Are there any good libraries for solving cubic splines in C++?

Question

I'm looking for a good C++ library to give me functions to solve for large cubic splines (on the order of 1000 points) anyone know one?

Answer 1

Write your own. Here is spline() function I wrote based on excellent wiki algorithm:

#include<iostream>
#include<vector>
#include<algorithm>
#include<cmath>
using namespace std;

using vec = vector<double>;

struct SplineSet{
    double a;
    double b;
    double c;
    double d;
    double x;
};

vector<SplineSet> spline(vec &x, vec &y)
{
    int n = x.size()-1;
    vec a;
    a.insert(a.begin(), y.begin(), y.end());
    vec b(n);
    vec d(n);
    vec h;

    for(int i = 0; i < n; ++i)
        h.push_back(x[i+1]-x[i]);

    vec alpha;
    alpha.push_back(0);
    for(int i = 1; i < n; ++i)
        alpha.push_back( 3*(a[i+1]-a[i])/h[i] - 3*(a[i]-a[i-1])/h[i-1]  );

    vec c(n+1);
    vec l(n+1);
    vec mu(n+1);
    vec z(n+1);
    l[0] = 1;
    mu[0] = 0;
    z[0] = 0;

    for(int i = 1; i < n; ++i)
    {
        l[i] = 2 *(x[i+1]-x[i-1])-h[i-1]*mu[i-1];
        mu[i] = h[i]/l[i];
        z[i] = (alpha[i]-h[i-1]*z[i-1])/l[i];
    }

    l[n] = 1;
    z[n] = 0;
    c[n] = 0;

    for(int j = n-1; j >= 0; --j)
    {
        c[j] = z [j] - mu[j] * c[j+1];
        b[j] = (a[j+1]-a[j])/h[j]-h[j]*(c[j+1]+2*c[j])/3;
        d[j] = (c[j+1]-c[j])/3/h[j];
    }

    vector<SplineSet> output_set(n);
    for(int i = 0; i < n; ++i)
    {
        output_set[i].a = a[i];
        output_set[i].b = b[i];
        output_set[i].c = c[i];
        output_set[i].d = d[i];
        output_set[i].x = x[i];
    }
    return output_set;
}

int main()
{
    vec x(11);
    vec y(11);
    for(int i = 0; i < x.size(); ++i)
    {
        x[i] = i;
        y[i] = sin(i);
    }

    vector<SplineSet> cs = spline(x, y);
    for(int i = 0; i < cs.size(); ++i)
        cout << cs[i].d << "\t" << cs[i].c << "\t" << cs[i].b << "\t" << cs[i].a << endl;
}

Answer 2

Try the Cubic B-Spline library:

• https://github.com/NCAR/bspline

and ALGLIB:

• http://www.alglib.net/interpolation/spline3.php

Answer 3

I had to write spline routine for an "entity" that was following a path (series of connected waypoints) in a game I am working on.

I created a base class to handle a "SplineInterface" and the created two derived classes, one based on the classic spline technique (e.g. Sedgewick/Algorithms) an a second one based on Bezier Splines.

Here is the code. It is a single header file, which contains all the splining classes:

#ifndef __SplineCommon__
#define __SplineCommon__

#include "CommonSTL.h"
#include "CommonProject.h"
#include "MathUtilities.h"

/* A Spline base class. */
class SplineBase
{
private:
   vector<Vec2> _points;
   bool _elimColinearPoints;

protected:


protected:
   /* OVERRIDE THESE FUNCTIONS */
   virtual void ResetDerived() = 0;

   enum
   {
      NOM_SIZE = 32,
   };

public:

   SplineBase()
   {
      _points.reserve(NOM_SIZE);
      _elimColinearPoints = true;
   }

   const vector<Vec2>& GetPoints() { return _points; }
   bool GetElimColinearPoints() { return _elimColinearPoints; }
   void SetElimColinearPoints(bool elim) { _elimColinearPoints = elim; }


   /* OVERRIDE THESE FUNCTIONS */
   virtual Vec2 Eval(int seg, double t) = 0;
   virtual bool ComputeSpline() = 0;
   virtual void DumpDerived() {}

