text: 
  /*****************************************************************************
   * \file  
   *      class for automatic differentiation on scalar values and 1st 
   *      derivatives and 2nd derivative.
   *       
   *      Erwin Aertbelien, Div. PMA, Dep. of Mech. Eng., K.U.Leuven
   *
   * \version 
   *      ORO_Geometry V0.2
   *
   * \par Note
   *      VC6++ contains a bug, concerning the use of inlined friend functions 
   *      in combination with namespaces.  So, try to avoid inlined friend 
   *      functions !  
   *
   * \par History
   *      - $log$ 
   *
   * \par Release
   *      $Name:  $ 
   ****************************************************************************/
  
  #ifndef Rall2D_H
  #define Rall2D_H
  
  #include <math.h>
  #include <assert.h>
  #include "utility.h"
  
  
  namespace KDL {
  
  /**
   * Rall2d contains a value, and its gradient and its 2nd derivative, and defines an algebraic 
   * structure on this pair.
   *  This template class has 3 template parameters :
   *  -   T contains the type of the value.
   *  -   V contains the type of the gradient (can be a vector-like type).  
   *  -   S defines a scalar type that can operate on Rall1d.  This is the type that 
   *      is used to give back values of Norm() etc. 
   *
   * S is usefull when you recurse a Rall1d object into itself to create a 2nd, 3th, 4th,.. 
   * derivatives. (e.g. Rall1d< Rall1d<double>, Rall1d<double>, double> ).
   *
   * S is always passed by value. 
   *
   * \par Class Type
   * Concrete implementation
   */
  template <class T,class V=T,class S=T>
  class Rall2d                      
      {
      public :
          T t;    //!< value
          V d;    //!< 1st derivative
          V dd;   //!< 2nd derivative
      public :
          // = Constructors
          INLINE Rall2d() {}
  
          explicit INLINE  Rall2d(typename TI<T>::Arg c)
              {t=c;SetToZero(d);SetToZero(dd);}
  
          INLINE Rall2d(typename TI<T>::Arg tn,const V& afg):t(tn),d(afg) {SetToZero(dd);}
  
          INLINE Rall2d(typename TI<T>::Arg tn,const V& afg,const V& afg2):t(tn),d(afg),dd(afg2) {}
  
          // = Copy Constructor
          INLINE Rall2d(const Rall2d<T,V,S>& r):t(r.t),d(r.d),dd(r.dd) {}
              //if one defines this constructor, it's better optimized then the
              //automatically generated one ( that one set's up a loop to copy
              // word by word.
          
          // = Member functions to access internal structures :
          INLINE T& Value() {
              return t;
          }
  
          INLINE V& D() {
              return d;
          }
  
          INLINE V& DD() {
              return dd;
          }
          INLINE static Rall2d<T,V,S> Zero() {
              Rall2d<T,V,S> tmp;
              SetToZero(tmp);
              return tmp;
          }
          INLINE static Rall2d<T,V,S> Identity() {
              Rall2d<T,V,S> tmp;
              SetToIdentity(tmp);
              return tmp;
          }
  
          // = assignment operators
          INLINE Rall2d<T,V,S>& operator =(S c)
              {t=c;SetToZero(d);SetToZero(dd);return *this;}
  
          INLINE Rall2d<T,V,S>& operator =(const Rall2d<T,V,S>& r)
              {t=r.t;d=r.d;dd=r.dd;return *this;}
  
          INLINE Rall2d<T,V,S>& operator /=(const Rall2d<T,V,S>& rhs)
              {
              t /= rhs.t;
              d = (d-t*rhs.d)/rhs.t;
              dd= (dd - S(2)*d*rhs.d-t*rhs.dd)/rhs.t;
              return *this;
              }
  
          INLINE Rall2d<T,V,S>& operator *=(const Rall2d<T,V,S>& rhs)
              {
              t *= rhs.t;
              d  = (d*rhs.t+t*rhs.d);
              dd = (dd*rhs.t+S(2)*d*rhs.d+t*rhs.dd);
              return *this;
              }
  
