/*
    This file is part of "sphereEversion",
    a program by Michael McGuffin.
    The code in this file was almost entirely taken
    (with slight adaptations) from the source code of
    "evert", a program written by Nathaniel Thurston.
    evert's source code can be down loaded from
        http://www.geom.umn.edu/docs/outreach/oi/software.html
        http://www.geom.uiuc.edu/docs/outreach/oi/software.html

    Grateful acknowledgements go out to Nathaniel Thurston,
    Silvio Levy, and the Geometry Center (University of Minnesota)
    for making evert's source code freely available to the public.

    Update: June 2016: Bruno Levy - ported to GLUP and Doxygen documentation.
             Made many small optimizations: using references whenever possible.
             Removed all unnecessary memory allocations. Create vertices in
             one single contiguous chunk of memory.

    It seems that the original geometry center webpage no longer exists, but
    some information is still available on Michael McGuffin's webpage:
        http://www.dgp.toronto.edu/~mjmcguff/eversion/
*/

#include "generateGeometry.h"

#include <math.h>
#include <stdio.h>
#include <stdlib.h>

// ----------------------------------------

class TwoJet;

/**
 * \brief Stores the value and the derivatives of a bivariate function.
 * \details  A jet is the collection of all of the low-order
 * derivatives of a function up to a certain point.  For instance, the
 * two-jet of f(x), a function of one variable, can be represented
 * by the triple (f, df/dx, d^2f/dx^2)
 */
class TwoJet {
public: 
    
    TwoJet() {
    }

    TwoJet(double d, double du, double dv) {
        f = d;
        fu = du;
        fv = dv;
        fuv = 0.0;
    }

    TwoJet(double d, double du, double dv, double duv) {
        f = d;
        fu = du;
        fv = dv;
        fuv = duv;
    }

    double as_double() const {
        return f;
    }
    
    bool operator<(double d) const {
        return f < d;
    }

    bool operator>(double d) const {
        return f > d;
    }

    bool operator<=(double d) const {
        return f <= d;
    }

    bool operator>=(double d) const {
        return f >= d;
    }

    double df_du() const {
        return fu;
    }

    double df_dv() const {
        return fv;
    }

    double d2f_dudv() const {
        return fuv;
    }

    void operator +=(const TwoJet& x) {
        f += x.f;
        fu += x.fu;
        fv += x.fv;
        fuv += x.fuv;
    }

    void operator +=(double d) {
        f += d;
    }

    void operator *=(const TwoJet& x) {
        fuv = f*x.fuv + fu*x.fv + fv*x.fu + fuv*x.f;
        fu = f*x.fu + fu*x.f;
        fv = f*x.fv + fv*x.f;
        f *= x.f;
    }

    void operator *=(double d) {
        f *= d;
        fu *= d;
        fv *= d;
        fuv *= d;
    }
    
    void operator %=(double d) {
        f = fmod(f, d);
        if (f < 0.0) {
            f += d; }
    }

    void operator ^=(double n) {
        if (f > 0.0) {
            double x0 = pow(f, n);
            double x1 = n * x0/f;
            double x2 = (n-1)*x1/f;
            fuv = x1*fuv + x2*fu*fv;
            fu = x1*fu;
            fv = x1*fv;
            f = x0;
        }
    }

    void Annihilate(int index) {
        if (index == 0) {
            fu = 0;
        } else if (index == 1) {
            fv = 0;
        }
        fuv = 0;
    }

    void TakeSin() {
        *this *= 2*M_PI;
        double s = sin(f), c = cos(f);
        f = s;
        fu = fu*c;
        fv = fv*c;
        fuv = c*fuv - s*fu*fv;
    }
    
    void TakeCos() {
        *this *= 2*M_PI;
        double s = cos(f), c = -sin(f);
        f = s; fu = fu*c; fv = fv*c; fuv = c*fuv - s*fu*fv;
    }

    friend TwoJet operator+(const TwoJet& x, const TwoJet& y);
    friend TwoJet operator*(const TwoJet& x, const TwoJet& y);
    friend TwoJet operator+(const TwoJet& x, double d);
    friend TwoJet operator*(const TwoJet& x, double d);
    friend TwoJet Sin(const TwoJet& x);
    friend TwoJet Cos(const TwoJet& x);
    friend TwoJet operator^(const TwoJet& x, double n);
    friend TwoJet Annihilate(const TwoJet& x, int index);
    friend TwoJet Interpolate(
        const TwoJet& v1, const TwoJet& v2, const TwoJet& weight
    );
    friend class TwoJet D(const class ThreeJet& x, int index);
    friend class ThreeJet;
    
private:
    double f;
    double fu, fv;
    double fuv;
};

