NewtonTopographyFractal.cs

using System;
using System.Numerics;
using System.Text;
using SkiaSharp;
using Waher.Script.Abstraction.Elements;
using Waher.Script.Model;
using Waher.Script.Exceptions;
using Waher.Script.Objects.VectorSpaces;
using Waher.Script.Fractals.Exceptions;
 
namespace Waher.Script.Fractals.ComplexFractals
{
    public class NewtonTopographyFractal : FunctionMultiVariate
    {
        public NewtonTopographyFractal(ScriptNode z, ScriptNode dr, ScriptNode R, ScriptNode c, ScriptNode Palette,
            ScriptNode DimX, ScriptNode DimY, int Start, int Length, Expression Expression)
            : base(new ScriptNode[] { z, dr, R, c, Palette, DimX, DimY },
                  new ArgumentType[] { ArgumentType.Scalar, ArgumentType.Scalar, ArgumentType.Scalar,
                  ArgumentType.Normal, ArgumentType.Vector, ArgumentType.Scalar, ArgumentType.Scalar},
                  Start, Length, Expression)
        {
        }
 
        public NewtonTopographyFractal(ScriptNode z, ScriptNode dr, ScriptNode R, ScriptNode c, ScriptNode Palette,
            ScriptNode DimX, int Start, int Length, Expression Expression)
            : base(new ScriptNode[] { z, dr, R, c, Palette, DimX },
                  new ArgumentType[] { ArgumentType.Scalar, ArgumentType.Scalar, ArgumentType.Scalar,
                  ArgumentType.Normal, ArgumentType.Vector, ArgumentType.Scalar},
                  Start, Length, Expression)
        {
        }
 
        public NewtonTopographyFractal(ScriptNode z, ScriptNode dr, ScriptNode R, ScriptNode c, ScriptNode Palette,
            int Start, int Length, Expression Expression)
            : base(new ScriptNode[] { z, dr, R, c, Palette },
                  new ArgumentType[] { ArgumentType.Scalar, ArgumentType.Scalar, ArgumentType.Scalar,
                  ArgumentType.Normal, ArgumentType.Vector },
                  Start, Length, Expression)
        {
        }
 
        public NewtonTopographyFractal(ScriptNode z, ScriptNode dr, ScriptNode R, ScriptNode c,
            int Start, int Length, Expression Expression)
            : base(new ScriptNode[] { z, dr, R, c },
                  new ArgumentType[] { ArgumentType.Scalar, ArgumentType.Scalar, ArgumentType.Scalar,
                  ArgumentType.Normal },
                  Start, Length, Expression)
        {
        }
 
        public override string[] DefaultArgumentNames
        {
            get
            {
                return new string[] { "z", "dr", "R", "c", "Palette", "DimX", "DimY" };
            }
        }
 
        public override IElement Evaluate(IElement[] Arguments, Variables Variables)
        {
            string ColorExpression = null;
            SKColor[] Palette;
            double[] Coefficients = null;
            Complex[] CoefficientsZ = null;
            ILambdaExpression f = null;
            ScriptNode fDef = null;
            double rc, ic;
            double dr;
            Complex R;
            int dimx, dimy;
            int c = Arguments.Length;
            int i = 0;
            object Obj;
 
            Obj = Arguments[i++].AssociatedObjectValue;
            if (Obj is Complex z)
            {
                rc = z.Real;
                ic = z.Imaginary;
            }
            else
            {
                rc = Expression.ToDouble(Obj);
                ic = Expression.ToDouble(Arguments[i++].AssociatedObjectValue);
            }
 
            if (i >= c)
                throw new ScriptRuntimeException("Insufficient parameters in call to NewtonTopographyFractal().", this);
 
            dr = Expression.ToDouble(Arguments[i++].AssociatedObjectValue);
 
            if (i < c && ((Obj = Arguments[i].AssociatedObjectValue) is double || Obj is Complex))
            {
                R = Expression.ToComplex(Obj);
                i++;
            }
            else
            {
                R = Complex.One;
 
                if (i < c && this.Arguments[i] is null)
                    i++;
            }
 
            if (i < c)
            {
                if (Arguments[i] is DoubleVector)
                    Coefficients = (double[])Arguments[i++].AssociatedObjectValue;
                else if (Arguments[i] is ComplexVector)
                    CoefficientsZ = (Complex[])Arguments[i++].AssociatedObjectValue;
                /*else if (Parameters[i] is RealPolynomial)
                    Coefficients = ((RealPolynomial)Arguments[i++].AssociatedObjectValue).Coefficients;
                else if (Parameters[i] is ComplexPolynomial)
                    CoefficientsZ = ((ComplexPolynomial)Arguments[i++].AssociatedObjectValue).Coefficients;*/
                else if (Arguments[i] is IVector)
                {
                    IVector Vector = (IVector)Arguments[i++];
                    int j, d = Vector.Dimension;
 
