Curvature and Evolute

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• Lines doesn't curve
• Circles curves constantly
• Small circles curves more than big Circles
• Tangent: Best linear Aproximation
• Oscullating Circle: Best circular Aproximation
  • Through point [; \underline{r}(t) ;]
  • Same Tangent [; \underline{r}'(t) ;]
  • Center of Curvature (Hurrycane's eye)
  • Curvature: [; \kappa(t)=\frac{ \widehat{\underline{r}}'(t) \cdot \underline{r}''(t) }{ |\underline{r}'(t)|^3} ;]
  • Oscillating Vector: [; \underline{\rho}(t)=\frac{ \widehat{\underline{r}}'(t) \cdot \underline{r}''(t) }{ |\underline{r}'(t)|^2} \underline{t}(t) ;]
• Evolute: Curve of Centers of Curvatures [; \underline{c}(t)= \underline{r}(t)+\underline{\rho}(t);]