// ----------------------------------------

TwoJet operator+(const TwoJet& x, const TwoJet& y) {
  return TwoJet(x.f+y.f, x.fu+y.fu, x.fv+y.fv, x.fuv + y.fuv);
}

TwoJet operator*(const TwoJet& x, const TwoJet& y) {
  return TwoJet(
    x.f*y.f,
    x.f*y.fu + x.fu*y.f,
    x.f*y.fv + x.fv*y.f,
    x.f*y.fuv + x.fu*y.fv + x.fv*y.fu + x.fuv*y.f
  );
}

TwoJet operator+(const TwoJet& x, double d) {
  return TwoJet( x.f + d, x.fu, x.fv, x.fuv);
}

TwoJet operator*(const TwoJet& x, double d) {
  return TwoJet( d*x.f, d*x.fu, d*x.fv, d*x.fuv);
}

TwoJet Sin(const TwoJet& x) {
  TwoJet t = x*(2*M_PI);
  double s = sin(t.f);
  double c = cos(t.f);
  return TwoJet(s, c*t.fu, c*t.fv, c*t.fuv - s*t.fu*t.fv);
}

TwoJet Cos(const TwoJet& x) {
  TwoJet t = x*(2*M_PI);
  double s = cos(t.f);
  double c = -sin(t.f);
  return TwoJet(s, c*t.fu, c*t.fv, c*t.fuv - s*t.fu*t.fv);
}

TwoJet operator^(const TwoJet& x, double n) {
  double x0 = pow(x.f, n);
  double x1 = (x.f == 0) ? 0 : n * x0/x.f;
  double x2 = (x.f == 0) ? 0 : (n-1)*x1/x.f;
  return TwoJet(x0, x1*x.fu, x1*x.fv, x1*x.fuv + x2*x.fu*x.fv);
}

TwoJet Annihilate(const TwoJet& x, int index) {
  return TwoJet(x.f, index == 1 ? x.fu : 0, index == 0 ? x.fv : 0, 0);
}

TwoJet Interpolate(const TwoJet& v1, const TwoJet& v2, const TwoJet& weight) {
  return (v1) * ((weight) * (-1) + 1) + v2*weight;
}


// ----------------------------------------

/**
 * \brief Stores the value and the derivatives of a bivariate function.
 * \details  A jet is the collection of all of the low-order
 * derivatives of a function up to a certain point.  For instance, the
 * two-jet of f(x), a function of one variable, can be represented
 * by the triple (f, df/dx, d^2f/dx^2)
 */
class ThreeJet {
public:
    ThreeJet(
        double d,
        double du, double dv,
        double duu, double duv, double dvv,
        double duuv, double duvv
    ) {
        f = d;
        fu = du;
        fv = dv;
        fuu = duu;
        fuv = duv;
        fvv = dvv;
        fuuv = duuv;
        fuvv = duvv;
    }

    ThreeJet() {
    }

    ThreeJet(double d, double du, double dv) {
        f = d;
        fu = du;
        fv = dv;
        fuu = 0.0;
        fuv = 0.0;
        fvv = 0.0;
        fuuv = 0.0;
        fuvv = 0.0;
    }

    double as_double() const {
        return f;
    }

    operator TwoJet() const {
        return TwoJet(f, fu, fv, fuv);
    }

    bool operator<(double d) const {
        return f < d;
    }

    bool operator>(double d) const {
        return f > d;
    }

    bool operator<=(double d) const {
        return f <= d;
    }

    bool operator>=(double d) const {
        return f >= d;
    }

    void operator %=(double d) {
        f = fmod(f, d);
        if (f < 0.0) {
            f += d;
        }
    }

private:
    
    friend ThreeJet operator+(const ThreeJet& x, const ThreeJet& y);
    friend ThreeJet operator*(const ThreeJet& x, const ThreeJet& y);
    friend ThreeJet operator+(const ThreeJet& x, double d);
    friend ThreeJet operator*(const ThreeJet& x, double d);
    friend ThreeJet Sin(const ThreeJet& x);
    friend ThreeJet Cos(const ThreeJet& x);
    friend ThreeJet operator^(const ThreeJet& x, double n);
    friend ThreeJet Annihilate(const ThreeJet& x, int index);
    friend ThreeJet Interpolate(
        const ThreeJet& v1, const ThreeJet& v2, const ThreeJet& weight
    );
    friend TwoJet D(const ThreeJet& x, int index);
    
    double f;
    double fu, fv;
    double fuu, fuv, fvv;
    double fuuv, fuvv;
};