                    CoefficientsZ = new Complex[d];
                    for (j = 0; j < d; j++)
                        CoefficientsZ[j] = Expression.ToComplex(Vector.GetElement(j).AssociatedObjectValue);
                }
                else if (Arguments[i].AssociatedObjectValue is ILambdaExpression)
                {
                    f = (ILambdaExpression)Arguments[i];
                    if (f.NrArguments != 1)
                        throw new ScriptRuntimeException("Lambda expression in calls to NewtonTopographyFractal() must be of one variable.", this);
 
                    fDef = this.Arguments[i++];
                }
                else
                {
                    throw new ScriptRuntimeException("Parameter " + (i + 1).ToString() +
                        " in call to NewtonTopographyFractal has to be a vector of numbers, containing coefficients " +
                        "of the polynomial to use. Now it was of type " + Arguments[i].GetType().FullName,
                        this);
                }
            }
            else
                throw new ScriptRuntimeException("Missing coefficients or lambda expression.", this);
 
            if (i < c && !(this.Arguments[i] is null) && Arguments[i] is ObjectVector)
            {
                ColorExpression = this.Arguments[i].SubExpression;
                Palette = FractalGraph.ToPalette((ObjectVector)Arguments[i++]);
            }
            else
            {
                Palette = ColorModels.RandomLinearAnalogousHSL.CreatePalette(128, 4, out int Seed, this, Variables);
                ColorExpression = "RandomLinearAnalogousHSL(128,4," + Seed.ToString() + ")";
 
                if (i < c && this.Arguments[i] is null)
                    i++;
            }
 
            if (i < c)
                dimx = (int)Expression.ToDouble(Arguments[i++].AssociatedObjectValue);
            else
                dimx = 320;
 
            if (i < c)
                dimy = (int)Expression.ToDouble(Arguments[i++].AssociatedObjectValue);
            else
                dimy = 200;
 
            if (i < c)
            {
                throw new ScriptRuntimeException("Parameter mismatch in call to NewtonTopographyFractal(r,c,dr,Coefficients[,Palette][,dimx[,dimy]]).",
                    this);
            }
 
            if (dimx <= 0 || dimx > 5000 || dimy <= 0 || dimy > 5000)
                throw new FractalImageSizeScriptException(this);
 
            if (!(f is null))
            {
                return CalcNewtonTopography(rc, ic, dr, R, f, Variables, Palette, dimx, dimy, this, this.FractalZoomScript,
                    new object[] { Palette, dimx, dimy, R, fDef, ColorExpression });
            }
            else if (!(CoefficientsZ is null))
            {
                return CalcNewtonTopography(Variables, rc, ic, dr, R, CoefficientsZ, Palette, dimx, dimy, this, this.FractalZoomScript,
                    new object[] { Palette, dimx, dimy, R, CoefficientsZ, ColorExpression });
            }
            else
            {
                return CalcNewtonTopography(Variables, rc, ic, dr, R, Coefficients, Palette, dimx, dimy, this, this.FractalZoomScript,
                    new object[] { Palette, dimx, dimy, R, Coefficients, ColorExpression });
            }
        }
 
        private string FractalZoomScript(double r, double i, double Size, object State)
        {
            object[] Parameters = (object[])State;
            int DimX = (int)Parameters[1];
            int DimY = (int)Parameters[2];
            Complex R = (Complex)Parameters[3];
            double[] Coefficients = Parameters[4] as double[];
            string ColorExpression = (string)Parameters[5];
 