          INLINE Rall2d<T,V,S>& operator +=(const Rall2d<T,V,S>& rhs)
              {
              t    +=rhs.t;
              d +=rhs.d;
              dd+=rhs.dd;
              return *this;
              }
  
          INLINE Rall2d<T,V,S>& operator -=(const Rall2d<T,V,S>& rhs)
              {
              t     -= rhs.t;
              d     -= rhs.d;
              dd    -= rhs.dd;
              return *this;
              }
  
          INLINE Rall2d<T,V,S>& operator /=(S rhs)
              {
              t    /= rhs;
              d    /= rhs;
              dd   /= rhs;
              return *this;
              }
  
          INLINE Rall2d<T,V,S>& operator *=(S rhs)
              {
              t    *= rhs;
              d    *= rhs;
              dd   *= rhs;
              return *this;
              }
  
          INLINE Rall2d<T,V,S>& operator -=(S rhs)
              {
              t    -= rhs;
              return *this;
              }
  
          INLINE Rall2d<T,V,S>& operator +=(S rhs)
              {
              t    += rhs;
              return *this;
              }
  
          // = Operators between Rall2d objects
  /*
           friend INLINE Rall2d<T,V,S> operator /(const Rall2d<T,V,S>& lhs,const Rall2d<T,V,S>& rhs);
           friend INLINE Rall2d<T,V,S> operator *(const Rall2d<T,V,S>& lhs,const Rall2d<T,V,S>& rhs);
           friend INLINE Rall2d<T,V,S> operator +(const Rall2d<T,V,S>& lhs,const Rall2d<T,V,S>& rhs);
           friend INLINE Rall2d<T,V,S> operator -(const Rall2d<T,V,S>& lhs,const Rall2d<T,V,S>& rhs);
           friend INLINE Rall2d<T,V,S> operator -(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> operator *(S s,const Rall2d<T,V,S>& v);
           friend INLINE Rall2d<T,V,S> operator *(const Rall2d<T,V,S>& v,S s);
           friend INLINE Rall2d<T,V,S> operator +(S s,const Rall2d<T,V,S>& v);
           friend INLINE Rall2d<T,V,S> operator +(const Rall2d<T,V,S>& v,S s);
           friend INLINE Rall2d<T,V,S> operator -(S s,const Rall2d<T,V,S>& v);
           friend INLINE INLINE Rall2d<T,V,S> operator -(const Rall2d<T,V,S>& v,S s);
           friend INLINE Rall2d<T,V,S> operator /(S s,const Rall2d<T,V,S>& v);
           friend INLINE Rall2d<T,V,S> operator /(const Rall2d<T,V,S>& v,S s);
  
          // = Mathematical functions that operate on Rall2d objects
  
           friend INLINE Rall2d<T,V,S> exp(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> log(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> sin(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> cos(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> tan(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> sinh(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> cosh(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> tanh(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> sqr(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> pow(const Rall2d<T,V,S>& arg,double m) ;
           friend INLINE Rall2d<T,V,S> sqrt(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> asin(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> acos(const Rall2d<T,V,S>& arg);
           friend INLINE Rall2d<T,V,S> atan(const Rall2d<T,V,S>& x);
           friend INLINE Rall2d<T,V,S> atan2(const Rall2d<T,V,S>& y,const Rall2d<T,V,S>& x);
           friend INLINE Rall2d<T,V,S> abs(const Rall2d<T,V,S>& x);
           friend INLINE Rall2d<T,V,S> hypot(const Rall2d<T,V,S>& y,const Rall2d<T,V,S>& x);
          // returns sqrt(y*y+x*x), but is optimized for accuracy and speed.
           friend INLINE S Norm(const Rall2d<T,V,S>& value) ;
          // returns Norm( value.Value() ).
  