// ----------------------------------------

ThreeJet operator+(const ThreeJet& x, const ThreeJet& y) {
    ThreeJet result;
    result.f = x.f + y.f;
    result.fu = x.fu + y.fu;
    result.fv = x.fv + y.fv;
    result.fuu = x.fuu + y.fuu;
    result.fuv = x.fuv + y.fuv;
    result.fvv = x.fvv + y.fvv;
    result.fuuv = x.fuuv + y.fuuv;
    result.fuvv = x.fuvv + y.fuvv;
    return result;
}

ThreeJet operator*(const ThreeJet& x, const ThreeJet& y) {
    ThreeJet result;
    result.f = x.f*y.f;
    result.fu = x.f*y.fu + x.fu*y.f;
    result.fv = x.f*y.fv + x.fv*y.f;
    result.fuu = x.f*y.fuu + 2*x.fu*y.fu + x.fuu*y.f;
    result.fuv = x.f*y.fuv + x.fu*y.fv + x.fv*y.fu + x.fuv*y.f;
    result.fvv = x.f*y.fvv + 2*x.fv*y.fv + x.fvv*y.f;
    result.fuuv = x.f*y.fuuv + 2*x.fu*y.fuv + x.fv*y.fuu
        + 2*x.fuv*y.fu + x.fuu*y.fv + x.fuuv*y.f;
    result.fuvv = x.f*y.fuvv + 2*x.fv*y.fuv + x.fu*y.fvv
        + 2*x.fuv*y.fv + x.fvv*y.fu + x.fuvv*y.f;
    return result;
}

ThreeJet operator+(const ThreeJet& x, double d) {
    ThreeJet result;
    result = x;
    result.f += d;
    return result;
}

ThreeJet operator*(const ThreeJet& x, double d) {
    ThreeJet result;
    result.f = d*x.f;
    result.fu = d*x.fu;
    result.fv = d*x.fv;
    result.fuu = d*x.fuu;
    result.fuv = d*x.fuv;
    result.fvv = d*x.fvv;
    result.fuuv = d*x.fuuv;
    result.fuvv = d*x.fuvv;
    return result;
}

ThreeJet Sin(const ThreeJet& x) {
    ThreeJet result;
    ThreeJet t = x*(2*M_PI);
    double s = sin(t.f);
    double c = cos(t.f);
    result.f = s;
    result.fu = c*t.fu;
    result.fv = c*t.fv;
    result.fuu = c*t.fuu - s*t.fu*t.fu;
    result.fuv = c*t.fuv - s*t.fu*t.fv;
    result.fvv = c*t.fvv - s*t.fv*t.fv;
    result.fuuv = c*t.fuuv - s*(2*t.fu*t.fuv + t.fv*t.fuu) - c*t.fu*t.fu*t.fv;
    result.fuvv = c*t.fuvv - s*(2*t.fv*t.fuv + t.fu*t.fvv) - c*t.fu*t.fv*t.fv;
    return result;
}

ThreeJet Cos(const ThreeJet& x) {
    ThreeJet result;
    ThreeJet t = x*(2*M_PI);
    double s = cos(t.f);
    double c = -sin(t.f);
    result.f = s;
    result.fu = c*t.fu;
    result.fv = c*t.fv;
    result.fuu = c*t.fuu - s*t.fu*t.fu;
    result.fuv = c*t.fuv - s*t.fu*t.fv;
    result.fvv = c*t.fvv - s*t.fv*t.fv;
    result.fuuv = c*t.fuuv - s*(2*t.fu*t.fuv + t.fv*t.fuu) - c*t.fu*t.fu*t.fv;
    result.fuvv = c*t.fuvv - s*(2*t.fv*t.fuv + t.fu*t.fvv) - c*t.fu*t.fv*t.fv;
    return result;
}

ThreeJet operator^(const ThreeJet& x, double n) {
    double x0 = pow(x.f, n);
    double x1 = (x.f == 0) ? 0 : n * x0/x.f;
    double x2 = (x.f == 0) ? 0 : (n-1) * x1/x.f;
    double x3 = (x.f == 0) ? 0 : (n-2) * x2/x.f;
    ThreeJet result;
    result.f = x0;
    result.fu = x1*x.fu;
    result.fv = x1*x.fv;
    result.fuu = x1*x.fuu + x2*x.fu*x.fu;
    result.fuv = x1*x.fuv + x2*x.fu*x.fv;
    result.fvv = x1*x.fvv + x2*x.fv*x.fv;
    result.fuuv =
        x1*x.fuuv + x2*(2*x.fu*x.fuv + x.fv*x.fuu) + x3*x.fu*x.fu*x.fv;
    result.fuvv =
        x1*x.fuvv + x2*(2*x.fv*x.fuv + x.fu*x.fvv) + x3*x.fu*x.fv*x.fv;
    return result;
}

