MathUtil.cs

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// Copyright (c) 2014 Silicon Studio Corp. (http://siliconstudio.co.jp)
// This file is distributed under MIT License. See LICENSE.md for details.
//
// -----------------------------------------------------------------------------
// Original code from SlimMath project. http://code.google.com/p/slimmath/
// Greetings to SlimDX Group. Original code published with the following license:
// -----------------------------------------------------------------------------
/*
* Copyright (c) 2007-2011 SlimDX Group
* 
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using System;
namespace SiliconStudio.Core.Mathematics
{
    public static class MathUtil
    {
        /// <summary>
        /// The value for which all absolute numbers smaller than are considered equal to zero.
        /// </summary>
        public const float ZeroTolerance = 1e-6f; // Value a 8x higher than 1.19209290E-07F

        /// <summary>
        /// The value for which all absolute numbers smaller than are considered equal to zero.
        /// </summary>
        public const double ZeroToleranceDouble = Double.Epsilon*8;

        /// <summary>
        /// A value specifying the approximation of π which is 180 degrees.
        /// </summary>
        public const float Pi = (float)Math.PI;

        /// <summary>
        /// A value specifying the approximation of 2π which is 360 degrees.
        /// </summary>
        public const float TwoPi = (float)(2 * Math.PI);

        /// <summary>
        /// A value specifying the approximation of π/2 which is 90 degrees.
        /// </summary>
        public const float PiOverTwo = (float)(Math.PI / 2);

        /// <summary>
        /// A value specifying the approximation of π/4 which is 45 degrees.
        /// </summary>
        public const float PiOverFour = (float)(Math.PI / 4);

        /// <summary>
        /// Checks if a and b are almost equals, taking into account the magnitude of floating point numbers (unlike <see cref="WithinEpsilon"/> method). See Remarks.
        /// See remarks.
        /// </summary>
        /// <param name="a">The left value to compare.</param>
        /// <param name="b">The right value to compare.</param>
        /// <returns><c>true</c> if a almost equal to b, <c>false</c> otherwise</returns>
        /// <remarks>
        /// The code is using the technique described by Bruce Dawson in 
        /// <a href="http://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/">Comparing Floating point numbers 2012 edition</a>. 
        /// </remarks>
        public unsafe static bool NearEqual(float a, float b)
        {
            // Check if the numbers are really close -- needed
            // when comparing numbers near zero.
            if (IsZero(a - b))
                return true;

            // Original from Bruce Dawson: http://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/
            int aInt = *(int*)&a;
            int bInt = *(int*)&b;

            // Different signs means they do not match.
            if ((aInt < 0) != (bInt < 0))
                return false;

            // Find the difference in ULPs.
            int ulp = Math.Abs(aInt - bInt);

            // Choose of maxUlp = 4
            // according to http://code.google.com/p/googletest/source/browse/trunk/include/gtest/internal/gtest-internal.h
            const int maxUlp = 4;
            return (ulp <= maxUlp);
        }

        /// <summary>
        /// Determines whether the specified value is close to zero (0.0f).
        /// </summary>
        /// <param name="a">The floating value.</param>
        /// <returns><c>true</c> if the specified value is close to zero (0.0f); otherwise, <c>false</c>.</returns>
        public static bool IsZero(float a)
        {
            return Math.Abs(a) < ZeroTolerance;
        }

        /// <summary>
        /// Determines whether the specified value is close to zero (0.0f).
        /// </summary>
        /// <param name="a">The floating value.</param>
        /// <returns><c>true</c> if the specified value is close to zero (0.0f); otherwise, <c>false</c>.</returns>
        public static bool IsZero(double a)
        {
            return Math.Abs(a) < ZeroToleranceDouble;
        }

        /// <summary>
        /// Determines whether the specified value is close to one (1.0f).
        /// </summary>
        /// <param name="a">The floating value.</param>
        /// <returns><c>true</c> if the specified value is close to one (1.0f); otherwise, <c>false</c>.</returns>
        public static bool IsOne(float a)
        {
            return IsZero(a - 1.0f);
        }