   /* Clear out all the data.
    */
   void Reset()
   {
      _points.clear();
      ResetDerived();
   }

   void AddPoint(const Vec2& pt)
   {
      // If this new point is colinear with the two previous points,
      // pop off the last point and add this one instead.
      if(_elimColinearPoints && _points.size() > 2)
      {
         int N = _points.size()-1;
         Vec2 p0 = _points[N-1] - _points[N-2];
         Vec2 p1 = _points[N] - _points[N-1];
         Vec2 p2 = pt - _points[N];
         // We test for colinearity by comparing the slopes
         // of the two lines.  If the slopes are the same,
         // we assume colinearity.
         float32 delta = (p2.y-p1.y)*(p1.x-p0.x)-(p1.y-p0.y)*(p2.x-p1.x);
         if(MathUtilities::IsNearZero(delta))
         {
            _points.pop_back();
         }
      }
      _points.push_back(pt);
   }

   void Dump(int segments = 5)
   {
      assert(segments > 1);

      cout << "Original Points (" << _points.size() << ")" << endl;
      cout << "-----------------------------" << endl;
      for(int idx = 0; idx < _points.size(); ++idx)
      {
         cout << "[" << idx << "]" << "  " << _points[idx] << endl;
      }

      cout << "-----------------------------" << endl;
      DumpDerived();

      cout << "-----------------------------" << endl;
      cout << "Evaluating Spline at " << segments << " points." << endl;
      for(int idx = 0; idx < _points.size()-1; idx++)
      {
         cout << "---------- " << "From " <<  _points[idx] << " to " << _points[idx+1] << "." << endl;
         for(int tIdx = 0; tIdx < segments+1; ++tIdx)
         {
            double t = tIdx*1.0/segments;
            cout << "[" << tIdx << "]" << "   ";
            cout << "[" << t*100 << "%]" << "   ";
            cout << " --> " << Eval(idx,t);
            cout << endl;
         }
      }
   }
};

class ClassicSpline : public SplineBase
{
private:
   /* The system of linear equations found by solving
    * for the 3 order spline polynomial is given by:
    * A*x = b.  The "x" is represented by _xCol and the
    * "b" is represented by _bCol in the code.
    *
    * The "A" is formulated with diagonal elements (_diagElems) and
    * symmetric off-diagonal elements (_offDiagElemns).  The
    * general structure (for six points) looks like:
    *
    *
    *  |  d1  u1   0   0   0  |      | p1 |    | w1 |
    *  |  u1  d2   u2  0   0  |      | p2 |    | w2 |
    *  |  0   u2   d3  u3  0  |   *  | p3 |  = | w3 |
    *  |  0   0    u3  d4  u4 |      | p4 |    | w4 |
    *  |  0   0    0   u4  d5 |      | p5 |    | w5 |
    *
    *
    *  The general derivation for this can be found
    *  in Robert Sedgewick's "Algorithms in C++".
    *
    */
   vector<double> _xCol;
   vector<double> _bCol;
   vector<double> _diagElems;
   vector<double> _offDiagElems;
public:
   ClassicSpline()
   {
      _xCol.reserve(NOM_SIZE);
      _bCol.reserve(NOM_SIZE);
      _diagElems.reserve(NOM_SIZE);
      _offDiagElems.reserve(NOM_SIZE);
   }

   /* Evaluate the spline for the ith segment
    * for parameter.  The value of parameter t must
    * be between 0 and 1.
    */
   inline virtual Vec2 Eval(int seg, double t)
   {
      const vector<Vec2>& points = GetPoints();

      assert(t >= 0);
      assert(t <= 1.0);
      assert(seg >= 0);
      assert(seg < (points.size()-1));

      const double ONE_OVER_SIX = 1.0/6.0;
      double oneMinust = 1.0 - t;
      double t3Minust = t*t*t-t;
      double oneMinust3minust = oneMinust*oneMinust*oneMinust-oneMinust;
      double deltaX = points[seg+1].x - points[seg].x;
      double yValue = t * points[seg + 1].y +
      oneMinust*points[seg].y +
      ONE_OVER_SIX*deltaX*deltaX*(t3Minust*_xCol[seg+1] - oneMinust3minust*_xCol[seg]);
      double xValue = t*(points[seg+1].x-points[seg].x) + points[seg].x;
      return Vec2(xValue,yValue);
   }


   /* Clear out all the data.
    */
   virtual void ResetDerived()
   {
      _diagElems.clear();
      _bCol.clear();
      _xCol.clear();
      _offDiagElems.clear();
   }


   virtual bool ComputeSpline()
   {
      const vector<Vec2>& p = GetPoints();


