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Functions.cpp
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Functions.cpp
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/*
FUNCTIONS.CPP a module for the per pixel calculations of fractals.
Written in Microsoft Visual 'C++' by Paul de Leeuw.
This program is written in "standard" C. Hardware dependant code
(console drivers & serial I/O) is in separate machine libraries.
*/
#include <math.h>
#include "manp.h"
#include "fractype.h"
#include "Complex.h"
#include "pixel.h"
/**************************************************************************
Initialise functions for each pixel
**************************************************************************/
int CPixel::InitFunctions(WORD type, Complex *z, Complex *q)
{
switch (type)
{
case MANDELFP: // Mandelbrot
case MANDEL: // to handle fractint par files
case JULIA: // to handle fractint par files
case JULIAFP: // like he said
/**************************************************************************
Run thorn type fractals
Written by Paul Bourke
Fractal attributed to Andrew Wayne Graff, alternatively named the "Secant Sea"
Contribution by Dane Vandeputte
Original: May 2004, Updated: May 2012
Sample C source code -- Contribution by Adam Majewski
This fractal is created by iterating the functions (xn,yn) and shading each pixel (x0,y0)
depending on whether or how fast the series escapes to infinity.
The parameter (cx,cy) is gives (slightly) different fractal images.
**************************************************************************/
case THORN: // Thorn Fractal
/**************************************************************************
The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function
in the complex plane which will either escape or remain bounded. The difference between this calculation and that for the Mandelbrot set is
that the real and imaginary components are set to their respective absolute values before squaring at each iteration. The mapping is non-analytic
because its real and imaginary parts do not obey the Cauchy–Riemann equations.
***************************************************************************/
case BURNINGSHIP: // Burning Ship
if (!juliaflag)
{
z->x = q->x + param[0];
z->y = q->y + param[1];
}
sqr = 0;
real_imag = 0.0;
break;
/**************************************************************************
The Burning Ship fractal for Higher Powers
The function that is iterated to generate this fractal can be generalized
to any arbitrary power:
z = [|Re(z)| + i |Im(z)|]N + c
As with the Mandelbrot set, this creates shapes that resemble the original
shape but have increasingly many attached arms or buds. Below is a table
showing the first few of these.
With N=2 we get the Burning Ship. The N=3 version has been called the "Bird of Prey."
The odd powers are symmetric around an axes that are oriented at +/- 45 degrees.
This is probably due to the symmetry of the absolute value functions. This
property has been mentioned at this site (external link) back in 1999.
These images are all centered at (0,0) in the complex number plane, and all
have a size of 5 horizontally and 4 vertically. This size is a bit too big
and results in a lot of blank space in the images, but the N=2 fractal is
offset from the origin, so the larger size is needed to have all the images
drawn with a consistent location and size so they can be compared accurately.
Here are a few images of the Cubic Burning Ship. The "bug" images are named
because they look like actual bugs, not because of a software error! These
are the first deep zoom images ever published of the cubic Burning Ship fractal.
The left image size is 1.0e-19, and the right image is 1.0e-23.