            StringBuilder sb = new StringBuilder();
 
            sb.Append("NewtonTopographyFractal((");
            sb.Append(Expression.ToString(r));
            sb.Append(',');
            sb.Append(Expression.ToString(i));
            sb.Append("),");
            sb.Append(Expression.ToString(Size / 4));
            sb.Append(',');
            sb.Append(Expression.ToString(R));
            sb.Append(',');
 
            if (Parameters[4] is ScriptNode fDef)
                sb.Append(fDef.SubExpression);
            else if (Parameters[4] is Complex[] CoefficientsZ)
                sb.Append(Expression.ToString(CoefficientsZ));
            else
                sb.Append(Expression.ToString(Coefficients));
 
            if (!string.IsNullOrEmpty(ColorExpression))
            {
                sb.Append(',');
                sb.Append(ColorExpression);
            }
 
            sb.Append(',');
            sb.Append(DimX.ToString());
            sb.Append(',');
            sb.Append(DimY.ToString());
            sb.Append(')');
 
            return sb.ToString();
        }
 
        public static FractalGraph CalcNewtonTopography(Variables Variables, double rCenter, double iCenter, double rDelta, Complex R,
            double[] Coefficients, SKColor[] Palette, int Width, int Height, ScriptNode Node, FractalZoomScript FractalZoomScript, object State)
        {
            double RRe = R.Real;
            double RIm = R.Imaginary;
            double r0, i0, r1, i1;
            double dr, di;
            double r, i;
            double zr, zi, zr2, zi2, zr3, zi3, zr4, zi4;
            double aspect;
            double Temp;
            int x, y;
            int n, N;
            int Degree = Coefficients.Length - 1;
 
            N = Palette.Length;
 
            if (Degree < 2)
            {
                Array.Resize(ref Coefficients, 3);
                while (Degree < 2)
                    Coefficients[++Degree] = 0;
            }
 
            if (Width <= 2 || Height <= 2)
                throw new ScriptRuntimeException("Width and Height has to be greater than 2.", Node);
 
            double[] Prim = new double[Degree];
            for (x = 1; x <= Degree; x++)
                Prim[x - 1] = x * Coefficients[x];
 
            Array.Reverse(Prim);
            Coefficients = (double[])Coefficients.Clone();
            Array.Reverse(Coefficients);
 
            int Size = Width * Height;
            double Conv = 1e-10;
            double Div = 1e10;
            int[] ColorIndex = new int[Size];
            int Index = 0;
 
            rDelta *= 0.5;
            r0 = rCenter - rDelta;
            r1 = rCenter + rDelta;
 
            aspect = ((double)Width) / Height;
 
            i0 = iCenter - rDelta / aspect;
            i1 = iCenter + rDelta / aspect;
 
            dr = (r1 - r0) / Width;
            di = (i1 - i0) / Height;
 
            for (y = 0, i = i0; y < Height; y++, i += di)
            {
                for (x = 0, r = r0; x < Width; x++, r += dr)
                {
                    zr = r;
                    zi = i;
 
                    n = 0;
                    do
                    {
                        // f:
                        zr2 = zi2 = 0;
                        foreach (double C in Coefficients)
                        {
                            Temp = zr2 * zr - zi2 * zi + C;
                            zi2 = zr2 * zi + zi2 * zr;
                            zr2 = Temp;
                        }
 
                        // f':
                        zr3 = zi3 = 0;
                        foreach (double C in Prim)
                        {
                            Temp = zr3 * zr - zi3 * zi + C;
                            zi3 = zr3 * zi + zi3 * zr;
                            zr3 = Temp;
                        }
 
                        // f/f':
 
                        Temp = 1.0 / (zr3 * zr3 + zi3 * zi3);
                        zr4 = (zr2 * zr3 + zi2 * zi3) * Temp;
                        zi4 = (zi2 * zr3 - zr2 * zi3) * Temp;
 
                        // R*f/f'
                        Temp = zr4 * RRe - zi4 * RIm;
                        zi4 = zr4 * RIm + zi4 * RRe;
                        zr4 = Temp;
 
                        zr -= zr4;
                        zi -= zi4;
 
                        Temp = Math.Sqrt(zr4 * zr4 + zi4 * zi4);
                    }
                    while ((Temp > Conv) && (Temp < Div) && (n++ < N));
 
                    if (Temp < Conv && n < N)
                        ColorIndex[Index++] = n;
                    else
                        ColorIndex[Index++] = N;
                }
            }
 