      _bCol.resize(p.size());
      _xCol.resize(p.size());
      _diagElems.resize(p.size());

      for(int idx = 1; idx < p.size(); ++idx)
      {
         _diagElems[idx] = 2*(p[idx+1].x-p[idx-1].x);
      }
      for(int idx = 0; idx < p.size(); ++idx)
      {
         _offDiagElems[idx] = p[idx+1].x - p[idx].x;
      }
      for(int idx = 1; idx < p.size(); ++idx)
      {
         _bCol[idx] = 6.0*((p[idx+1].y-p[idx].y)/_offDiagElems[idx] -
                           (p[idx].y-p[idx-1].y)/_offDiagElems[idx-1]);
      }
      _xCol[0] = 0.0;
      _xCol[p.size()-1] = 0.0;
      for(int idx = 1; idx < p.size()-1; ++idx)
      {
         _bCol[idx+1] = _bCol[idx+1] - _bCol[idx]*_offDiagElems[idx]/_diagElems[idx];
         _diagElems[idx+1] = _diagElems[idx+1] - _offDiagElems[idx]*_offDiagElems[idx]/_diagElems[idx];
      }
      for(int idx = (int)p.size()-2; idx > 0; --idx)
      {
         _xCol[idx] = (_bCol[idx] - _offDiagElems[idx]*_xCol[idx+1])/_diagElems[idx];
      }
      return true;
   }
};

/* Bezier Spline Implementation
 * Based on this article:
 * http://www.particleincell.com/blog/2012/bezier-splines/
 */
class BezierSpine : public SplineBase
{
private:
   vector<Vec2> _p1Points;
   vector<Vec2> _p2Points;
public:
   BezierSpine()
   {
      _p1Points.reserve(NOM_SIZE);
      _p2Points.reserve(NOM_SIZE);
   }

   /* Evaluate the spline for the ith segment
    * for parameter.  The value of parameter t must
    * be between 0 and 1.
    */
   inline virtual Vec2 Eval(int seg, double t)
   {
      assert(seg < _p1Points.size());
      assert(seg < _p2Points.size());

      double omt = 1.0 - t;

      Vec2 p0 = GetPoints()[seg];
      Vec2 p1 = _p1Points[seg];
      Vec2 p2 = _p2Points[seg];
      Vec2 p3 = GetPoints()[seg+1];

      double xVal = omt*omt*omt*p0.x + 3*omt*omt*t*p1.x +3*omt*t*t*p2.x+t*t*t*p3.x;
      double yVal = omt*omt*omt*p0.y + 3*omt*omt*t*p1.y +3*omt*t*t*p2.y+t*t*t*p3.y;
      return Vec2(xVal,yVal);
   }

   /* Clear out all the data.
    */
   virtual void ResetDerived()
   {
      _p1Points.clear();
      _p2Points.clear();
   }


   virtual bool ComputeSpline()
   {
      const vector<Vec2>& p = GetPoints();

      int N = (int)p.size()-1;
      _p1Points.resize(N);
      _p2Points.resize(N);
      if(N == 0)
         return false;

      if(N == 1)
      {  // Only 2 points...just create a straight line.
         // Constraint:  3*P1 = 2*P0 + P3
         _p1Points[0] = (2.0/3.0*p[0] + 1.0/3.0*p[1]);
         // Constraint:  P2 = 2*P1 - P0
         _p2Points[0] = 2.0*_p1Points[0] - p[0];
         return true;
      }

      /*rhs vector*/
      vector<Vec2> a(N);
      vector<Vec2> b(N);
      vector<Vec2> c(N);
      vector<Vec2> r(N);

      /*left most segment*/
      a[0].x = 0;
      b[0].x = 2;
      c[0].x = 1;
      r[0].x = p[0].x+2*p[1].x;

      a[0].y = 0;
      b[0].y = 2;
      c[0].y = 1;
      r[0].y = p[0].y+2*p[1].y;

      /*internal segments*/
      for (int i = 1; i < N - 1; i++)
      {
         a[i].x=1;
         b[i].x=4;
         c[i].x=1;
         r[i].x = 4 * p[i].x + 2 * p[i+1].x;

         a[i].y=1;
         b[i].y=4;
         c[i].y=1;
         r[i].y = 4 * p[i].y + 2 * p[i+1].y;
      }

