Question

Write the fomula of radius of curvature for parametric curves and polar curves

Answer 1

The radius of curvature is given by



(1)

where  is the curvature. At a given point on a curve,  is the radius of the osculating circle. The symbol  is sometimes used instead of  to denote the radius of curvature (e.g., Lawrence 1972, p. 4).
Let  and  be given parametrically by


(2)



(3)

then



(4)

where  and . Similarly, if the curve is written in the form , then the radius of curvature is given by



(5)

In polar coordinates , the radius of curvature is given by



(6)

where  and (Gray 1997, p. 89).