spline.h

//  ---------------------------------------------------------------------------
//  This file is part of reSID, a MOS6581 SID emulator engine.
//  Copyright (C) 2004  Dag Lem <resid@nimrod.no>
//
//  This program is free software; you can redistribute it and/or modify
//  it under the terms of the GNU General Public License as published by
//  the Free Software Foundation; either version 2 of the License, or
//  (at your option) any later version.
//
//  This program is distributed in the hope that it will be useful,
//  but WITHOUT ANY WARRANTY; without even the implied warranty of
//  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
//  GNU General Public License for more details.
//
//  You should have received a copy of the GNU General Public License
//  along with this program; if not, write to the Free Software
//  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
//  ---------------------------------------------------------------------------

#ifndef __SPLINE_H__
#define __SPLINE_H__

// Our objective is to construct a smooth interpolating single-valued function
// y = f(x).
//
// Catmull-Rom splines are widely used for interpolation, however these are
// parametric curves [x(t) y(t) ...] and can not be used to directly calculate
// y = f(x).
// For a discussion of Catmull-Rom splines see Catmull, E., and R. Rom,
// "A Class of Local Interpolating Splines", Computer Aided Geometric Design.
//
// Natural cubic splines are single-valued functions, and have been used in
// several applications e.g. to specify gamma curves for image display.
// These splines do not afford local control, and a set of linear equations
// including all interpolation points must be solved before any point on the
// curve can be calculated. The lack of local control makes the splines
// more difficult to handle than e.g. Catmull-Rom splines, and real-time
// interpolation of a stream of data points is not possible.
// For a discussion of natural cubic splines, see e.g. Kreyszig, E., "Advanced
// Engineering Mathematics".
//
// Our approach is to approximate the properties of Catmull-Rom splines for
// piecewice cubic polynomials f(x) = ax^3 + bx^2 + cx + d as follows:
// Each curve segment is specified by four interpolation points,
// p0, p1, p2, p3.
// The curve between p1 and p2 must interpolate both p1 and p2, and in addition
//   f'(p1.x) = k1 = (p2.y - p0.y)/(p2.x - p0.x) and
//   f'(p2.x) = k2 = (p3.y - p1.y)/(p3.x - p1.x).
//
// The constraints are expressed by the following system of linear equations
//
//   [ 1  xi    xi^2    xi^3 ]   [ d ]    [ yi ]
//   [     1  2*xi    3*xi^2 ] * [ c ] =  [ ki ]
//   [ 1  xj    xj^2    xj^3 ]   [ b ]    [ yj ]
//   [     1  2*xj    3*xj^2 ]   [ a ]    [ kj ]
//
// Solving using Gaussian elimination and back substitution, setting
// dy = yj - yi, dx = xj - xi, we get
// 
//   a = ((ki + kj) - 2*dy/dx)/(dx*dx);
//   b = ((kj - ki)/dx - 3*(xi + xj)*a)/2;
//   c = ki - (3*xi*a + 2*b)*xi;
//   d = yi - ((xi*a + b)*xi + c)*xi;
//
// Having calculated the coefficients of the cubic polynomial we have the
// choice of evaluation by brute force
//
//   for (x = x1; x <= x2; x += res) {
//     y = ((a*x + b)*x + c)*x + d;
//     plot(x, y);
//   }
//
// or by forward differencing
//
//   y = ((a*x1 + b)*x1 + c)*x1 + d;
//   dy = (3*a*(x1 + res) + 2*b)*x1*res + ((a*res + b)*res + c)*res;
//   d2y = (6*a*(x1 + res) + 2*b)*res*res;
//   d3y = 6*a*res*res*res;
//     
//   for (x = x1; x <= x2; x += res) {
//     plot(x, y);
//     y += dy; dy += d2y; d2y += d3y;
//   }
//
// See Foley, Van Dam, Feiner, Hughes, "Computer Graphics, Principles and
// Practice" for a discussion of forward differencing.
//
// If we have a set of interpolation points p0, ..., pn, we may specify
// curve segments between p0 and p1, and between pn-1 and pn by using the
// following constraints:
//   f''(p0.x) = 0 and
//   f''(pn.x) = 0.
//
// Substituting the results for a and b in
//
//   2*b + 6*a*xi = 0
//
// we get
//
//   ki = (3*dy/dx - kj)/2;
//
// or by substituting the results for a and b in
//
//   2*b + 6*a*xj = 0
//
// we get
//
//   kj = (3*dy/dx - ki)/2;
//
// Finally, if we have only two interpolation points, the cubic polynomial
// will degenerate to a straight line if we set
//
//   ki = kj = dy/dx;
//


#if SPLINE_BRUTE_FORCE
#define interpolate_segment interpolate_brute_force
#else
#define interpolate_segment interpolate_forward_difference
#endif