http://www.hpdz.net/StillImages/BurningShip.htm
***************************************************************************/
case BURNINGSHIPPOWER: // Burning Ship to higher power
case POWER: // Power
case JULIA4FP:
case JULIA4:
case MANDEL4FP:
case MANDEL4:
*degree = (int)param[0];
if (*degree < 1)
*degree = 1;
if (type == JULIA4FP || type == JULIA4 || type == MANDEL4FP || type == MANDEL4) // handle legacy Fractint types
*degree = 4;
if (!juliaflag)
{
z->x = q->x + param[1];
z->y = q->y + param[2];
}
break;
case CUBIC: // Art Matrix Cubic
switch ((int)param[0])
{
case 0:
subtype = 'B';
break;
case 1:
subtype = 'C';
break;
case 2:
subtype = 'F';
break;
case 3:
subtype = 'K';
break;
default:
subtype = 'B';
break;
}
if ((int)param[1] < 0)
special = 0;
else
special = (int)param[1];
period_level = FALSE; // no periodicity checking
if (subtype == 'B') // CBIN
{
t3 = *q * 3; // T3 = 3*T
t2 = q->CSqr(); // T2 = T*T
a = (t2 + 1) / t3; // A = (T2 + 1)/T3
// B = 2*A*A*A + (T2 - 2)/T3
temp = a.CCube() * 2; // 2*A*A*A
b = (t2 - 2) / t3 + temp; // B = 2*A*A*A + (T2 - 2)/T3
}
else if (subtype == 'C' || subtype == 'F') // CCIN or CFIN
{
a = *q; // A = T
// find B = T + 2*T*T*T
temp = q->CCube(); // B = T*T*T
if (subtype == 'C')
b = temp + temp + *q; // B = B * 2 + T
else
{
b = (temp - *q) * 2; // B = B * 2 - 2 * T
a2 = a + a;
}
}
else if (subtype == 'K') // CKIN
{
a = 0;
v = 0;
b = *q; // B = T
}
aa3 = a.CSqr() * 3; // AA3 = A*A*3
if (!juliaflag)
*z = -a; // Z = -A
break;
case SPECIALNEWT: // Art Matrix Newton
l2 = q->CSqr(); // L2 = L*L
a = -l2 + 0.25; // A = ( .25,0) - L2
b = -l2 - 0.75; // B = (-.75,0) - L2
lm5 = *q - 0.5; // LM5 = L - (.5,0)
lp5 = *q + 0.5; // LP5 = L + (.5,0)
break;
case MATEIN: // Art Matriuc Matein fractal
if ((absolute = q->CSumSqr()) > 1.0)
return(-1); // not inside set
if (!juliaflag)
*z = 1;
for (int i = 0; i < 100; ++i) // DO 300 I = 1,100
{
temp = z->CInvert(); // 300 Z = L*(Z + 1/Z)
*z = *q * (*z + temp);
}
distance = 1.0; // D = 1
oz = z->CInvert(); // OZ = 1/Z
break;
case SINFRACTAL: // Sine
if (!juliaflag)
*z = param[3];
break;
case REDSHIFTRIDER: // RedShiftRider a*z^2 +/- z^n + c
a.x = param[0];
a.y = param[1];
*degree = (int)param[2];
if (!juliaflag)
{
z->x = q->x + param[3];
z->y = q->y + param[4];
}
break;
case TALIS: // Talis Power Z = Z^N/(M + Z^(N-1)) + C
*degree = (int)param[0];
if (*degree < 1)
*degree = 1;
if (!juliaflag)
{
z->x = q->x + param[2];
z->y = q->y + param[3];
}
break;
case POLYNOMIAL: // Polynomial
if (!juliaflag)
{
z->x = q->x + param[0];
z->y = q->y + param[1];
}
for (int i = 0; i < MAXPOLYDEG; i++) // find highest order of polynomial to help with forward differencing calculations
{
if (param[2 + i] != 0.0)
{
fractalspecific[type].SlopeDegree = MAXPOLYDEG - i;
break;
}
}
break;
case RATIONALMAP: // Art Matrix Rational Map
switch ((int)param[0])
{
case 0:
subtype = 'A';
break;
case 1:
subtype = 'B';
break;
default:
subtype = 'A';
break;
}
if ((int)param[1] < 0)
special = 0;
else
special = (int)param[1];
if (threshold != OldThreshold)
{
OldThreshold = threshold;
int gap = threshold / 16; // split the colour map into 16 equal parts
for (int i = 0; i < 4; i++)
