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417 lines
14 KiB
C#
417 lines
14 KiB
C#
//
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// Author: Ryan Seghers
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//
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// Copyright (C) 2013-2014 Ryan Seghers
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//
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// Permission is hereby granted, free of charge, to any person obtaining
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// a copy of this software and associated documentation files (the
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// "Software"), to deal in the Software without restriction, including
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// without limitation the irrevocable, perpetual, worldwide, and royalty-free
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// rights to use, copy, modify, merge, publish, distribute, sublicense,
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// display, perform, create derivative works from and/or sell copies of
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// the Software, both in source and object code form, and to
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// permit persons to whom the Software is furnished to do so, subject to
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// the following conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
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// LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
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// OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
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// WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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//
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using System;
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using OpenTK;
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namespace SCJMapper_V2.OGL
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{
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/// <summary>
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/// Cubic spline interpolation.
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/// Call Fit (or use the corrector constructor) to compute spline coefficients, then Eval to evaluate the spline at other X coordinates.
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/// </summary>
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/// <remarks>
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/// <para>
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/// This is implemented based on the wikipedia article:
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/// http://en.wikipedia.org/wiki/Spline_interpolation
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/// I'm not sure I have the right to include a copy of the article so the equation numbers referenced in
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/// comments will end up being wrong at some point.
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/// </para>
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/// <para>
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/// This is not optimized, and is not MT safe.
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/// This can extrapolate off the ends of the splines.
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/// You must provide points in X sort order.
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/// </para>
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/// </remarks>
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public class CubicSpline
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{
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#region Fields
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// N-1 spline coefficients for N points
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private float[] a;
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private float[] b;
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// Save the original x and y for Eval
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private float[] xOrig;
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private float[] yOrig;
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#endregion
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#region Ctor
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/// <summary>
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/// Default ctor.
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/// </summary>
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public CubicSpline()
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{
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}
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/// <summary>
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/// Construct and call Fit.
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/// </summary>
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/// <param name="x">Input. X coordinates to fit.</param>
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/// <param name="y">Input. Y coordinates to fit.</param>
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/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
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/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
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public CubicSpline(float[] x, float[] y, float startSlope = float.NaN, float endSlope = float.NaN, bool debug = false)
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{
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Fit(x, y, startSlope, endSlope );
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}
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/// <summary>
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/// Construct and call Fit.
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/// </summary>
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/// <param name="xyPts">Input. XY coordinates to fit</param>
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/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
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/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
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public CubicSpline( Vector2[] xyPts, float startSlope = float.NaN, float endSlope = float.NaN, bool debug = false )
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{
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float[] x = new float[xyPts.Length];
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float[] y = new float[xyPts.Length];
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for ( int i = 0; i < xyPts.Length; i++ ) {
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x[i] = xyPts[i].X; y[i] = xyPts[i].Y;
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}
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Fit( x, y, startSlope, endSlope );
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}
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#endregion
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#region Private Methods
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/// <summary>
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/// Throws if Fit has not been called.
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/// </summary>
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private void CheckAlreadyFitted( )
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{
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if (a == null) throw new Exception("Fit must be called before you can evaluate.");
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}
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private int _lastIndex = 0;
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/// <summary>
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/// Find where in xOrig the specified x falls, by simultaneous traverse.
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/// This allows xs to be less than x[0] and/or greater than x[n-1]. So allows extrapolation.
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/// This keeps state, so requires that x be sorted and xs called in ascending order, and is not multi-thread safe.
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/// </summary>
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private int GetNextXIndex(float x)
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{
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if (x < xOrig[_lastIndex])
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{
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throw new ArgumentException("The X values to evaluate must be sorted.");
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}
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while ((_lastIndex < xOrig.Length - 2) && (x > xOrig[_lastIndex + 1]))
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{
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_lastIndex++;
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}
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return _lastIndex;
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}
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/// <summary>
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/// Evaluate the specified x value using the specified spline.
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/// </summary>
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/// <param name="x">The x value.</param>
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/// <param name="j">Which spline to use.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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/// <returns>The y value.</returns>
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private float EvalSpline(float x, int j)
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{
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float dx = xOrig[j + 1] - xOrig[j];
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float t = (x - xOrig[j]) / dx;
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float y = (1 - t) * yOrig[j] + t * yOrig[j + 1] + t * (1 - t) * (a[j] * (1 - t) + b[j] * t); // equation 9
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return y;
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}
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#endregion
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#region Fit*
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/// <summary>
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/// Fit x,y and then eval at points xs and return the corresponding y's.
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/// This does the "natural spline" style for ends.