            ColorIndex = FractalGraph.FindBoundaries(ColorIndex, Width, Height);
 
            return new FractalGraph(Variables, FractalGraph.ToPixels(ColorIndex, Width, Height, Palette),
                r0, i0, r1, i1, rDelta * 2, true, Node, FractalZoomScript, State);
        }
 
        public static FractalGraph CalcNewtonTopography(Variables Variables, double rCenter, double iCenter, double rDelta, Complex R,
            Complex[] Coefficients, SKColor[] Palette, int Width, int Height, ScriptNode Node, FractalZoomScript FractalZoomScript, 
            object State)
        {
            double RRe = R.Real;
            double RIm = R.Imaginary;
            double r0, i0, r1, i1;
            double dr, di;
            double r, i;
            double zr, zi, zr2, zi2, zr3, zi3, zr4, zi4;
            double aspect;
            double Temp;
            int x, y;
            int n, N;
            int Degree = Coefficients.Length - 1;
 
            N = Palette.Length;
 
            if (Degree < 2)
            {
                Array.Resize(ref Coefficients, 3);
                while (Degree < 2)
                    Coefficients[++Degree] = Complex.Zero;
            }
 
            if (Width <= 2 || Height <= 2)
                throw new ScriptRuntimeException("Width and Height has to be greater than 2.", Node);
 
            Complex[] Prim = new Complex[Degree];
            for (x = 1; x <= Degree; x++)
                Prim[x - 1] = x * Coefficients[x];
 
            Array.Reverse(Prim);
            Coefficients = (Complex[])Coefficients.Clone();
            Array.Reverse(Coefficients);
 
            int j, c = Prim.Length;
            double[] ReC = new double[c + 1];
            double[] ImC = new double[c + 1];
            double[] RePrim = new double[c];
            double[] ImPrim = new double[c];
            Complex z;
 
            for (j = 0; j < c; j++)
            {
                z = Coefficients[j];
                ReC[j] = z.Real;
                ImC[j] = z.Imaginary;
 
                z = Prim[j];
                RePrim[j] = z.Real;
                ImPrim[j] = z.Imaginary;
            }
 
            z = Coefficients[j];
            ReC[j] = z.Real;
            ImC[j] = z.Imaginary;
 
            int Size = Width * Height;
            double Conv = 1e-10;
            double Div = 1e10;
            int[] ColorIndex = new int[Size];
            int Index = 0;
 
            rDelta *= 0.5;
            r0 = rCenter - rDelta;
            r1 = rCenter + rDelta;
 
            aspect = ((double)Width) / Height;
 
            i0 = iCenter - rDelta / aspect;
            i1 = iCenter + rDelta / aspect;
 
            dr = (r1 - r0) / Width;
            di = (i1 - i0) / Height;
 
            for (y = 0, i = i0; y < Height; y++, i += di)
            {
                for (x = 0, r = r0; x < Width; x++, r += dr)
                {
                    zr = r;
                    zi = i;
 
                    n = 0;
                    do
                    {
                        // f:
                        zr2 = zi2 = 0;
                        for (j = 0; j <= c; j++)
                        {
                            Temp = zr2 * zr - zi2 * zi + ReC[j];
                            zi2 = zr2 * zi + zi2 * zr + ImC[j];
                            zr2 = Temp;
                        }
 
                        // f':
                        zr3 = zi3 = 0;
                        for (j = 0; j < c; j++)
                        {
                            Temp = zr3 * zr - zi3 * zi + RePrim[j];
                            zi3 = zr3 * zi + zi3 * zr + ImPrim[j];
                            zr3 = Temp;
                        }
 
                        // f/f':
 
                        Temp = 1.0 / (zr3 * zr3 + zi3 * zi3);
                        zr4 = (zr2 * zr3 + zi2 * zi3) * Temp;
                        zi4 = (zi2 * zr3 - zr2 * zi3) * Temp;
 
                        // R*f/f'
                        Temp = zr4 * RRe - zi4 * RIm;
                        zi4 = zr4 * RIm + zi4 * RRe;
                        zr4 = Temp;
 
                        zr -= zr4;
                        zi -= zi4;
 
                        Temp = Math.Sqrt(zr4 * zr4 + zi4 * zi4);
                    }
                    while ((Temp > Conv) && (Temp < Div) && (n++ < N));
 
                    if (Temp < Conv && n < N)
                        ColorIndex[Index++] = n;
                    else
                        ColorIndex[Index++] = N;
                }
            }
 