TwoJet D(const ThreeJet& x, int index) {
    TwoJet result;
    if (index == 0) {
        result.f = x.fu;
        result.fu = x.fuu;
        result.fv = x.fuv;
        result.fuv = x.fuuv;
    } else if (index == 1) {
        result.f = x.fv;
        result.fu = x.fuv;
        result.fv = x.fvv;
        result.fuv = x.fuvv;
    } else {
        result.f = 0.0;
        result.fu = 0.0;
        result.fv = 0.0;
        result.fuv = 0.0;
    }
    return result;
}

ThreeJet Annihilate(const ThreeJet& x, int index) {
    ThreeJet result = ThreeJet(x.f,0,0);
    if (index == 0) {
        result.fv = x.fv;
        result.fvv = x.fvv;
    } else if (index == 1) {
        result.fu = x.fu;
        result.fuu = x.fuu;
    }
    return result;
}

ThreeJet Interpolate(
    const ThreeJet& v1, const ThreeJet& v2, const ThreeJet& weight
) {
    return (v1) * ((weight) * (-1) + 1) + v2*weight;
}

// ----------------------------------------

/**
 * \brief A vector with automatic differentiation.
 * \details Each coordinate of the vector stores a function
 *  value and derivatives.
 */
struct TwoJetVec {
    TwoJet x;
    TwoJet y;
    TwoJet z;
    TwoJetVec() {
    }
    TwoJetVec(
        TwoJet a, TwoJet b, TwoJet c
    ) {
        x = a;
        y = b;
        z = c;
    }
};

TwoJetVec operator+(const TwoJetVec& v, const TwoJetVec& w);
TwoJetVec operator*(const TwoJetVec& v, const TwoJet& a);
TwoJetVec operator*(const TwoJetVec& v, double a);
TwoJetVec AnnihilateVec(const TwoJetVec& v, int index);
TwoJetVec Cross(const TwoJetVec& v, const TwoJetVec& w);
TwoJet Dot(const TwoJetVec& v, const TwoJetVec& w);
TwoJetVec Normalize(const TwoJetVec& v);
TwoJetVec RotateZ(const TwoJetVec& v, const TwoJet& angle);
TwoJetVec RotateY(const TwoJetVec& v, const TwoJet& angle);
TwoJetVec RotateX(const TwoJetVec& v, const TwoJet& angle);
TwoJetVec InterpolateVec(
    const TwoJetVec& v1, const TwoJetVec& v2, const TwoJet& weight
);
TwoJet Length(const TwoJetVec& v);

// ----------------------------------------

TwoJetVec operator+(const TwoJetVec& v, const TwoJetVec& w) {
    TwoJetVec result;
    result.x = v.x + w.x;
    result.y = v.y + w.y;
    result.z = v.z + w.z;
    return result;
}

TwoJetVec operator*(const TwoJetVec& v, const TwoJet& a) {
    TwoJetVec result;
    result.x = v.x*a;
    result.y = v.y*a;
    result.z = v.z*a;
    return result;
}

TwoJetVec operator*(const TwoJetVec& v, double a) {
    TwoJetVec result;
    result.x = v.x*a;
    result.y = v.y*a;
    result.z = v.z*a;
    return result;
}

TwoJetVec AnnihilateVec(const TwoJetVec& v, int index) {
    TwoJetVec result;
    result.x = Annihilate(v.x, index);
    result.y = Annihilate(v.y, index);
    result.z = Annihilate(v.z, index);
    return result;
}

TwoJetVec Cross(const TwoJetVec& v, const TwoJetVec& w) {
    TwoJetVec result;
    result.x = v.y*w.z + v.z*w.y*-1;
    result.y = v.z*w.x + v.x*w.z*-1;
    result.z = v.x*w.y + v.y*w.x*-1;
    return result;
}

TwoJet Dot(const TwoJetVec& v, const TwoJetVec& w) {
    return v.x*w.x + v.y*w.y + v.z*w.z;
}

TwoJetVec Normalize(const TwoJetVec& v) {
    TwoJet a;
    a = Dot(v,v);
    if (a > 0) {
        a = a^-0.5;
    } else {
        a = TwoJet(0, 0, 0);
    }
    return v*a;
}

TwoJetVec RotateZ(const TwoJetVec& v, const TwoJet& angle) {
    TwoJetVec result;
    TwoJet s,c;
    s = Sin (angle);
    c = Cos (angle);
    result.x =          v.x*c + v.y*s;
    result.y = v.x*s*-1 + v.y*c;
    result.z = v.z;
    return result;
}

