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  "body": "\n## Summary\n\nThe osculating circle of a curve at a point is the circle that fits the curve best at that point.\nIt touches the curve there, with the same tangent line and the same [[curvature|7JQS6VKFEZ]].\nIts radius is the radius of curvature $1/\\kappa$, and its centre is the centre of curvature.\n\n## Explanations\n\n### L1 human\n\nStand at a point of a curved road and try to draw a circle that follows the road as long as possible near you.\nOn a gentle bend the best circle is huge, and on a sharp bend it is small.\nThat best circle is the osculating circle; the word comes from the Latin for \"kissing\".\nFor the parabola $y = x^2$ at the origin, the osculating circle has radius $1/2$ and centre $(0, 1/2)$.\n",
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