          // = Some utility functions to improve performance
          // (should also be declared on primitive types to improve uniformity
           friend INLINE Rall2d<T,V,S> LinComb(S alfa,const Rall2d<T,V,S>& a,
              TI<T>::Arg beta,const Rall2d<T,V,S>& b );
           friend INLINE void LinCombR(S alfa,const Rall2d<T,V,S>& a,
              TI<T>::Arg beta,const Rall2d<T,V,S>& b,Rall2d<T,V,S>& result );
          // = Setting value of a Rall2d object to 0 or 1
           friend INLINE void SetToZero(Rall2d<T,V,S>& value);
           friend INLINE void SetToOne(Rall2d<T,V,S>& value);
          // = Equality in an eps-interval
           friend INLINE bool Equal(const Rall2d<T,V,S>& y,const Rall2d<T,V,S>& x,double eps);
           */
      };
  
  
  
  
  
  // = Operators between Rall2d objects
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator /(const Rall2d<T,V,S>& lhs,const Rall2d<T,V,S>& rhs)
      {
      Rall2d<T,V,S> tmp;
      tmp.t = lhs.t/rhs.t;
      tmp.d = (lhs.d-tmp.t*rhs.d)/rhs.t;
      tmp.dd= (lhs.dd-S(2)*tmp.d*rhs.d-tmp.t*rhs.dd)/rhs.t;
      return tmp;
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator *(const Rall2d<T,V,S>& lhs,const Rall2d<T,V,S>& rhs)
      {
      Rall2d<T,V,S> tmp;
      tmp.t  = lhs.t*rhs.t;
      tmp.d  = (lhs.d*rhs.t+lhs.t*rhs.d);
      tmp.dd = (lhs.dd*rhs.t+S(2)*lhs.d*rhs.d+lhs.t*rhs.dd);
      return tmp;
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator +(const Rall2d<T,V,S>& lhs,const Rall2d<T,V,S>& rhs)
      {
      return Rall2d<T,V,S>(lhs.t+rhs.t,lhs.d+rhs.d,lhs.dd+rhs.dd);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator -(const Rall2d<T,V,S>& lhs,const Rall2d<T,V,S>& rhs)
      {
      return Rall2d<T,V,S>(lhs.t-rhs.t,lhs.d-rhs.d,lhs.dd-rhs.dd);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator -(const Rall2d<T,V,S>& arg)
      {
      return Rall2d<T,V,S>(-arg.t,-arg.d,-arg.dd);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator *(S s,const Rall2d<T,V,S>& v)
      {
      return Rall2d<T,V,S>(s*v.t,s*v.d,s*v.dd);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator *(const Rall2d<T,V,S>& v,S s)
      {
      return Rall2d<T,V,S>(v.t*s,v.d*s,v.dd*s);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator +(S s,const Rall2d<T,V,S>& v)
      {
      return Rall2d<T,V,S>(s+v.t,v.d,v.dd);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator +(const Rall2d<T,V,S>& v,S s)
      {
      return Rall2d<T,V,S>(v.t+s,v.d,v.dd);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator -(S s,const Rall2d<T,V,S>& v)
      {
      return Rall2d<T,V,S>(s-v.t,-v.d,-v.dd);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator -(const Rall2d<T,V,S>& v,S s)
      {
      return Rall2d<T,V,S>(v.t-s,v.d,v.dd);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator /(S s,const Rall2d<T,V,S>& rhs)
      {
      Rall2d<T,V,S> tmp;
      tmp.t = s/rhs.t;
      tmp.d = (-tmp.t*rhs.d)/rhs.t;
      tmp.dd= (-S(2)*tmp.d*rhs.d-tmp.t*rhs.dd)/rhs.t;
      return tmp;
  }
  
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> operator /(const Rall2d<T,V,S>& v,S s)
      {
      return Rall2d<T,V,S>(v.t/s,v.d/s,v.dd/s);
      }
  