TwoJetVec RotateY(const TwoJetVec& v, const TwoJet& angle) {
    TwoJetVec result;
    TwoJet s, c;
    s = Sin (angle);
    c = Cos (angle);
    result.x = v.x*c + v.z*s*-1;
    result.y = v.y;
    result.z = v.x*s + v.z*c    ;
    return result;
}

TwoJetVec RotateX(const TwoJetVec& v, const TwoJet& angle) {
    TwoJetVec result;
    TwoJet s,c;
    s = Sin (angle);
    c = Cos (angle);
    result.x = v.x;
    result.y = v.y*c + v.z*s;
    result.z = v.y*s*-1 + v.z*c;
    return result;
}

TwoJetVec InterpolateVec(
    const TwoJetVec& v1, const TwoJetVec& v2, const TwoJet& weight
) {
    return (v1) * (weight*-1 + 1) + v2*weight;
}

TwoJet Length(const TwoJetVec& v) {
    return (TwoJet(v.x^2) + TwoJet(v.y^2)) ^ (.5);
}

// ----------------------------------------

/**
 * \brief A vector with automatic differentiation.
 * \details Each coordinate of the vector stores a function
 *  value and derivatives.
 */
struct ThreeJetVec {
    ThreeJet x;
    ThreeJet y;
    ThreeJet z;
    operator TwoJetVec() {
        return TwoJetVec(x,y,z);
    }
};

ThreeJetVec operator+(const ThreeJetVec& v, const ThreeJetVec& w);
ThreeJetVec operator*(const ThreeJetVec& v, const ThreeJet& a);
ThreeJetVec operator*(const ThreeJetVec& v, double a);
ThreeJetVec AnnihilateVec(const ThreeJetVec& v, int index);
ThreeJetVec Cross(const ThreeJetVec& v, const ThreeJetVec& w);
ThreeJet Dot(const ThreeJetVec& v, const ThreeJetVec& w);
TwoJetVec D(const ThreeJetVec& x, int index);
ThreeJetVec Normalize(const ThreeJetVec& v);
ThreeJetVec RotateZ(const ThreeJetVec& v, const ThreeJet& angle);
ThreeJetVec RotateY(const ThreeJetVec& v, const ThreeJet& angle);
ThreeJetVec RotateX(const ThreeJetVec& v, const ThreeJet& angle);
ThreeJetVec InterpolateVec(
    const ThreeJetVec& v1, const ThreeJetVec& v2, const ThreeJet& weight
);
ThreeJet Length(const ThreeJetVec& v);

// ----------------------------------------

ThreeJetVec operator+(const ThreeJetVec& v, const ThreeJetVec& w) {
    ThreeJetVec result;
    result.x = v.x + w.x;
    result.y = v.y + w.y;
    result.z = v.z + w.z;
    return result;
}

ThreeJetVec operator*(const ThreeJetVec& v, const ThreeJet& a) {
    ThreeJetVec result;
    result.x = v.x*a;
    result.y = v.y*a;
    result.z = v.z*a;
    return result;
}

ThreeJetVec operator*(const ThreeJetVec& v, double a) {
    ThreeJetVec result;
    result.x = v.x*a;
    result.y = v.y*a;
    result.z = v.z*a;
    return result;
}

ThreeJetVec AnnihilateVec(const ThreeJetVec& v, int index) {
    ThreeJetVec result;
    result.x = Annihilate(v.x, index);
    result.y = Annihilate(v.y, index);
    result.z = Annihilate(v.z, index);
    return result;
}

TwoJetVec D(const ThreeJetVec& x, int index) {
    TwoJetVec result;
    result.x = D(x.x, index);
    result.y = D(x.y, index);
    result.z = D(x.z, index);
    return result;
}

ThreeJetVec Cross(const ThreeJetVec& v, const ThreeJetVec& w) {
    ThreeJetVec result;
    result.x = v.y*w.z + v.z*w.y*-1;
    result.y = v.z*w.x + v.x*w.z*-1;
    result.z = v.x*w.y + v.y*w.x*-1;
    return result;
}

ThreeJet Dot(const ThreeJetVec& v, const ThreeJetVec& w) {
    return v.x*w.x + v.y*w.y + v.z*w.z;
}

ThreeJetVec Normalize(const ThreeJetVec& v) {
    ThreeJet a;
    a = Dot(v,v);
    if (a > 0) {
        a = a^-0.5;
    } else {
        a = ThreeJet(0, 0, 0);
    }
    return v*a;
}

ThreeJetVec RotateZ(const ThreeJetVec& v, const ThreeJet& angle) {
    ThreeJetVec result;
    ThreeJet s,c;
    s = Sin (angle);
    c = Cos (angle);
    result.x =          v.x*c + v.y*s;
    result.y = v.x*s*-1 + v.y*c;
    result.z = v.z;
    return result;
}