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/// This can extrapolate off the ends of the splines.
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/// You must provide points in X sort order.
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/// </summary>
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/// <param name="x">Input. X coordinates to fit.</param>
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/// <param name="y">Input. Y coordinates to fit.</param>
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/// <param name="xs">Input. X coordinates to evaluate the fitted curve at.</param>
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/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
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/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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/// <returns>The computed y values for each xs.</returns>
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public float[] FitAndEval(float[] x, float[] y, float[] xs, float startSlope = float.NaN, float endSlope = float.NaN)
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{
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Fit(x, y, startSlope, endSlope);
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return Eval(xs);
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}
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/// <summary>
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/// Compute spline coefficients for the specified x,y points.
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/// This does the "natural spline" style for ends.
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/// This can extrapolate off the ends of the splines.
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/// You must provide points in X sort order.
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/// </summary>
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/// <param name="x">Input. X coordinates to fit.</param>
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/// <param name="y">Input. Y coordinates to fit.</param>
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/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
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/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
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/// <param name="debug">Turn on console output. Default is false.</param>
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public void Fit(float[] x, float[] y, float startSlope = float.NaN, float endSlope = float.NaN)
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{
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if (Single.IsInfinity(startSlope) || Single.IsInfinity(endSlope))
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{
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throw new Exception("startSlope and endSlope cannot be infinity.");
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}
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// Save x and y for eval
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this.xOrig = x;
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this.yOrig = y;
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int n = x.Length;
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float[] r = new float[n]; // the right hand side numbers: wikipedia page overloads b
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TriDiagonalMatrixF m = new TriDiagonalMatrixF(n);
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float dx1, dx2, dy1, dy2;
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// First row is different (equation 16 from the article)
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if (float.IsNaN(startSlope))
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{
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dx1 = x[1] - x[0];
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m.C[0] = 1.0f / dx1;
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m.B[0] = 2.0f * m.C[0];
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r[0] = 3 * (y[1] - y[0]) / (dx1 * dx1);
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}
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else
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{
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m.B[0] = 1;
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r[0] = startSlope;
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}
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// Body rows (equation 15 from the article)
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for (int i = 1; i < n - 1; i++)
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{
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dx1 = x[i] - x[i - 1];
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dx2 = x[i + 1] - x[i];
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m.A[i] = 1.0f / dx1;
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m.C[i] = 1.0f / dx2;
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m.B[i] = 2.0f * (m.A[i] + m.C[i]);
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dy1 = y[i] - y[i - 1];
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dy2 = y[i + 1] - y[i];
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r[i] = 3 * (dy1 / (dx1 * dx1) + dy2 / (dx2 * dx2));
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}
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// Last row also different (equation 17 from the article)
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if (float.IsNaN(endSlope))
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{
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dx1 = x[n - 1] - x[n - 2];
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dy1 = y[n - 1] - y[n - 2];
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m.A[n - 1] = 1.0f / dx1;
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m.B[n - 1] = 2.0f * m.A[n - 1];
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r[n - 1] = 3 * (dy1 / (dx1 * dx1));
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}
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else
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{
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m.B[n - 1] = 1;
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r[n - 1] = endSlope;
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}
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// k is the solution to the matrix
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float[] k = m.Solve(r);
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// a and b are each spline's coefficients
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this.a = new float[n - 1];
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this.b = new float[n - 1];
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for (int i = 1; i < n; i++)
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{
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dx1 = x[i] - x[i - 1];
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dy1 = y[i] - y[i - 1];
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a[i - 1] = k[i - 1] * dx1 - dy1; // equation 10 from the article
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b[i - 1] = -k[i] * dx1 + dy1; // equation 11 from the article
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}
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}
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#endregion
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#region Eval*
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/// <summary>
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/// Evaluate the spline at the specified x coordinate.
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/// This can extrapolate off the ends of the splines.
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/// The spline must already be computed before calling this, meaning you must have already called Fit() or FitAndEval().
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/// </summary>
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/// <param name="x">Input. X coordinate to evaluate the fitted curve at.</param>
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/// <returns>The computed y values for x.</returns>
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public float Eval( float x )
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{
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CheckAlreadyFitted( );
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float y;
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_lastIndex = 0; // Reset simultaneous traversal in case there are multiple calls
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int j = GetNextXIndex( x );
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// Evaluate using j'th spline
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y = EvalSpline( x, j );
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return y;
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}
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/// <summary>
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/// Evaluate the spline at the specified x coordinates.
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/// This can extrapolate off the ends of the splines.
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/// You must provide X's in ascending order.
