In this problem, we define what we mean by \"the circle that fits the parametriz

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In this problem, we define what we mean by "the circle that fits the parametrized curve C best near r0". For convenience, let r(s) be a parametrization of C by arc length with r(0) = r0. Define That, and kappa by That = dr / ds (0), kappa = |d2r / ds2(0)| and kappa = d2r / ds2(0). In this problem, we only consider the case that kappa > 0, so that is a well defined unit vector that is perpendicular to That. Pick any c epsilon IR3, any p? > 0 and any two mutually perpendicular unit vectors That? and ?. Then R(s) = c - p? cos 2 / rho? ? + rho? sin s / rho? That? is a circle, parametrized by arc length. We may parametrize any circle by choosing c, p?, That? and ? appropriately. Set D(s) = |R(s) - r(s)|2. It is, of course, the square of the distance from the point R(s) on the circle to the point r(s) on C. We'll say that the circle above fits C best near r0 if D(0) = D?(0) = D?(0) = D(3)(0) = D(4) (0) = 0. Prove that D(0) = D?(0) = D?(0) = D(3)(0) = D(4)(0) = 0 if and only if That? = That, ? = , rho? = 1 / kappa and c = r0 + 1 / kappa .

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