ThreeJetVec RotateY(const ThreeJetVec& v, const ThreeJet& angle) {
    ThreeJetVec result;
    ThreeJet s, c;
    s = Sin (angle);
    c = Cos (angle);
    result.x = v.x*c + v.z*s*-1;
    result.y = v.y;
    result.z = v.x*s + v.z*c    ;
    return result;
}

ThreeJetVec RotateX(const ThreeJetVec& v, const ThreeJet& angle) {
    ThreeJetVec result;
    ThreeJet s,c;
    s = Sin (angle);
    c = Cos (angle);
    result.x = v.x;
    result.y = v.y*c + v.z*s;
    result.z = v.y*s*-1 + v.z*c;
    return result;
}

ThreeJetVec InterpolateVec(
    const ThreeJetVec& v1, const ThreeJetVec& v2, const ThreeJet& weight
) {
    return (v1) * (weight*-1 + 1) + v2*weight;
}

ThreeJet Length(const ThreeJetVec& v) {
    return (ThreeJet(v.x^2) + ThreeJet(v.y^2)) ^ (.5);
}

// ----------------------------------------

static TwoJetVec FigureEight(
    const TwoJetVec& w, TwoJetVec h,
    const TwoJetVec& bend, const TwoJet& form, TwoJet v
) {
    TwoJet height;
    v %= 1;
    height = (Cos (v*2) + -1) * (-1);
    if (v > 0.25 && v < 0.75)
        height = height*-1 + 4;
    height = height*0.6;
    h = h + bend*(height*height*(1/64.0));
    return w*Sin (v*2) +
        (h) * (Interpolate((Cos (v) + -1) * (-2), height, form)) ;
}

static TwoJetVec AddFigureEight(
    ThreeJetVec p, const ThreeJet& u, const TwoJet& v,
    ThreeJet form, const ThreeJet& scale, int numStrips
) {
    ThreeJet size = form * scale;
    form = form*2 + form*form*-1;
    TwoJetVec dv = AnnihilateVec(D(p, 1), 1);
    p = AnnihilateVec(p, 1);
    TwoJetVec du = Normalize(D(p, 0));
    TwoJetVec h = Normalize(Cross(du, dv))*TwoJet(size);
    TwoJetVec w = Normalize(Cross(h, du))*(TwoJet(size)*1.1);
    return RotateZ(
        TwoJetVec(p) +
        FigureEight(w, h, du*D(size, 0)*(D(u, 0)^(-1)), form, v),
        v*(1.0/numStrips)
    );
}

// ----------------------------------------

static ThreeJetVec Arc(
    ThreeJet u, const ThreeJet& v,
    double xsize, double ysize, double zsize
) {
   ThreeJetVec result;
   u = u*0.25;
   result.x = Sin (u) * Sin (v) * xsize;
   result.y = Sin (u) * Cos (v) * ysize;
   result.z = Cos (u) * zsize;
   return result;
}

static ThreeJetVec Straight(
    ThreeJet u, const ThreeJet& v,
    double xsize, double ysize, double zsize
) {
   ThreeJetVec result;
   u = u*0.25;
   result.x = Sin (v) * xsize;
   result.y = Cos (v) * ysize;
   result.z = Cos (u) * zsize;
   return result;
}

static ThreeJet Param1(ThreeJet x) {
    double offset = 0;
    x %= 4;
    if (x > 2) {
        x = x+(-2);
        offset = 2;
    }
    if (x <= 1) {
        return x*2 + (x^2)*(-1) + offset;
    }
    return (x^2) + x*(-2) + (2 + offset);
}

static ThreeJet Param2(ThreeJet x) {
    double offset = 0;
    x %= 4;
    if (x > 2) {
        x = x+(-2);
        offset = 2;
    }
    if (x <= 1) {
        return (x^2) + offset;
    }
    return (x^2)*(-1) + x*4 + (-2 + offset);
}

inline ThreeJet TInterp(double x) {
    return ThreeJet(x,0,0);
}

static ThreeJet UInterp(ThreeJet x) {
    x %= 2;
    if (x > 1) {
        x = x*(-1) + 2;
    }
    return (x^2)*3 + (x^3) * (-2);
}

static const int FFPOW = 3;

static ThreeJet FFInterp(ThreeJet x) {
    x %= 2;
    if (x > 1) {
        x = x*(-1) + 2;
    }
    x = x*1.06 + -0.05;
    if (x < 0) {
        return ThreeJet(0, 0, 0);
    }
    if (x > 1) {
        return ThreeJet(0, 0, 0) + 1;
    }
    return (x ^ (FFPOW-1)) * (FFPOW) + (x^FFPOW) * (-FFPOW+1);
}