{
penp[i] = penpref[i] * gap;
penn[i] = pennref[i] * gap;
}
}
if (subtype == 'A')
{
cmcc = *q - q->CSqr(); // CMCC = C - C*C
temp = -*q + 2;
a = temp / cmcc; // A = (2 - C)/CMCC
temp = cmcc + 1;
b = -temp / cmcc; // B = -(CMCC + 1)/CMCC
// ALPHA = 1/(C*C * (B + B + B*B/A) * (2*A*C + B))
temp = a * *q * 2 + b; // 2*A*C + B
temp1 = b.CSqr() / a + b + b; // (B + B + B*B/A)
temp3 = q->CSqr()*temp1*temp;
alpha = temp3.CInvert();
}
else if (subtype == 'B')
{
a = *q;
b = a + 1;
temp = a.CSqr() - 1;
alpha = a / temp; // ALPHA = A/(A*A - 1)
}
else
return(ERROR); // unknown subtype
// ESCAPE = 4/ABS(ALPHA)
// ESCAPE = ESCAPE*ESCAPE
if (alpha.x != 0.0 || alpha.y != 0.0)
escape = 16.0 / alpha.CSumSqr();
else
return(FALSE); // no naughty division
epsilon = 0.000001 / escape; // EPSILN = 0.000001/ESCAPE
der = 1.0; // DER = 1.0
if (!juliaflag)
{ // Z = -B/(A + A)
temp = -a * 2;
*z = b / temp;
}
// iterating Z = 1/(A*Z*Z + B*Z + 1) has various proterties:
// Z = 1/(A*Z*Z + B*Z + 1)
// Julia Z = 1/(A*Z*Z + B*Z + 1)
// Julia Z = 1/(A*Z*Z + B*Z + 1)
// ????? Z = 1/(A*Z*Z + B*Z + 1)
int zcount;
switch (subtype)
{
case 'A':
if (juliaflag)
zcount = 4;
else
zcount = 2;
break;
case 'B':
zcount = 3;
break;
}
for (int i = 0; i < zcount; ++i)
{
// 1/(A*Z*Z + B*Z + 1)
temp = b * *z + 1; // B*Z + 1
temp1 = z->CSqr()*a + temp; // (A*Z*Z + B*Z + 1)
*z = temp1.CInvert(); // Z = 1/(A*Z*Z + B*Z + 1)
}
break;
case MANDELDERIVATIVES: // a group of Mandelbrot Derivatives
InitManDerFunctions(subtype, z, q);
break;
case TIERAZON: // a group of Tierazon fractals
InitTierazonFunctions(subtype, z, q);
break;
case NEWTONAPPLE: // a specific Tierazon fractal
InitTierazonFunctions(55, z, q);
break;
case NEWTONFLOWER: // a specific Tierazon fractal
InitTierazonFunctions(35, z, q);
break;
case NEWTONMSET: // a specific Tierazon fractal
InitTierazonFunctions(52, z, q);
break;
case NEWTONPOLYGON: // a specific Tierazon fractal
InitTierazonFunctions(31, z, q);
break;
case NEWTONCROSS: // a specific Tierazon fractal
InitTierazonFunctions(59, z, q);
break;
case NEWTONJULIANOVA: // a specific Tierazon fractal
InitTierazonFunctions(2, z, q);
break;
case NEWTONVARIATION: // a specific Tierazon fractal
InitTierazonFunctions(87, z, q);
break;
case QUARTET1: // a specific Tierazon fractal
InitTierazonFunctions(85, z, q);
break;
case QUARTET2: // a specific Tierazon fractal
InitTierazonFunctions(85, z, q);
break;
case QUARTET3: // a specific Tierazon fractal
InitTierazonFunctions(96, z, q);
break;
case NOVA:
if (*degree < 2)
*degree = 2;
z->x = 1.0 + param[1];
z->y = 0.0 + param[2];
break;
case QUAD: // a specific Tierazon fractal
InitTierazonFunctions(90, z, q);
break;
case RAMONSIN: // a specific Tierazon fractal
InitTierazonFunctions(116, z, q);
break;
case RAMONCOS: // a specific Tierazon fractal
InitTierazonFunctions(117, z, q);
break;
case FORMULA05: // a specific Tierazon fractal
InitTierazonFunctions(5, z, q);
break;
case TEDDY: // a specific Tierazon fractal
InitTierazonFunctions(104, z, q);
break;
/*
#define NEWTONFLOWER 186
#define NEWTONMSET 190
#define NEWTONPOLYGON 174
#define NEWTONCROSS 193
#define NEWTONJULIANOVA 191
