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If a curve resides only in the xy-plane and is defined by the function y = f(t) then there is an easier formula for the curvature. We can parameterize the curve by r(t) = tˆi + f(t)ˆj.

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Well, since we are given the curve in vector form, we will use our first curvature formula of: κ = ‖ r → ′ ( t) × r → ′ ′ ( t) ‖ ‖ r → ′ ( t) ‖ 3. So, first we will need to calculate r → ′ ( t)

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If the curve is expressed in the polar form [r = f (θ)], then the radius of curvature is given by, ρ = (r2 + r2 1)3 2 (r2 + 2r2 1 − rr2) ρ = ( r 2 + r 1 2) 3 2 ( r 2 + 2 r 1 2 - r r 2) Where, r1 = dr dθ r 1 = d r d