static const int FSPOW = 3;

static ThreeJet FSInterp(ThreeJet x) {
   x %= 2;
   if (x > 1) {
       x = x*(-1) + 2;
   }
   return ((x ^ (FSPOW-1)) * (FSPOW) + (x^FSPOW) * (-FSPOW+1)) * (-0.2);
}

static ThreeJetVec Stage0(const ThreeJet& u, const ThreeJet& v) {
    return Straight(u, v, 1, 1, 1);
}

static ThreeJetVec Stage1(const ThreeJet& u, const ThreeJet& v) {
   return Arc(u, v, 1, 1, 1);
}

static ThreeJetVec Stage2(const ThreeJet& u, const ThreeJet& v) {
    return InterpolateVec(
        Arc(Param1(u), v, 0.9, 0.9, -1),
        Arc(Param2(u), v, 1, 1, 0.5),
        UInterp(u)
    );
}

static ThreeJetVec Stage3(const ThreeJet& u, const ThreeJet& v) {
    return InterpolateVec(
        Arc(Param1(u), v,-0.9,-0.9,-1),
        Arc(Param2(u), v,-1, 1,-0.5),
        UInterp(u)
    );
}

static ThreeJetVec Stage4(const ThreeJet& u, const ThreeJet& v) {
   return Arc(u, v, -1,-1, -1);
}

static ThreeJetVec Scene01(const ThreeJet& u, const ThreeJet& v, double t) {
    return InterpolateVec(Stage0(u,v), Stage1(u,v), TInterp(t));
}

static ThreeJetVec Scene12(const ThreeJet& u, const ThreeJet& v, double t) {
    return InterpolateVec(Stage1(u,v), Stage2(u,v), TInterp(t));
}

static ThreeJetVec Scene23(const ThreeJet& u, const ThreeJet& v, double t) {
    ThreeJet tmp = TInterp(t);
    t = tmp.as_double() * 0.5;
    double tt = (u <= 1) ? t : -t;
    return InterpolateVec(
        RotateZ(Arc(Param1(u), v, 0.9, 0.9,-1), ThreeJet(tt,0,0)),
        RotateY(Arc(Param2(u), v, 1, 1, 0.5), ThreeJet(t,0,0)),
        UInterp(u)
    );
}

static ThreeJetVec Scene34(const ThreeJet& u, const ThreeJet& v, double t) {
    return InterpolateVec(Stage3(u,v), Stage4(u,v), TInterp(t));
}

static TwoJetVec BendIn(
    const ThreeJet& u, const ThreeJet& v, double t, int numStrips
) {
    ThreeJet tmp = TInterp(t);
    t = tmp.as_double();
    return AddFigureEight(
        Scene01(u, ThreeJet(0, 0, 1), t),
        u, v, ThreeJet(0, 0, 0), FSInterp(u),
        numStrips
    );
}

static TwoJetVec Corrugate(
    const ThreeJet& u, const ThreeJet& v, double t, int numStrips
) {
   ThreeJet tmp = TInterp(t);
   t = tmp.as_double();
   return AddFigureEight(
      Stage1(u, ThreeJet(0, 0, 1)),
      u, v, FFInterp(u) * ThreeJet(t,0,0), FSInterp(u),
      numStrips
   );
}

static TwoJetVec PushThrough(
    const ThreeJet& u, const ThreeJet& v, double t, int numStrips
) {
   return AddFigureEight(
      Scene12(u,ThreeJet(0, 0, 1),t),
      u, v, FFInterp(u), FSInterp(u),
      numStrips
   );
}

static TwoJetVec Twist(
    const ThreeJet& u, const ThreeJet& v, double t, int numStrips
) {
    return AddFigureEight(
        Scene23(u,ThreeJet(0, 0, 1),t),
        u, v, FFInterp(u), FSInterp(u),
        numStrips
    );
}

static TwoJetVec UnPush(
    const ThreeJet& u, const ThreeJet& v, double t, int numStrips
) {
    return AddFigureEight(
        Scene34(u,ThreeJet(0, 0, 1),t),
        u, v, FFInterp(u), FSInterp(u),
        numStrips
    );
}

static TwoJetVec UnCorrugate(
    const ThreeJet& u, const ThreeJet& v, double t, int numStrips
) {
    ThreeJet tmp;
    tmp = TInterp((t) * (-1) + 1);
    t = tmp.as_double();
    return AddFigureEight(
        Stage4(u,ThreeJet(0, 0, 1)),
        u, v, FFInterp(u) * ThreeJet(t,0,0), FSInterp(u),
        numStrips
    );
}

