SCJMapper-V2/OGL/CubicSpline.cs

//
// Author: Ryan Seghers
//
// Copyright (C) 2013-2014 Ryan Seghers
//
// Permission is hereby granted, free of charge, to any person obtaining
// a copy of this software and associated documentation files (the
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// display, perform, create derivative works from and/or sell copies of 
// the Software, both in source and object code form, and to
// permit persons to whom the Software is furnished to do so, subject to
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// 
// The above copyright notice and this permission notice shall be
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// 
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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using System;
using OpenTK;

namespace SCJMapper_V2
{
	/// <summary>
	/// Cubic spline interpolation.
	/// Call Fit (or use the corrector constructor) to compute spline coefficients, then Eval to evaluate the spline at other X coordinates.
	/// </summary>
	/// <remarks>
	/// <para>
	/// This is implemented based on the wikipedia article:
	/// http://en.wikipedia.org/wiki/Spline_interpolation
	/// I'm not sure I have the right to include a copy of the article so the equation numbers referenced in 
	/// comments will end up being wrong at some point.
	/// </para>
	/// <para>
	/// This is not optimized, and is not MT safe.
	/// This can extrapolate off the ends of the splines.
	/// You must provide points in X sort order.
	/// </para>
	/// </remarks>
	public class CubicSpline
	{
		#region Fields

		// N-1 spline coefficients for N points
		private float[] a;
		private float[] b;

		// Save the original x and y for Eval
		private float[] xOrig;
		private float[] yOrig;

		#endregion

		#region Ctor

		/// <summary>
		/// Default ctor.
		/// </summary>
		public CubicSpline()
		{
		}

		/// <summary>
		/// Construct and call Fit.
		/// </summary>
		/// <param name="x">Input. X coordinates to fit.</param>
		/// <param name="y">Input. Y coordinates to fit.</param>
		/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
		/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
		public CubicSpline(float[] x, float[] y, float startSlope = float.NaN, float endSlope = float.NaN, bool debug = false)
		{
			Fit(x, y, startSlope, endSlope );
		}

    /// <summary>
    /// Construct and call Fit.
    /// </summary>
    /// <param name="xyPts">Input. XY coordinates to fit</param>
    /// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
    /// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
    public CubicSpline( Vector2[] xyPts, float startSlope = float.NaN, float endSlope = float.NaN, bool debug = false )
    {
      float[] x = new float[xyPts.Length];
      float[] y = new float[xyPts.Length];
      for ( int i = 0; i < xyPts.Length; i++ ) {
        x[i] = xyPts[i].X; y[i] = xyPts[i].Y;
      }
      Fit( x, y, startSlope, endSlope );
    }


    #endregion

    #region Private Methods

    /// <summary>
    /// Throws if Fit has not been called.
    /// </summary>
    private void CheckAlreadyFitted( )
		{
			if (a == null) throw new Exception("Fit must be called before you can evaluate.");
		}

		private int _lastIndex = 0;

		/// <summary>
		/// Find where in xOrig the specified x falls, by simultaneous traverse.
		/// This allows xs to be less than x[0] and/or greater than x[n-1]. So allows extrapolation.
		/// This keeps state, so requires that x be sorted and xs called in ascending order, and is not multi-thread safe.
		/// </summary>
		private int GetNextXIndex(float x)
		{
			if (x < xOrig[_lastIndex])
			{
				throw new ArgumentException("The X values to evaluate must be sorted.");
			}

			while ((_lastIndex < xOrig.Length - 2) && (x > xOrig[_lastIndex + 1]))
			{
				_lastIndex++;
			}

			return _lastIndex;
		}

		/// <summary>
		/// Evaluate the specified x value using the specified spline.
		/// </summary>
		/// <param name="x">The x value.</param>
		/// <param name="j">Which spline to use.</param>
		/// <param name="debug">Turn on console output. Default is false.</param>
		/// <returns>The y value.</returns>
		private float EvalSpline(float x, int j)
		{
			float dx = xOrig[j + 1] - xOrig[j];
			float t = (x - xOrig[j]) / dx;
			float y = (1 - t) * yOrig[j] + t * yOrig[j + 1] + t * (1 - t) * (a[j] * (1 - t) + b[j] * t); // equation 9
			return y;
		}