// ----------------------------------------------------------------------------
// Calculation of coefficients.
// ----------------------------------------------------------------------------
inline
void cubic_coefficients(double x1, double y1, double x2, double y2,
			double k1, double k2,
			double& a, double& b, double& c, double& d)
{
  double dx = x2 - x1, dy = y2 - y1;

  a = ((k1 + k2) - 2*dy/dx)/(dx*dx);
  b = ((k2 - k1)/dx - 3*(x1 + x2)*a)/2;
  c = k1 - (3*x1*a + 2*b)*x1;
  d = y1 - ((x1*a + b)*x1 + c)*x1;
}

// ----------------------------------------------------------------------------
// Evaluation of cubic polynomial by brute force.
// ----------------------------------------------------------------------------
template<class PointPlotter>
inline
void interpolate_brute_force(double x1, double y1, double x2, double y2,
			     double k1, double k2,
			     PointPlotter plot, double res)
{
  double a, b, c, d;
  cubic_coefficients(x1, y1, x2, y2, k1, k2, a, b, c, d);
  
  // Calculate each point.
  for (double x = x1; x <= x2; x += res) {
    double y = ((a*x + b)*x + c)*x + d;
    plot(x, y);
  }
}

// ----------------------------------------------------------------------------
// Evaluation of cubic polynomial by forward differencing.
// ----------------------------------------------------------------------------
template<class PointPlotter>
inline
void interpolate_forward_difference(double x1, double y1, double x2, double y2,
				    double k1, double k2,
				    PointPlotter plot, double res)
{
  double a, b, c, d;
  cubic_coefficients(x1, y1, x2, y2, k1, k2, a, b, c, d);
  
  double y = ((a*x1 + b)*x1 + c)*x1 + d;
  double dy = (3*a*(x1 + res) + 2*b)*x1*res + ((a*res + b)*res + c)*res;
  double d2y = (6*a*(x1 + res) + 2*b)*res*res;
  double d3y = 6*a*res*res*res;
    
  // Calculate each point.
  for (double x = x1; x <= x2; x += res) {
    plot(x, y);
    y += dy; dy += d2y; d2y += d3y;
  }
}

template<class PointIter>
inline
double x(PointIter p)
{
  return (*p)[0];
}

template<class PointIter>
inline
double y(PointIter p)
{
  return (*p)[1];
}

// ----------------------------------------------------------------------------
// Evaluation of complete interpolating function.
// Note that since each curve segment is controlled by four points, the
// end points will not be interpolated. If extra control points are not
// desirable, the end points can simply be repeated to ensure interpolation.
// Note also that points of non-differentiability and discontinuity can be
// introduced by repeating points.
// ----------------------------------------------------------------------------
template<class PointIter, class PointPlotter>
inline
void interpolate(PointIter p0, PointIter pn, PointPlotter plot, double res)
{
  double k1, k2;

  // Set up points for first curve segment.
  PointIter p1 = p0; ++p1;
  PointIter p2 = p1; ++p2;
  PointIter p3 = p2; ++p3;

  // Draw each curve segment.
  for (; p2 != pn; ++p0, ++p1, ++p2, ++p3) {
    // p1 and p2 equal; single point.
    if (x(p1) == x(p2)) {
      continue;
    }
    // Both end points repeated; straight line.
    if (x(p0) == x(p1) && x(p2) == x(p3)) {
      k1 = k2 = (y(p2) - y(p1))/(x(p2) - x(p1));
    }
    // p0 and p1 equal; use f''(x1) = 0.
    else if (x(p0) == x(p1)) {
      k2 = (y(p3) - y(p1))/(x(p3) - x(p1));
      k1 = (3*(y(p2) - y(p1))/(x(p2) - x(p1)) - k2)/2;
    }
    // p2 and p3 equal; use f''(x2) = 0.
    else if (x(p2) == x(p3)) {
      k1 = (y(p2) - y(p0))/(x(p2) - x(p0));
      k2 = (3*(y(p2) - y(p1))/(x(p2) - x(p1)) - k1)/2;
    }
    // Normal curve.
    else {
      k1 = (y(p2) - y(p0))/(x(p2) - x(p0));
      k2 = (y(p3) - y(p1))/(x(p3) - x(p1));
    }

    interpolate_segment(x(p1), y(p1), x(p2), y(p2), k1, k2, plot, res);
  }
}

// ----------------------------------------------------------------------------
// Class for plotting integers into an array.
// ----------------------------------------------------------------------------
template<class F>
class PointPlotter
{
 protected:
  F* f;

 public:
  PointPlotter(F* arr) : f(arr)
  {
  }

  void operator ()(double x, double y)
  {
    // Clamp negative values to zero.
    if (y < 0) {
      y = 0;
    }

    f[F(x)] = F(y);
  }
};


#endif // not __SPLINE_H__