ++j)
      {
        for (DenseIndex k = 0; k < derivativeIndices.size(); ++k)
        {
          if (parameterIndices[j + 1] == derivativeIndices[k]
              && parameterIndices[j + 1] != 0
              && parameterIndices[j + 1] != numParameters - 1)
          {
            derivativeKnots[++newKnotIndex] = temporaryParameters.segment(j, 3).mean();
            break;
          }
        }
      }
    }
    
    KnotVectorType temporaryKnots(averageKnots.size() + derivativeKnots.size());
 
    std::merge(averageKnots.data(), averageKnots.data() + averageKnots.size(),
               derivativeKnots.data(), derivativeKnots.data() + derivativeKnots.size(),
               temporaryKnots.data());
 
    // Number of knots (one for each point and derivative) plus spline order.
    DenseIndex numKnots = numParameters + numDerivatives + degree + 1;
    knots.resize(numKnots);
 
    knots.head(degree).fill(temporaryKnots[0]);
    knots.tail(degree).fill(temporaryKnots.template tail<1>()[0]);
    knots.segment(degree, temporaryKnots.size()) = temporaryKnots;
  }