		#endregion

		#region Fit*

		/// <summary>
		/// Fit x,y and then eval at points xs and return the corresponding y's.
		/// This does the "natural spline" style for ends.
		/// This can extrapolate off the ends of the splines.
		/// You must provide points in X sort order.
		/// </summary>
		/// <param name="x">Input. X coordinates to fit.</param>
		/// <param name="y">Input. Y coordinates to fit.</param>
		/// <param name="xs">Input. X coordinates to evaluate the fitted curve at.</param>
		/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
		/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
		/// <param name="debug">Turn on console output. Default is false.</param>
		/// <returns>The computed y values for each xs.</returns>
		public float[] FitAndEval(float[] x, float[] y, float[] xs, float startSlope = float.NaN, float endSlope = float.NaN)
		{
			Fit(x, y, startSlope, endSlope);
			return Eval(xs);
		}

		/// <summary>
		/// Compute spline coefficients for the specified x,y points.
		/// This does the "natural spline" style for ends.
		/// This can extrapolate off the ends of the splines.
		/// You must provide points in X sort order.
		/// </summary>
		/// <param name="x">Input. X coordinates to fit.</param>
		/// <param name="y">Input. Y coordinates to fit.</param>
		/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
		/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
		/// <param name="debug">Turn on console output. Default is false.</param>
		public void Fit(float[] x, float[] y, float startSlope = float.NaN, float endSlope = float.NaN)
		{
			if (Single.IsInfinity(startSlope) || Single.IsInfinity(endSlope))
			{
				throw new Exception("startSlope and endSlope cannot be infinity.");
			}

			// Save x and y for eval
			this.xOrig = x;
			this.yOrig = y;

			int n = x.Length;
			float[] r = new float[n]; // the right hand side numbers: wikipedia page overloads b

			TriDiagonalMatrixF m = new TriDiagonalMatrixF(n);
			float dx1, dx2, dy1, dy2;

			// First row is different (equation 16 from the article)
			if (float.IsNaN(startSlope))
			{
				dx1 = x[1] - x[0];
				m.C[0] = 1.0f / dx1;
				m.B[0] = 2.0f * m.C[0];
				r[0] = 3 * (y[1] - y[0]) / (dx1 * dx1);
			}
			else
			{
				m.B[0] = 1;
				r[0] = startSlope;
			}

			// Body rows (equation 15 from the article)
			for (int i = 1; i < n - 1; i++)
			{
				dx1 = x[i] - x[i - 1];
				dx2 = x[i + 1] - x[i];

				m.A[i] = 1.0f / dx1;
				m.C[i] = 1.0f / dx2;
				m.B[i] = 2.0f * (m.A[i] + m.C[i]);

				dy1 = y[i] - y[i - 1];
				dy2 = y[i + 1] - y[i];
				r[i] = 3 * (dy1 / (dx1 * dx1) + dy2 / (dx2 * dx2));
			}

			// Last row also different (equation 17 from the article)
			if (float.IsNaN(endSlope))
			{
				dx1 = x[n - 1] - x[n - 2];
				dy1 = y[n - 1] - y[n - 2];
				m.A[n - 1] = 1.0f / dx1;
				m.B[n - 1] = 2.0f * m.A[n - 1];
				r[n - 1] = 3 * (dy1 / (dx1 * dx1));
			}
			else
			{
				m.B[n - 1] = 1;
				r[n - 1] = endSlope;
			}

			// k is the solution to the matrix
			float[] k = m.Solve(r);

			// a and b are each spline's coefficients
			this.a = new float[n - 1];
			this.b = new float[n - 1];

			for (int i = 1; i < n; i++)
			{
				dx1 = x[i] - x[i - 1];
				dy1 = y[i] - y[i - 1];
				a[i - 1] = k[i - 1] * dx1 - dy1; // equation 10 from the article
				b[i - 1] = -k[i] * dx1 + dy1; // equation 11 from the article
			}
		}

		#endregion

		#region Eval*

    /// <summary>
    /// Evaluate the spline at the specified x coordinate.
    /// This can extrapolate off the ends of the splines.
    /// The spline must already be computed before calling this, meaning you must have already called Fit() or FitAndEval().
    /// </summary>
    /// <param name="x">Input. X coordinate to evaluate the fitted curve at.</param>
    /// <returns>The computed y values for x.</returns>
    public float Eval( float x )
    {
      CheckAlreadyFitted( );

      float y;
      _lastIndex = 0; // Reset simultaneous traversal in case there are multiple calls

      int j = GetNextXIndex( x );