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> exp(const Rall2d<T,V,S>& arg)
      {
      Rall2d<T,V,S> tmp;
      tmp.t  = exp(arg.t);
      tmp.d  = tmp.t*arg.d;
      tmp.dd = tmp.d*arg.d+tmp.t*arg.dd;
      return tmp;
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> log(const Rall2d<T,V,S>& arg)
      {
      Rall2d<T,V,S> tmp;
      tmp.t  = log(arg.t);
      tmp.d  = arg.d/arg.t;
      tmp.dd = (arg.dd-tmp.d*arg.d)/arg.t;
      return tmp;
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> sin(const Rall2d<T,V,S>& arg)
      {
      T v1 = sin(arg.t);
      T v2 = cos(arg.t);
      return Rall2d<T,V,S>(v1,v2*arg.d,v2*arg.dd - (v1*arg.d)*arg.d );
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> cos(const Rall2d<T,V,S>& arg)
      {
      T v1 = cos(arg.t);
      T v2 = -sin(arg.t);
      return Rall2d<T,V,S>(v1,v2*arg.d, v2*arg.dd - (v1*arg.d)*arg.d);
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> tan(const Rall2d<T,V,S>& arg)
      {
      T v1 = tan(arg.t);
      T v2 = S(1)+sqr(v1);
      return Rall2d<T,V,S>(v1,v2*arg.d, v2*(arg.dd+(S(2)*v1*sqr(arg.d))));
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> sinh(const Rall2d<T,V,S>& arg)
      {
      T v1 = sinh(arg.t);
      T v2 = cosh(arg.t);
      return Rall2d<T,V,S>(v1,v2*arg.d,v2*arg.dd + (v1*arg.d)*arg.d );
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> cosh(const Rall2d<T,V,S>& arg)
      {
      T v1 = cosh(arg.t);
      T v2 = sinh(arg.t);
      return Rall2d<T,V,S>(v1,v2*arg.d,v2*arg.dd + (v1*arg.d)*arg.d );
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> tanh(const Rall2d<T,V,S>& arg)
      {       
      T v1 = tanh(arg.t);
      T v2 = S(1)-sqr(v1);
      return Rall2d<T,V,S>(v1,v2*arg.d, v2*(arg.dd-(S(2)*v1*sqr(arg.d))));
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> sqr(const Rall2d<T,V,S>& arg)
      {
      return Rall2d<T,V,S>(arg.t*arg.t,
                          (S(2)*arg.t)*arg.d,
                          S(2)*(sqr(arg.d)+arg.t*arg.dd)
                          );
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> pow(const Rall2d<T,V,S>& arg,double m) 
      {
      Rall2d<T,V,S> tmp;
      tmp.t = pow(arg.t,m);
      T v2  = (m/arg.t)*tmp.t;
      tmp.d  = v2*arg.d;
      tmp.dd = (S((m-1))/arg.t)*tmp.d*arg.d + v2*arg.dd;
      return tmp;
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> sqrt(const Rall2d<T,V,S>& arg)
      {
      /* By inversion of sqr(x) :*/
      Rall2d<T,V,S> tmp;
      tmp.t  = sqrt(arg.t);
      tmp.d  = (S(0.5)/tmp.t)*arg.d;
      tmp.dd = (S(0.5)*arg.dd-sqr(tmp.d))/tmp.t;
      return tmp;
      }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> asin(const Rall2d<T,V,S>& arg)
  {
      /* By inversion of sin(x) */
      Rall2d<T,V,S> tmp;
      tmp.t  = asin(arg.t);
      T v = cos(tmp.t);
      tmp.d  = arg.d/v;
      tmp.dd = (arg.dd+arg.t*sqr(tmp.d))/v;
      return tmp;
  }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> acos(const Rall2d<T,V,S>& arg)
  {
      /* By inversion of cos(x) */
      Rall2d<T,V,S> tmp;
      tmp.t  = acos(arg.t);
      T v = -sin(tmp.t);
      tmp.d  = arg.d/v;
      tmp.dd = (arg.dd+arg.t*sqr(tmp.d))/v;
      return tmp;
  
