Let [math]\vec{c}\left(t\right)[/math] be a twice differentiable regular curve with unit normal [math]\vec{N}\left(t\right)[/math] and curvature [math]\kappa\left(t\right)[/math] and let [math]t_0[/math] be in the domain of [math]\vec{c}[/math]. [br]The osculating circle should contain the point [math]\vec{c}\left(t_0\right)[/math] and have curvature [math]\kappa\left(t_0\right)[/math]. Since the curvature of a circle is constant and is the reciprocal of the radius, we know the radius of the osculating circle should be [math]\frac{1}{\kappa\left(t_0\right)}[/math].[br]We also know that the unit normal is parallel to the radius of the osculating circle and pointing towards the center. Thus we can find the center of the osculating circle via:[br][math]\left(h,k\right)=\frac{1}{\kappa\left(t_0\right)}\vec{N}\left(t_0\right)+\vec{c}\left(t_0\right)[/math][br]Now we can parameterize the osculating circle by scaling a unit circle and then translating the center to [math]\left(h,k\right)[/math]:[br][br]Osculating Circle = [math]\frac{1}{\kappa\left(t_0\right)}\left(\cos t,\sin t\right)+\frac{1}{\kappa\left(t_0\right)}\vec{N}\left(t_0\right)+\vec{c}\left(t_0\right)[/math]