the osculating circle
/ OSS-kew-lay-ting /
At any point of a curve, among all the circles you could draw, exactly one hugs the curve more snugly than any other — matching not just the point and the direction but also the rate of bending. That best-fitting circle is the osculating circle. The name comes from the Latin osculari, 'to kiss': it is the circle that kisses the curve, sharing the most intimate contact possible at that spot. Where the tangent line is the best straight-line approximation, the osculating circle is the best circular approximation.
Precisely, at a point P on a curve with curvature kappa (assume kappa is not zero), the osculating circle is the unique circle that has the same tangent line as the curve at P and the same curvature there. Its radius is R = 1/kappa, called the radius of curvature, and its center lies a distance R from P in the direction of the principal normal N — that is, on the concave (inside) side of the bend, at the point P + R N. So a sharply curved spot (large kappa) gets a small kissing circle, a nearly straight spot (small kappa) gets a huge one, and a perfectly straight stretch (kappa = 0) gets a 'circle of infinite radius', namely the tangent line itself. To find it concretely: compute kappa at P, set R = 1/kappa, find N, and place the center at P + R N.
The osculating circle makes curvature visible and tangible: kappa = 1/R literally says curvature is the reciprocal of the radius of the kissing circle, so you can see a curve's curvature by watching how tight its osculating circles are. The locus traced by all the osculating-circle centers as P moves along the curve is a new curve called the evolute. One honest subtlety about the word 'kissing': the osculating circle generally crosses the curve at the point of contact rather than staying on one side, because it matches the curve to second order — agreement of position, slope, and bending — which is a far closer fit than the tangent line and is enough to make it cross through rather than merely touch.
For the parabola y = x^2 at its vertex (0, 0), the curvature is kappa = 2, so the radius of curvature is R = 1/2. The principal normal points straight up, so the osculating circle has center (0, 1/2) and radius 1/2 — the circle x^2 + (y - 1/2)^2 = 1/4 nestled inside the parabola's bend at the bottom.
The kissing circle: same tangent, same curvature, radius R = 1/kappa, center at P + R N.
Despite 'kissing', the osculating circle usually crosses the curve at the contact point rather than touching from one side, because it agrees to second order. Where the curvature is zero there is no finite osculating circle — the role is played by the tangent line.