Lecture 3 Let c, d be two unit speed curves. We say that c and d osculate at t0 if they are both deﬁned at that point and satisfy c(t0 ) = d(t0 ), c� (t0 ) = d� (t0 ), c�� (t0 ) = d�� (t0 ). Because the curves are unit speed, c�� (t0 ) = d�� (t0 ) is equivalent to saying that κc (t0 ) = κd (t0 ). Proposition 3.1. Let c be a unit speed curve, and t0 a point where κ(t0 ) �= 0. Then there is a unique circle which osculates c at t0 (the osculating circle). The curvature |κ(t0 )| is then the inverse radius of the osculating circle at that point. If the curvature is zero, there is no osculating circle, and instead the curve osculates its tangent line. Proposition 3.2. Let f : U → R be a smooth function, deﬁned on an open subset U ⊂ R2 . Let c : I → U be a regular curve, which is contained in its level set {f (x) = a}. Then, at every point t such that x = c(t) satisﬁes �f (x) = � 0, we have ±κ(t) = �J�f (x), D2 f (x)J�f (x)� . ��f (x)�3 Here, D2 f (x) is the Hessian (the matrix of second derivatives). The sign is determined as follows. If det(�f (x), c� (t)) > 0, then κ(t) is the right hand side of the equation above. Otherwise, −κ(t) is the right hand side. Example 3.3. Let f : R2 → R be a function with f (0) = 0, Df (0) = 0, and D2 f (0) positive deﬁnite (so that the origin is a local minimum). Then as one gets closer and closer to the origin, the curvature of the level sets goes to inﬁnity.