        /// <summary>
        /// Checks if a - b are almost equals within a float epsilon.
        /// </summary>
        /// <param name="a">The left value to compare.</param>
        /// <param name="b">The right value to compare.</param>
        /// <param name="epsilon">Epsilon value</param>
        /// <returns><c>true</c> if a almost equal to b within a float epsilon, <c>false</c> otherwise</returns>
        public static bool WithinEpsilon(float a, float b, float epsilon)
        {
            float num = a - b;
            return ((-epsilon <= num) && (num <= epsilon));
        }

        /// <summary>
        /// Does something with arrays.
        /// </summary>
        /// <typeparam name="T">Most likely the type of elements in the array.</typeparam>
        /// <param name="value">Who knows what this is for.</param>
        /// <param name="count">Probably the length of the array.</param>
        /// <returns>An array of who knows what.</returns>
        public static T[] Array<T>(T value, int count)
        {
            T[] result = new T[count];
            for (int i = 0; i < count; i++)
                result[i] = value;

            return result;
        }

        /// <summary>
        /// Converts revolutions to degrees.
        /// </summary>
        /// <param name="revolution">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float RevolutionsToDegrees(float revolution)
        {
            return revolution * 360.0f;
        }

        /// <summary>
        /// Converts revolutions to radians.
        /// </summary>
        /// <param name="revolution">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float RevolutionsToRadians(float revolution)
        {
            return revolution * TwoPi;
        }

        /// <summary>
        /// Converts revolutions to gradians.
        /// </summary>
        /// <param name="revolution">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float RevolutionsToGradians(float revolution)
        {
            return revolution * 400.0f;
        }

        /// <summary>
        /// Converts degrees to revolutions.
        /// </summary>
        /// <param name="degree">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float DegreesToRevolutions(float degree)
        {
            return degree / 360.0f;
        }

        /// <summary>
        /// Converts degrees to radians.
        /// </summary>
        /// <param name="degree">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float DegreesToRadians(float degree)
        {
            return degree * (Pi / 180.0f);
        }

        /// <summary>
        /// Converts radians to revolutions.
        /// </summary>
        /// <param name="radian">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float RadiansToRevolutions(float radian)
        {
            return radian / TwoPi;
        }

        /// <summary>
        /// Converts radians to gradians.
        /// </summary>
        /// <param name="radian">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float RadiansToGradians(float radian)
        {
            return radian * (200.0f / Pi);
        }

        /// <summary>
        /// Converts gradians to revolutions.
        /// </summary>
        /// <param name="gradian">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float GradiansToRevolutions(float gradian)
        {
            return gradian / 400.0f;
        }

        /// <summary>
        /// Converts gradians to degrees.
        /// </summary>
        /// <param name="gradian">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float GradiansToDegrees(float gradian)
        {
            return gradian * (9.0f / 10.0f);
        }

        /// <summary>
        /// Converts gradians to radians.
        /// </summary>
        /// <param name="gradian">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float GradiansToRadians(float gradian)
        {
            return gradian * (Pi / 200.0f);
        }

        /// <summary>
        /// Converts radians to degrees.
        /// </summary>
        /// <param name="radian">The value to convert.</param>
        /// <returns>The converted value.</returns>
        public static float RadiansToDegrees(float radian)
        {
            return radian * (180.0f / Pi);
        }

        /// <summary>
        /// Clamps the specified value.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <param name="min">The min.</param>
        /// <param name="max">The max.</param>
        /// <returns>The result of clamping a value between min and max</returns>
        public static float Clamp(float value, float min, float max)
        {
            return value < min ? min : value > max ? max : value;
        }

        /// <summary>
        /// Clamps the specified value.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <param name="min">The min.</param>
        /// <param name="max">The max.</param>
        /// <returns>The result of clamping a value between min and max</returns>
        public static double Clamp(double value, double min, double max)
        {
            return value < min ? min : value > max ? max : value;
        }

        /// <summary>
        /// Clamps the specified value.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <param name="min">The min.</param>
        /// <param name="max">The max.</param>
        /// <returns>The result of clamping a value between min and max</returns>
        public static int Clamp(int value, int min, int max)
        {
            return value < min ? min : value > max ? max : value;
        }