      /*right segment*/
      a[N-1].x = 2;
      b[N-1].x = 7;
      c[N-1].x = 0;
      r[N-1].x = 8*p[N-1].x+p[N].x;

      a[N-1].y = 2;
      b[N-1].y = 7;
      c[N-1].y = 0;
      r[N-1].y = 8*p[N-1].y+p[N].y;


      /*solves Ax=b with the Thomas algorithm (from Wikipedia)*/
      for (int i = 1; i < N; i++)
      {
         double m;

         m = a[i].x/b[i-1].x;
         b[i].x = b[i].x - m * c[i - 1].x;
         r[i].x = r[i].x - m * r[i-1].x;

         m = a[i].y/b[i-1].y;
         b[i].y = b[i].y - m * c[i - 1].y;
         r[i].y = r[i].y - m * r[i-1].y;
      }

      _p1Points[N-1].x = r[N-1].x/b[N-1].x;
      _p1Points[N-1].y = r[N-1].y/b[N-1].y;
      for (int i = N - 2; i >= 0; --i)
      {
         _p1Points[i].x = (r[i].x - c[i].x * _p1Points[i+1].x) / b[i].x;
         _p1Points[i].y = (r[i].y - c[i].y * _p1Points[i+1].y) / b[i].y;
      }

      /*we have p1, now compute p2*/
      for (int i=0;i<N-1;i++)
      {
         _p2Points[i].x=2*p[i+1].x-_p1Points[i+1].x;
         _p2Points[i].y=2*p[i+1].y-_p1Points[i+1].y;
      }

      _p2Points[N-1].x = 0.5 * (p[N].x+_p1Points[N-1].x);
      _p2Points[N-1].y = 0.5 * (p[N].y+_p1Points[N-1].y);

      return true;
   }

   virtual void DumpDerived()
   {
      cout << " Control Points " << endl;
      for(int idx = 0; idx < _p1Points.size(); idx++)
      {
         cout << "[" << idx << "]  ";
         cout << "P1: " << _p1Points[idx];
         cout << "   ";
         cout << "P2: " << _p2Points[idx];
         cout << endl;
      }
   }
};


#endif /* defined(__SplineCommon__) */

Some Notes

• The classic spline will crash if you give it a vertical set of points. That is why I created the Bezier...I have lots of vertical lines/paths to follow. It could be modified to just give a straight line.
• The base class has an option to remove collinear points as you add them. This uses a simple slope comparison of two lines to figure out if they are on the same line. You don't have to do this, but for long paths that are straight lines, it cuts down on cycles. When you do a lot of pathfinding on a regular-spaced graph, you tend to get a lot of continuous segments.

Here is an example of using the Bezier Spline:

/* Smooth the points on the path so that turns look
 * more natural.  We'll only smooth the first few 
 * points.  Most of the time, the full path will not
 * be executed anyway...why waste cycles.
 */
void SmoothPath(vector<Vec2>& path, int32 divisions)
{
   const int SMOOTH_POINTS = 6;

   BezierSpine spline;

   if(path.size() < 2)
      return;

   // Cache off the first point.  If the first point is removed,
   // the we occasionally run into problems if the collision detection
   // says the first node is occupied but the splined point is too
   // close, so the FSM "spins" trying to find a sensor cell that is
   // not occupied.
   //   Vec2 firstPoint = path.back();
   //   path.pop_back();
   // Grab the points.
   for(int idx = 0; idx < SMOOTH_POINTS && path.size() > 0; idx++)
   {
      spline.AddPoint(path.back());
      path.pop_back();
   }
   // Smooth them.
   spline.ComputeSpline();
   // Push them back in.
   for(int idx = spline.GetPoints().size()-2; idx >= 0; --idx)
   {
      for(int division = divisions-1; division >= 0; --division)
      {
         double t = division*1.0/divisions;
         path.push_back(spline.Eval(idx, t));
      }
   }
   // Push back in the original first point.
   //   path.push_back(firstPoint);
}

Notes

• While the whole path could be smoothed, in this application, since the path was changing every so often, it was better to just smooth the first points and then connect it up.
• The points are loaded in "reverse" order into the path vector. This may or may not save cycles (I've slept since then).

This code is part of a much larger code base, but you can download it all on github and see a blog entry about it here.

You can look at this in action in this video.

Answer 4

Take a look at David Eberly's GeometricTools.com. I'm just starting, but code and doc are so far of outstanding quality.
(He has books too: Geometric tools for computer graphics, 3D game engine design.)