#define NEWTONVARIATION 207
#define QUARTET1 194
#define QUARTET2 195
#define QUARTET3 196
#define NOVA 208
#define QUAD 201
#define RAMONSIN 197
#define RAMONCOS 198
#define FORMULA05 199
#define TALIS 192
#define TEDDY 189
*/
case TETRATION: // a specific Tierazon fractal
InitTetration(z, q);
break;
case KLEINIAN: // a specific Tierazon fractal
InitKleinian(z, q);
break;
}
return 0;
}
/**************************************************************************
Run functions for each iteration
**************************************************************************/
int CPixel::RunFunctions(WORD type, Complex *z, Complex *q, BYTE *SpecialFlag, long *iteration)
{
switch (type)
{
case MANDELFP: // Mandelbrot
case MANDEL: // to handle fractint par files
case JULIA: // to handle fractint par files
case JULIAFP: // like he said
sqr.x = z->x * z->x;
sqr.y = z->y * z->y;
real_imag = z->x * z->y;
z->x = q->x + sqr.x - sqr.y;
z->y = q->y + real_imag + real_imag;
return BailoutTest(z, sqr);
/*
{
Complex z2 = {z->x * z->x, z->y * z->y};
Complex one = {1.0,0.0};
*z = (*z * *z * *z) / (one + z2) + *q;
return (z->CSumSqr() >= rqlim);
}
{
z->y += q->y * sin(z->x);
z->x += q->x * z->y;
return (z->CSumSqr() >= rqlim);
}
*/
case POWER: // Power
case JULIA4FP:
case JULIA4:
case MANDEL4FP:
case MANDEL4:
*z = z->CPolynomial(*degree);
*z = *z + *q;
return FractintBailoutTest(z);
// return (z->CSumSqr() >= rqlim);
case BURNINGSHIP: // Burning Ship
sqr.x = z->x * z->x;
sqr.y = z->y * z->y;
real_imag = fabs(z->x * z->y);
z->x = sqr.x - sqr.y + q->x;
z->y = real_imag + real_imag - q->y;
return BailoutTest(z, sqr);
case BURNINGSHIPPOWER: // Burning Ship to higher power
z->x = fabs(z->x);
z->y = -fabs(z->y);
*z = z->CPolynomial(*degree);
*z = *z + *q;
return FractintBailoutTest(z);
case CUBIC: // Art Matrix Cubic
if (subtype == 'K') // CKIN
{
*z = z->CCube() + b; // Z = Z*Z*Z + B
z->x += param[2];
z->y += param[3];
}
else
{
temp = z->CCube() + b; // Z = Z*Z*Z + B
*z = temp - aa3 * *z; // Z = Z*Z*Z - AA3*Z + B
z->x += param[2];
z->y += param[3];
}
if (z->CSumSqr() > 100.0)
return (TRUE);
else
{
if (subtype == 'F')
{
if (q->CSumSqr() < 0.111111)
{
*iteration = special;
*SpecialFlag = TRUE; // for decomp and biomorph
return (TRUE);
}
v = *z + a2;
}
else if (subtype == 'K')
v = *z - v;
else
v = *z - a;
if (v.CSumSqr() <= 0.000001)
{
*iteration = special;
*SpecialFlag = TRUE; /* for decomp and biomorph */
return (TRUE);
}
return (FALSE);
}
case SPECIALNEWT: // Art Matrix Newton
{
if ((int)param[0] < 0)
special = 2;
else
special = (int)param[0];
Complex z2 = z->CSqr(); // z2 = z*z
// Z = (2*Z*Z2 + A)/(3*Z2 + B)
Complex top = z2 * *z * 2 + a;
Complex bottom = z2 * 3 + b;
*z = top / bottom;
z->x += param[1];
z->y += param[2];
v = *z - 1;
// v.x += param[1];
// v.y += param[2];
if (v.CSumSqr() <= 0.000001)
{
phaseflag = 0; // first phase
return(TRUE);
}
v = *z - lm5;
if (v.CSumSqr() <= 0.000001)
{
phaseflag = 1; // second phase
return(TRUE);
}
v = *z + lp5;
if (v.CSumSqr() <= 0.000001)
{
phaseflag = 2; // third phase
return(TRUE);
}
return(FALSE);
}
case MATEIN: // Art Matriuc Matein fractal
epsilon = 0.01;
escape = 10.0E20;