            ColorIndex = FractalGraph.FindBoundaries(ColorIndex, Width, Height);
 
            return new FractalGraph(Variables, FractalGraph.ToPixels(ColorIndex, Width, Height, Palette),
                r0, i0, r1, i1, rDelta * 2, true, Node, FractalZoomScript, State);
        }
 
        public static FractalGraph CalcNewtonTopography(double rCenter, double iCenter, double rDelta, Complex R,
            ILambdaExpression f, Variables Variables, SKColor[] Palette, int Width, int Height,
            ScriptNode Node, FractalZoomScript FractalZoomScript, object State)
        {
            double RRe = R.Real;
            double RIm = R.Imaginary;
            double r0, i0, r1, i1;
            double dr, di;
            double r, i;
            double aspect;
            int x, y;
            int n, N;
 
            N = Palette.Length;
 
            if (Width <= 2 || Height <= 2)
                throw new ScriptRuntimeException("Width and Height has to be greater than 2.", Node);
 
            Variables v = new Variables();
            Variables.CopyTo(v);
 
            if (!(f is IDifferentiable Differentiable) ||
                !(Differentiable.Differentiate(Differentiable.DefaultVariableName, v) is ILambdaExpression fPrim))
            {
                throw new ScriptRuntimeException("Lambda expression not differentiable.", Node);
            }
 
            int Size = Width * Height;
            double Conv = 1e-10;
            double Div = 1e10;
            int[] ColorIndex = new int[Size];
            int Index = 0;
 
            Complex[] Row;
            Complex[] Row2;
            Complex[] Row3;
            int[] Offset;
            IElement[] P = new IElement[1];
            int j, c, x2;
            IElement Obj, Obj2;
            double Mod;
            Complex z;
 
            rDelta *= 0.5;
            r0 = rCenter - rDelta;
            r1 = rCenter + rDelta;
 
            aspect = ((double)Width) / Height;
 
            i0 = iCenter - rDelta / aspect;
            i1 = iCenter + rDelta / aspect;
 
            dr = (r1 - r0) / Width;
            di = (i1 - i0) / Height;
 
            for (y = 0, i = i0; y < Height; y++, i += di)
            {
                Row = new Complex[Width];
                Offset = new int[Width];
 
                c = Width;
                for (x = 0, x2 = y * Width, r = r0; x < Width; x++, r += dr, x2++)
                {
                    Row[x] = new Complex(r, i);
                    Offset[x] = x2;
                }
 
                n = 0;
                while (n < N && c > 0)
                {
                    n++;
                    P[0] = new ComplexVector(Row);
                    Obj = f.Evaluate(P, v);
                    Obj2 = fPrim.Evaluate(P, v);
                    Row2 = Obj.AssociatedObjectValue as Complex[];
                    Row3 = Obj2.AssociatedObjectValue as Complex[];
 
                    if (Row2 is null || Row3 is null)
                    {
                        throw new ScriptRuntimeException("Lambda expression (and its first derivative) must be able to accept complex vectors, " +
                            "and return complex vectors of equal length. Type returned: " +
                            Obj.GetType().FullName + " and " + Obj2.GetType().FullName, Node);
                    }
                    else if (Row2.Length != c || Row3.Length != c)
                    {
                        throw new ScriptRuntimeException("Lambda expression (and its first derivative) must be able to accept complex vectors, " +
                            "and return complex vectors of equal length. Length returned: " +
                            Row2.Length.ToString() + " and " + Row3.Length.ToString() +
                            ". Expected: " + c.ToString(), Node);
                    }
 
                    for (x = x2 = 0; x < c; x++)
                    {
                        j = Offset[x];
                        z = R * Row2[x] / Row3[x];
                        Row[x] -= z;
 
                        Mod = z.Magnitude;
 
                        if (Mod > Conv && Mod < Div)
                        {
                            if (x != x2)
                                Offset[x2] = j;
 
                            x2++;
                        }
                        else
                        {
                            if (n >= N)
                                ColorIndex[Index++] = N;
                            else
                                ColorIndex[j++] = n;
                        }
                    }
 
                    if (x2 < x)
                    {
                        Array.Resize(ref Row, x2);
                        Array.Resize(ref Offset, x2);
                        c = x2;
                    }
                }
            }
 
            ColorIndex = FractalGraph.FindBoundaries(ColorIndex, Width, Height);
 
            return new FractalGraph(Variables, FractalGraph.ToPixels(ColorIndex, Width, Height, Palette),
                r0, i0, r1, i1, rDelta * 2, true, Node, FractalZoomScript, State);
        }
 
        public override string FunctionName => nameof(NewtonTopographyFractal);
    }
}