      // Evaluate using j'th spline
      y = EvalSpline( x, j );

      return y;
    }


    /// <summary>
    /// Evaluate the spline at the specified x coordinates.
    /// This can extrapolate off the ends of the splines.
    /// You must provide X's in ascending order.
    /// The spline must already be computed before calling this, meaning you must have already called Fit() or FitAndEval().
    /// </summary>
    /// <param name="x">Input. X coordinates to evaluate the fitted curve at.</param>
    /// <returns>The computed y values for each x.</returns>
    public float[] Eval( float[] x )
    {
      CheckAlreadyFitted( );

      int n = x.Length;
      float[] y = new float[n];
      _lastIndex = 0; // Reset simultaneous traversal in case there are multiple calls

      for ( int i = 0; i < n; i++ ) {
        // Find which spline can be used to compute this x (by simultaneous traverse)
        int j = GetNextXIndex( x[i] );

        // Evaluate using j'th spline
        y[i] = EvalSpline( x[i], j );
      }

      return y;
    }

    /// <summary>
		/// Evaluate (compute) the slope of the spline at the specified x coordinates.
		/// This can extrapolate off the ends of the splines.
		/// You must provide X's in ascending order.
		/// The spline must already be computed before calling this, meaning you must have already called Fit() or FitAndEval().
		/// </summary>
		/// <param name="x">Input. X coordinates to evaluate the fitted curve at.</param>
		/// <returns>The computed y values for each x.</returns>
		public float[] EvalSlope(float[] x )
		{
			CheckAlreadyFitted();

			int n = x.Length;
			float[] qPrime = new float[n];
			_lastIndex = 0; // Reset simultaneous traversal in case there are multiple calls

			for (int i = 0; i < n; i++)
			{
				// Find which spline can be used to compute this x (by simultaneous traverse)
				int j = GetNextXIndex(x[i]);

				// Evaluate using j'th spline
				float dx = xOrig[j + 1] - xOrig[j];
				float dy = yOrig[j + 1] - yOrig[j];
				float t = (x[i] - xOrig[j]) / dx;

				// From equation 5 we could also compute q' (qp) which is the slope at this x
				qPrime[i] = dy / dx
					+ (1 - 2 * t) * (a[j] * (1 - t) + b[j] * t) / dx
					+ t * (1 - t) * (b[j] - a[j]) / dx;

			}

			return qPrime;
		}

		#endregion

		#region Static Methods

		/// <summary>
		/// Static all-in-one method to fit the splines and evaluate at X coordinates.
		/// </summary>
		/// <param name="x">Input. X coordinates to fit.</param>
		/// <param name="y">Input. Y coordinates to fit.</param>
		/// <param name="xs">Input. X coordinates to evaluate the fitted curve at.</param>
		/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
		/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
		/// <returns>The computed y values for each xs.</returns>
		public static float[] Compute(float[] x, float[] y, float[] xs, float startSlope = float.NaN, float endSlope = float.NaN )
		{
			CubicSpline spline = new CubicSpline();
			return spline.FitAndEval(x, y, xs, startSlope, endSlope );
		}

		/// <summary>
		/// Fit the input x,y points using a 'geometric' strategy so that y does not have to be a single-valued
		/// function of x.
		/// </summary>
		/// <param name="x">Input x coordinates.</param>
		/// <param name="y">Input y coordinates, do not need to be a single-valued function of x.</param>
		/// <param name="nOutputPoints">How many output points to create.</param>
		/// <param name="xs">Output (interpolated) x values.</param>
		/// <param name="ys">Output (interpolated) y values.</param>
		public static void FitGeometric(float[] x, float[] y, int nOutputPoints, out float[] xs, out float[] ys)
		{
			// Compute distances
			int n = x.Length;
			float[] dists = new float[n]; // cumulative distance
			dists[0] = 0;
			float totalDist = 0;

			for (int i = 1; i < n; i++)
			{
				float dx = x[i] - x[i - 1];
				float dy = y[i] - y[i - 1];
				float dist = (float)Math.Sqrt(dx * dx + dy * dy);
				totalDist += dist;
				dists[i] = totalDist;
			}

			// Create 'times' to interpolate to
			float dt = totalDist / (nOutputPoints - 1);
			float[] times = new float[nOutputPoints];
			times[0] = 0;

			for (int i = 1; i < nOutputPoints; i++)
			{
				times[i] = times[i - 1] + dt;
			}

			// Spline fit both x and y to times
			CubicSpline xSpline = new CubicSpline();
			xs = xSpline.FitAndEval(dists, x, times);

			CubicSpline ySpline = new CubicSpline();
			ys = ySpline.FitAndEval(dists, y, times);
		}

		#endregion
	}
}