        /// <summary>
        /// Inverse-interpolates a value linearly.
        /// </summary>
        /// <param name="value">Value to get inverse interpolation.</param>
        /// <param name="min">Minimum value that takes place in inverse-interpolation.</param>
        /// <param name="max">Maximum value that takes place in inverse-interpolation.</param>
        /// <returns>Returns an inverse-linearly interpolated coeficient.</returns>
        public static float InverseLerp(float min, float max, float value)
        {
            if (IsZero(Math.Abs(max - min)))
                return float.NaN;
            return (value - min) / (max - min);
        }

        /// <summary>
        /// Inverse-interpolates a value linearly.
        /// </summary>
        /// <param name="value">Value to get inverse interpolation.</param>
        /// <param name="min">Minimum value that takes place in inverse-interpolation.</param>
        /// <param name="max">Maximum value that takes place in inverse-interpolation.</param>
        /// <returns>Returns an inverse-linearly interpolated coeficient.</returns>
        public static double InverseLerp(double min, double max, double value)
        {
            if (IsZero(Math.Abs(max - min)))
                return double.NaN;
            return (value - min)/(max - min);
        }

        /// <summary>
        /// Interpolates between two values using a linear function by a given amount.
        /// </summary>
        /// <remarks>
        /// See http://www.encyclopediaofmath.org/index.php/Linear_interpolation and
        /// http://fgiesen.wordpress.com/2012/08/15/linear-interpolation-past-present-and-future/
        /// </remarks>
        /// <param name="from">Value to interpolate from.</param>
        /// <param name="to">Value to interpolate to.</param>
        /// <param name="amount">Interpolation amount.</param>
        /// <returns>The result of linear interpolation of values based on the amount.</returns>
        public static double Lerp(double from, double to, double amount)
        {
            return (1 - amount) * from + amount * to;
        }

        /// <summary>
        /// Interpolates between two values using a linear function by a given amount.
        /// </summary>
        /// <remarks>
        /// See http://www.encyclopediaofmath.org/index.php/Linear_interpolation and
        /// http://fgiesen.wordpress.com/2012/08/15/linear-interpolation-past-present-and-future/
        /// </remarks>
        /// <param name="from">Value to interpolate from.</param>
        /// <param name="to">Value to interpolate to.</param>
        /// <param name="amount">Interpolation amount.</param>
        /// <returns>The result of linear interpolation of values based on the amount.</returns>
        public static float Lerp(float from, float to, float amount)
        {
            return (1 - amount) * from + amount * to;
        }

        /// <summary>
        /// Interpolates between two values using a linear function by a given amount.
        /// </summary>
        /// <remarks>
        /// See http://www.encyclopediaofmath.org/index.php/Linear_interpolation and
        /// http://fgiesen.wordpress.com/2012/08/15/linear-interpolation-past-present-and-future/
        /// </remarks>
        /// <param name="from">Value to interpolate from.</param>
        /// <param name="to">Value to interpolate to.</param>
        /// <param name="amount">Interpolation amount.</param>
        /// <returns>The result of linear interpolation of values based on the amount.</returns>
        public static byte Lerp(byte from, byte to, float amount)
        {
            return (byte)Lerp((float)from, (float)to, amount);
        }

        /// <summary>
        /// Performs smooth (cubic Hermite) interpolation between 0 and 1.
        /// </summary>
        /// <remarks>
        /// See https://en.wikipedia.org/wiki/Smoothstep
        /// </remarks>
        /// <param name="amount">Value between 0 and 1 indicating interpolation amount.</param>
        public static float SmoothStep(float amount)
        {
            return (amount <= 0) ? 0
                : (amount >= 1) ? 1
                : amount * amount * (3 - (2 * amount));
        }

        /// <summary>
        /// Performs a smooth(er) interpolation between 0 and 1 with 1st and 2nd order derivatives of zero at endpoints.
        /// </summary>
        /// <remarks>
        /// See https://en.wikipedia.org/wiki/Smoothstep
        /// </remarks>
        /// <param name="amount">Value between 0 and 1 indicating interpolation amount.</param>
        public static float SmootherStep(float amount)
        {
            return (amount <= 0) ? 0
                : (amount >= 1) ? 1
                : amount * amount * amount * (amount * ((amount * 6) - 15) + 10);
        }

