subroutine fd2(n,x,sx,h,dcut)
*     differentiate a periodic function. differed from fd1 by a factor
*     of 2 pi.
      integer i,n
      double precision x(0:8192),sx(0:8192)
      double precision h,dcut,p2
      p2 = 8.D0*datan(1.D0)
      call fd1(n,x,sx,h,dcut)
      do 10 i=0,n
         sx(i) = sx(i)/p2
 10   continue
      end

      subroutine fin2(n,h,p2,yi,y)
*     integrate a periodic function. differed from fin by a factor
*     of 2 pi.
      integer i,n
      double precision yi(0:8192),y(0:8192)
      double precision h,p2
      call fin(n,h,p2,yi,y)
      do 10 i=0,n
         yi(i) = p2*yi(i)
 10   continue
      end

      subroutine checkl(n,x,y)
*     ***********************************************************
*     check the equidistance condition of the parametrization
*     for the curve.
*     ***********************************************************
      integer i,n
      double precision x(0:8192),y(0:8192),dist(8192)
      double precision disave,err
      disave = 0.D0
      do 10 i=1,n
         dist(i) = dsqrt( (x(i)-x(i-1))**2 + (y(i)-y(i-1))**2 )
         disave = disave + dist(i)
 10   continue
      disave = disave/dble(n)
      err = 0.D0
      do 20 i=1,n
         err = err + (dist(i)-disave)**2
 20   continue
      err = dsqrt(err/dble(n))
      print *,'equidistance error =',err
      end

      subroutine checkv(n,velo)
*     ***********************************************************
*     obtain the average velocity of the curve.
*     ***********************************************************
      integer i,n
      double precision velo(0:8192)
      double precision vsum,vave
      vsum = 0.D0
      do 10 i=1,n
         vsum = vsum + velo(i)
 10   continue
      vave = vsum/dble(n)
      print *,'vave = ',vave
      end