First Order Derivatives

• Velocity Vector: [; \underline{v}(t)=\underline{r}'(t)= \left( \begin{array}{c} x'(t)\\y'(t) \end{array} \right) ;]
• Scalar Velocity: [; v(t)= \sqrt{ x'(t)^2+y'(t)^2} \geq 0 ;]
• Taylor's Formula 1st Order: [; \underline{r}(t)= \underline{r}(t_0) +(t-t_0) \underline{r}'(t_0) +(t-t_0) \underline{\epsilon}(t), \quod t \approx t_0 ;] [; \underline{\epsilon}(t) \rightarrow \underline{0},\quad t \rightarrow t_0 ;]
• Aproximation by Tangent: Linear Fit

    eps=1.0E-3
    eps2inv=1.0/(2.0*self.eps)

    dRs=[]
    
    def dR(t):
       return (
           self.R(t+self.eps)-self.R(t-self.eps)
       )*self.eps2inv
   
    ##!
    ##! Calculate and store derivatives.
    ##!
    
    def Calc_dRs(self,ts=[]):
        ts=self.Get_ts(ts)
        self.dRs=[]
        for i in range( len(ts )):
            self.dRs[i]=self.dR(ts[i])
            
        return self.dRs