Curvature

of y=f(x) #

Radius of Curvature = $$\frac{(1+f’^2)^{3/2}}{f’’}$$

At the origin #

• Newton’s Method
  • If x-axis is tangent, ROC=$$\lim_{(x,y)\rightarrow(0,0)}\left(\frac{x^2}{2y}\right)$$
  • If y-axis is tangent, ROC=$$\lim_{(x,y)\rightarrow(0,0)}\left(\frac{y^2}{2x}\right)$$
• Series Method: If none of x,y axes are tangent
  • Apply the regular ROC formula

of (x(t),y(t)) #

Radius of Curvature = $$\frac{(x’^2+y’^2)^{3/2}}{y’‘x’-x’‘y’}$$

of r=f(θ) #

Radius of Curvature = $$ \frac{(r^2+r_1^2)^{3/2}}{r^2+2r_1-rr_2}\\ where\\ r_1=r’,r_2=r’’ $$