*z = *q * (*z + oz); // Z = L*(Z + OZ)
z->x += param[0];
z->y += param[1];
oz = z->CInvert(); // OZ = 1/Z
temp = -oz / *z + 1; // T = 1 - OZ/Z
// D = D*ABSL*(REAL(T)*REAL(T) + IMAG(T)*IMAG(T))
distance = distance * absolute * temp.CSumSqr();
if (distance <= epsilon)
{
phaseflag = 0; // first phase
return(TRUE);
}
if (distance > escape)
{
phaseflag = 1; // second phase
return(TRUE);
}
return(FALSE);
/**************************************************************************
Determine count before 'Z' becomes unstable
Z = L*SIN(Z)
sin(x+iy) = sin(x)cosh(y) + icos(x)sinh(y)
***************************************************************************/
case SINFRACTAL: // Sine
if (param[2] == 0.0)
*z = *q * z->CSin();
else
*z = *q + z->CSin();
z->x += param[0];
z->y += param[1];
return FractintBailoutTest(z);
case EXPFRACTAL: // Exponential
{
int compare;
*degree = (int)(fabs(param[3]));
for (int i = 0; i < *degree; i++)
*z = z->CExp();
if (param[4] == 0.0)
*z = *q * *z; // Z = L*EXP(Z)
else
*z = *q + *z; // Z = L*EXP(Z)
// Complex Exponent: e^(x+iy) = (e^x) * cos(y) + i * (e^x) * sin(y)
switch ((int)param[0])
{
case 0:
subtype = 'R';
compare = (z->x >= rqlim);
break;
case 1:
subtype = 'I';
compare = (z->y >= rqlim);
break;
case 2:
subtype = 'M';
compare = (z->x >= rqlim || z->x <= -rqlim || z->y >= rqlim || z->y <= -rqlim);
break;
default:
subtype = 'R';
compare = (z->x >= rqlim);
break;
}
z->x += param[1];
z->y += param[2];
if (compare)
return(TRUE);
return(FALSE);
}
case THORN: // Thorn Fractal
{
double a1 = z->x;
double b1 = z->y;
z->x = a1 / cos(b1) + q->x;
z->y = b1 / sin(a1) + q->y;
return FractintBailoutTest(z);
}
case REDSHIFTRIDER: // RedShiftRider a*z^2 +/- z^n + c
*z = a * *z * *z + ((param[5] == 1.0) ? 1.0 : -1.0) * z->CPolynomial(*degree);
*z = *z + *q;
return FractintBailoutTest(z);
case TALIS: // Talis Power Z = Z^N/(M + Z^(N-1)) + C
{
double m = param[1];
a = z->CPolynomial(*degree - 1);
*z = (a * *z) / (m + a) + *q;
return FractintBailoutTest(z);
}
case POLYNOMIAL: // Polynomial
{
Complex InitialZ = *z;
Complex FinalZ = { 0.0, 0.0 };
{
for (int m = 0; m < MAXPOLYDEG; m++)
{
Complex BigComplexTemp = InitialZ;
if (param[2 + m] != 0.0)
{
for (int k = 0; k < MAXPOLYDEG - m - 1; k++)
BigComplexTemp *= InitialZ;
FinalZ += (BigComplexTemp * param[2 + m]);
}
}
*z = FinalZ + *q;
}
return FractintBailoutTest(z);
}
case RATIONALMAP: // Art Matrix Rational Map
{
Complex az = a * *z; // AZ = A*Z
temp = az * *z + 1;
temp1 = b * *z + temp;
*z = temp1.CInvert(); // Z = 1/(AZ*Z + B*Z + 1)
z->x += param[2];
z->y += param[3];
if (z->CSumSqr() > escape)
{
if ((alpha.x * z->y + alpha.y * z->x) <= 0.0)
*color = penp[*iteration % 4];
else
*color = penn[*iteration % 4];
*iteration = *color;
return (TRUE);
}
temp = az * 2 + b;
Complex d = z->CSqr()*temp; // D = (2*AZ + B)*Z*Z
double dist = d.CSumSqr();
der *= dist; // DER = DER*(REAL(D)*REAL(D) + IMAG(D)*IMAG(D))
if (der < epsilon)
{
*iteration = threshold;
return(TRUE);
}
if (*iteration >= threshold)
{
*iteration = special;
return(TRUE);
}
return(FALSE); // continue iterations
}
case MANDELDERIVATIVES: // a group of Mandelbrot Derivatives
