Parametrization & Derivatives

• [; \underline{r}(t)= r \left( \begin{array}{c} t-\sin{t}\\1-\cos{t}\end{array} \right) = r \left( t \underline{i}+\underline{j}- \underline{q}(t) \right) ;]
• [; \underline{r}'(t)= r \left( \begin{array}{c} 1-\cos{t}\\\sin{t}\end{array} \right) = r \left( \underline{i}- \underline{p}(t) \right) ;]
• [; \underline{r}''(t)= r \left( \begin{array}{c} \sin{t}\\\cos{t}\end{array} \right) = r \underline{q}(t) ;]
• Canonical Cycloid: [; r=1 ;]

class Cycloid(Curve):
    r=1.0
    
    def Cycloid(self,t):
        return Vector([
            t-sin(t),
            1.0-cos(t)
        ])*self.r
    
    def dCycloid(self,t):
        return Vector([
            1.0-cos(t),
            sin(t)
        ])*self.r
    
    def d2Cycloid(self,t):
        return Vector([
            sin(t),
            cos(t)
        ])*self.r