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/// The spline must already be computed before calling this, meaning you must have already called Fit() or FitAndEval().
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/// </summary>
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/// <param name="x">Input. X coordinates to evaluate the fitted curve at.</param>
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/// <returns>The computed y values for each x.</returns>
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public float[] Eval( float[] x )
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{
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CheckAlreadyFitted( );
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int n = x.Length;
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float[] y = new float[n];
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_lastIndex = 0; // Reset simultaneous traversal in case there are multiple calls
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for ( int i = 0; i < n; i++ ) {
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// Find which spline can be used to compute this x (by simultaneous traverse)
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int j = GetNextXIndex( x[i] );
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// Evaluate using j'th spline
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y[i] = EvalSpline( x[i], j );
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}
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return y;
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}
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/// <summary>
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/// Evaluate (compute) the slope of the spline at the specified x coordinates.
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/// This can extrapolate off the ends of the splines.
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/// You must provide X's in ascending order.
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/// The spline must already be computed before calling this, meaning you must have already called Fit() or FitAndEval().
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/// </summary>
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/// <param name="x">Input. X coordinates to evaluate the fitted curve at.</param>
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/// <returns>The computed y values for each x.</returns>
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public float[] EvalSlope(float[] x )
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{
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CheckAlreadyFitted();
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int n = x.Length;
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float[] qPrime = new float[n];
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_lastIndex = 0; // Reset simultaneous traversal in case there are multiple calls
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for (int i = 0; i < n; i++)
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{
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// Find which spline can be used to compute this x (by simultaneous traverse)
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int j = GetNextXIndex(x[i]);
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// Evaluate using j'th spline
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float dx = xOrig[j + 1] - xOrig[j];
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float dy = yOrig[j + 1] - yOrig[j];
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float t = (x[i] - xOrig[j]) / dx;
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// From equation 5 we could also compute q' (qp) which is the slope at this x
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qPrime[i] = dy / dx
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+ (1 - 2 * t) * (a[j] * (1 - t) + b[j] * t) / dx
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+ t * (1 - t) * (b[j] - a[j]) / dx;
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}
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return qPrime;
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}
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#endregion
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#region Static Methods
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/// <summary>
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/// Static all-in-one method to fit the splines and evaluate at X coordinates.
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/// </summary>
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/// <param name="x">Input. X coordinates to fit.</param>
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/// <param name="y">Input. Y coordinates to fit.</param>
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/// <param name="xs">Input. X coordinates to evaluate the fitted curve at.</param>
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/// <param name="startSlope">Optional slope constraint for the first point. Single.NaN means no constraint.</param>
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/// <param name="endSlope">Optional slope constraint for the final point. Single.NaN means no constraint.</param>
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/// <returns>The computed y values for each xs.</returns>
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public static float[] Compute(float[] x, float[] y, float[] xs, float startSlope = float.NaN, float endSlope = float.NaN )
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{
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CubicSpline spline = new CubicSpline();
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return spline.FitAndEval(x, y, xs, startSlope, endSlope );
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}
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/// <summary>
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/// Fit the input x,y points using a 'geometric' strategy so that y does not have to be a single-valued
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/// function of x.
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/// </summary>
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/// <param name="x">Input x coordinates.</param>
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/// <param name="y">Input y coordinates, do not need to be a single-valued function of x.</param>
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/// <param name="nOutputPoints">How many output points to create.</param>
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/// <param name="xs">Output (interpolated) x values.</param>
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/// <param name="ys">Output (interpolated) y values.</param>
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public static void FitGeometric(float[] x, float[] y, int nOutputPoints, out float[] xs, out float[] ys)
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{
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// Compute distances
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int n = x.Length;
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float[] dists = new float[n]; // cumulative distance
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dists[0] = 0;
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float totalDist = 0;
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for (int i = 1; i < n; i++)
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{
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float dx = x[i] - x[i - 1];
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float dy = y[i] - y[i - 1];
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float dist = (float)Math.Sqrt(dx * dx + dy * dy);
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totalDist += dist;
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dists[i] = totalDist;
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}
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// Create 'times' to interpolate to
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float dt = totalDist / (nOutputPoints - 1);
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float[] times = new float[nOutputPoints];
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times[0] = 0;
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for (int i = 1; i < nOutputPoints; i++)
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{
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times[i] = times[i - 1] + dt;
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}
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// Spline fit both x and y to times
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CubicSpline xSpline = new CubicSpline();
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xs = xSpline.FitAndEval(dists, x, times);
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CubicSpline ySpline = new CubicSpline();
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ys = ySpline.FitAndEval(dists, y, times);
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}
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#endregion
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}
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}
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