return (RunManDerFunctions(subtype, z, q, SpecialFlag, iteration));
case TIERAZON: // a group of Tierazon fractalsa
return (RunTierazonFunctions(subtype, z, q, SpecialFlag, iteration));
case NEWTONAPPLE: // a specific Tierazon fractal
return (RunTierazonFunctions(55, z, q, SpecialFlag, iteration));
case NEWTONFLOWER: // a specific Tierazon fractal
return (RunTierazonFunctions(35, z, q, SpecialFlag, iteration));
case NEWTONMSET: // a specific Tierazon fractal
return (RunTierazonFunctions(52, z, q, SpecialFlag, iteration));
case NEWTONPOLYGON: // a specific Tierazon fractal
return (RunTierazonFunctions(31, z, q, SpecialFlag, iteration));
case NEWTONCROSS: // a specific Tierazon fractal
return (RunTierazonFunctions(59, z, q, SpecialFlag, iteration));
case NEWTONJULIANOVA: // a specific Tierazon fractal
return (RunTierazonFunctions(2, z, q, SpecialFlag, iteration));
case NEWTONVARIATION: // a specific Tierazon fractal
return (RunTierazonFunctions(87, z, q, SpecialFlag, iteration));
case QUARTET1: // a specific Tierazon fractal
return (RunTierazonFunctions(85, z, q, SpecialFlag, iteration));
case QUARTET2: // a specific Tierazon fractal
return (RunTierazonFunctions(85, z, q, SpecialFlag, iteration));
case QUARTET3: // a specific Tierazon fractal
return (RunTierazonFunctions(96, z, q, SpecialFlag, iteration));
case NOVA:
{
double d;
Complex z1, zd, fn, f1n;
z1 = *z;
f1n = z->CPolynomial(*degree - 1); // z^(deg - 1) - first derivative power
fn = f1n * *z; // z^(deg)
*z = *z - (fn - 1) / (*degree*f1n) + *q;
zd = *z - z1;
d = zd.CSumSqr();
return (d < MINSIZE);
}
case QUAD: // a specific Tierazon fractal
return (RunTierazonFunctions(90, z, q, SpecialFlag, iteration));
case RAMONSIN: // a specific Tierazon fractal
return (RunTierazonFunctions(116, z, q, SpecialFlag, iteration));
case RAMONCOS: // a specific Tierazon fractal
return (RunTierazonFunctions(117, z, q, SpecialFlag, iteration));
case FORMULA05: // a specific Tierazon fractal
return (RunTierazonFunctions(5, z, q, SpecialFlag, iteration));
case TEDDY: // a specific Tierazon fractal
return (RunTierazonFunctions(104, z, q, SpecialFlag, iteration));
case TETRATION: // Tetration fractal
return(DoTetration(iteration));
break;
case KLEINIAN: // Kleinian fractal
return (CalculateKleinian(z));
break;
}
return 0;
}
/**************************************************************************
Bailout Test
**************************************************************************/
bool CPixel::BailoutTest(Complex *z, Complex SqrZ)
{
// Complex TempSqr;
double magnitude;
double manhmag;
double manrmag;
switch (BailoutTestType)
{
case BAIL_MOD:
magnitude = SqrZ.x + SqrZ.y;
return (magnitude >= rqlim);
case BAIL_REAL:
return (SqrZ.x >= rqlim);
case BAIL_IMAG:
return (SqrZ.y >= rqlim);
case BAIL_OR:
return (SqrZ.x >= rqlim || SqrZ.y >= rqlim);
case BAIL_AND:
return (SqrZ.x >= rqlim && SqrZ.y >= rqlim);
case MANH:
manhmag = fabs(z->x) + fabs(z->y);
return ((manhmag * manhmag) >= rqlim);
case MANR:
manrmag = z->x + z->y; // don't need abs() since we square it next
return ((manrmag * manrmag) >= rqlim);
default:
magnitude = SqrZ.x + SqrZ.y;
return (magnitude >= rqlim);
}
}