// ----------------------------------------

/**
 * \brief Converts a TwoJetVec into a point and a normal.
 * \param[in] p a const reference to the TwoJetVec
 * \param[out] point a pointer to the three floating point coordinates
 *  of the point
 * \param[out] normal a pointer to the three floating point coordinates
 *  of the normal
 */
static void getPoint(const TwoJetVec& p, float* point, float* normal) {
    double x = p.x.as_double() ;
    double y = p.y.as_double() ;
    double z = p.z.as_double() ;
    double nx = p.y.df_du()*p.z.df_dv()-p.z.df_du()*p.y.df_dv();
    double ny = p.z.df_du()*p.x.df_dv()-p.x.df_du()*p.z.df_dv();
    double nz = p.x.df_du()*p.y.df_dv()-p.y.df_du()*p.x.df_dv();
    double s = nx*nx + ny*ny + nz*nz;
    if (s > 0.0) {
        s = sqrt(1.0/s);
    }
    point[0] = float(x);
    point[1] = float(y);
    point[2] = float(z);
    normal[0] = float(-nx*s);
    normal[1] = float(-ny*s);
    normal[2] = float(-nz*s);
}

/**
 * \brief Converts a TwoJetVec into a point.
 * \param[in] p a const reference to the TwoJetVec
 * \param[out] point a pointer to the three floating point coordinates
 *  of the point
 */
static void getPoint(const TwoJetVec& p, float* point) {
    double x = p.x.as_double() ;
    double y = p.y.as_double() ;
    double z = p.z.as_double() ;
    point[0] = float(x);
    point[1] = float(y);
    point[2] = float(z);
}

// ----------------------------------------

typedef TwoJetVec SurfaceTimeFunction(
    const ThreeJet& u, const ThreeJet& v, double t, int numStrips
);

static void computePointGrid(
    SurfaceTimeFunction *func,
    double umin, double umax, int ucount,
    double vmin, double vmax, int vcount,
    double t,
    float* points,
    float* normals,
    int numStrips
) {
    double delta_u = (umax-umin) / double(ucount);
    double delta_v = (vmax-vmin) / double(vcount);
    for(int v=0; v<=vcount; ++v) {
        ThreeJet V(vmin + double(v)*delta_v, 0, 1);
        for(int u=0; u<=ucount; ++u) {
            ThreeJet U(umin + double(u)*delta_u, 1, 0);
            TwoJetVec P = func(U,V,t,numStrips);
            int offset = (v * (ucount + 1) + u)*3;
            if(normals == NULL) {
                getPoint(P, points + offset);
            } else {
                getPoint(P, points + offset, normals + offset);
            }
        }
    }
}

// ----------------------------------------

void generateGeometry(
    float* points,
    float* normals,
    
    double time,
    int numStrips,

    double u_min, int u_count, double u_max,
    double v_min, int v_count, double v_max,
    
    double bendtime,
    
    double corrStart,
    double pushStart,
    double twistStart,
    double unpushStart,
    double uncorrStart
) {
    if (NULL == points) {
        return;
    }

    if (bendtime >= 0.0) {
        computePointGrid(
            BendIn,
            u_min, u_max, u_count,
            v_min, v_max, v_count,
            bendtime,
            points, normals,
            numStrips
        );
   } else {

        /* time = (time - howfar) / chunk */

        if (time >= uncorrStart) {
            computePointGrid(
                UnCorrugate,
                u_min, u_max, u_count,
                v_min, v_max, v_count,
                (time - uncorrStart) / (1.0 - uncorrStart),
                points, normals,
                numStrips
            );
        } else if (time >= unpushStart) {
            computePointGrid(
                UnPush,
                u_min, u_max, u_count,
                v_min, v_max, v_count,
                (time - unpushStart) / (uncorrStart - unpushStart),
                points, normals,
                numStrips
            );
        } else if (time >= twistStart) {
            computePointGrid(
                Twist,
                u_min, u_max, u_count,
                v_min, v_max, v_count,
                (time - twistStart) / (unpushStart - twistStart),
                points, normals,
                numStrips
            );
        } else if (time >= pushStart) {
            computePointGrid(
                PushThrough,
                u_min, u_max, u_count,
                v_min, v_max, v_count,
                (time - pushStart) / (twistStart - pushStart),
                points, normals,
                numStrips
            );
        } else if (time >= corrStart) {
            computePointGrid(
                Corrugate,
                u_min, u_max, u_count,
                v_min, v_max, v_count,
                (time - corrStart) / (pushStart - corrStart),
                points, normals,
                numStrips
            );
        }
   }
}