  }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> atan(const Rall2d<T,V,S>& x)
  {
      /* By inversion of tan(x) */
      Rall2d<T,V,S> tmp;
      tmp.t  = atan(x.t);
      T v    = S(1)+sqr(x.t);
      tmp.d  = x.d/v;
      tmp.dd = x.dd/v-(S(2)*x.t)*sqr(tmp.d);
      return tmp;
  }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> atan2(const Rall2d<T,V,S>& y,const Rall2d<T,V,S>& x)
  {
      Rall2d<T,V,S> tmp;
      tmp.t  = atan2(y.t,x.t);
      T v    = sqr(y.t)+sqr(x.t);
      tmp.d  = (x.t*y.d-x.d*y.t)/v;
      tmp.dd = ( x.t*y.dd-x.dd*y.t-S(2)*(x.t*x.d+y.t*y.d)*tmp.d ) / v;
      return tmp;
  }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> abs(const Rall2d<T,V,S>& x)
  {
      T v(Sign(x));
      return Rall2d<T,V,S>(v*x,v*x.d,v*x.dd);
  }
  
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> hypot(const Rall2d<T,V,S>& y,const Rall2d<T,V,S>& x)
  {
      Rall2d<T,V,S> tmp;
      tmp.t  = hypot(y.t,x.t);
      tmp.d  = (x.t*x.d+y.t*y.d)/tmp.t;
      tmp.dd = (sqr(x.d)+x.t*x.dd+sqr(y.d)+y.t*y.dd-sqr(tmp.d))/tmp.t;
      return tmp;
  }
  // returns sqrt(y*y+x*x), but is optimized for accuracy and speed.
  
  template <class T,class V,class S>
  INLINE  S Norm(const Rall2d<T,V,S>& value) 
  {
      return Norm(value.t);
  }
  // returns Norm( value.Value() ).
  
  
  // (should also be declared on primitive types to improve uniformity
  template <class T,class V,class S>
  INLINE  Rall2d<T,V,S> LinComb(S alfa,const Rall2d<T,V,S>& a,
      const T& beta,const Rall2d<T,V,S>& b ) {
          return Rall2d<T,V,S>(
              LinComb(alfa,a.t,beta,b.t),
              LinComb(alfa,a.d,beta,b.d),
              LinComb(alfa,a.dd,beta,b.dd) 
          );
  }
  
  template <class T,class V,class S>
  INLINE  void LinCombR(S alfa,const Rall2d<T,V,S>& a,
      const T& beta,const Rall2d<T,V,S>& b,Rall2d<T,V,S>& result ) {
              LinCombR(alfa, a.t,       beta, b.t,      result.t);
              LinCombR(alfa, a.d,    beta, b.d,   result.d);
              LinCombR(alfa, a.dd,    beta, b.dd,   result.dd);
  }
  
  template <class T,class V,class S>
  INLINE  void SetToZero(Rall2d<T,V,S>& value)
      {
      SetToZero(value.t);
      SetToZero(value.d);
      SetToZero(value.dd);
      }
  
  template <class T,class V,class S>
  INLINE  void SetToIdentity(Rall2d<T,V,S>& value)
      {
      SetToZero(value.d);
      SetToIdentity(value.t);
      SetToZero(value.dd);
      }
  
  template <class T,class V,class S>
  INLINE  bool Equal(const Rall2d<T,V,S>& y,const Rall2d<T,V,S>& x,double eps=epsilon)
  {
      return (Equal(x.t,y.t,eps)&&
              Equal(x.d,y.d,eps)&&
              Equal(x.dd,y.dd,eps)
              );
  }
  
  
  }
  
  
  #endif
  