        /// <summary>
        /// Calculates the modulo of the specified value, handling negative values.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <param name="modulo">The modulo.</param>
        /// <returns>The result of the modulo applied to value</returns>
        public static int Mod(int value, int modulo)
        {
            if (modulo == 0)
            {
                return value;
            }
            value %= modulo;
            return value < 0 ? value + modulo : value;
        }

        /// <summary>
        /// Calculates the modulo of the specified value.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <param name="modulo">The modulo.</param>
        /// <returns>The result of the modulo applied to value</returns>
        public static float Mod(float value, float modulo)
        {
            if (modulo == 0.0f)
            {
                return value;
            }
            value %= modulo;
            return value < 0 ? value + modulo : value;
        }

        /// <summary>
        /// Calculates the modulo 2*PI of the specified value.
        /// </summary>
        /// <param name="value">The value.</param>
        /// <returns>The result of the modulo applied to value</returns>
        public static float Mod2PI(float value)
        {
            return Mod(value, TwoPi);
        }

        /// <summary>
        /// Wraps the specified value into a range [min, max]
        /// </summary>
        /// <param name="value">The value to wrap.</param>
        /// <param name="min">The min.</param>
        /// <param name="max">The max.</param>
        /// <returns>Result of the wrapping.</returns>
        /// <exception cref="ArgumentException">Is thrown when <paramref name="min"/> is greater than <paramref name="max"/>.</exception>
        public static int Wrap(int value, int min, int max)
        {
            if (min > max)
                throw new ArgumentException(string.Format("min {0} should be less than or equal to max {1}", min, max), "min");

            // Code from http://stackoverflow.com/a/707426/1356325
            int range_size = max - min + 1;

            if (value < min)
                value += range_size * ((min - value) / range_size + 1);

            return min + (value - min) % range_size;
        }

        /// <summary>
        /// Wraps the specified value into a range [min, max[
        /// </summary>
        /// <param name="value">The value.</param>
        /// <param name="min">The min.</param>
        /// <param name="max">The max.</param>
        /// <returns>Result of the wrapping.</returns>
        /// <exception cref="ArgumentException">Is thrown when <paramref name="min"/> is greater than <paramref name="max"/>.</exception>
        public static float Wrap(float value, float min, float max)
        {
            if (NearEqual(min, max)) return min;

            double mind = min;
            double maxd = max;
            double valued = value;

            if (mind > maxd)
                throw new ArgumentException(string.Format("min {0} should be less than or equal to max {1}", min, max), "min");

            var range_size = maxd - mind;
            return (float)(mind + (valued - mind) - (range_size * Math.Floor((valued - mind) / range_size)));
        }

        /// <summary>
        /// Gauss function.
        /// </summary>
        /// <param name="amplitude">Curve amplitude.</param>
        /// <param name="x">Position X.</param>
        /// <param name="y">Position Y</param>
        /// <param name="radX">Radius X.</param>
        /// <param name="radY">Radius Y.</param>
        /// <param name="sigmaX">Curve sigma X.</param>
        /// <param name="sigmaY">Curve sigma Y.</param>
        /// <returns>The result of Gaussian function.</returns>
        public static float Gauss(float amplitude, float x, float y, float radX, float radY, float sigmaX, float sigmaY)
        {
            return (float)Gauss((double)amplitude, x, y, radX, radY, sigmaX, sigmaY);
        }

        /// <summary>
        /// Gauss function.
        /// </summary>
        /// <param name="amplitude">Curve amplitude.</param>
        /// <param name="x">Position X.</param>
        /// <param name="y">Position Y</param>
        /// <param name="radX">Radius X.</param>
        /// <param name="radY">Radius Y.</param>
        /// <param name="sigmaX">Curve sigma X.</param>
        /// <param name="sigmaY">Curve sigma Y.</param>
        /// <returns>The result of Gaussian function.</returns>
        public static double Gauss(double amplitude, double x, double y, double radX, double radY, double sigmaX, double sigmaY)
        {
            return (amplitude * Math.E) -
                (
                    Math.Pow(x - (radX / 2), 2) / (2 * Math.Pow(sigmaX, 2)) +
                    Math.Pow(y - (radY / 2), 2) / (2 * Math.Pow(sigmaY, 2))
                );
        }
    }
}