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  "body": "\n## Summary\n\nHermite interpolation finds a polynomial that matches, at each node, a prescribed value and prescribed derivatives up to some order.\nIt extends [[polynomial interpolation|SS1YMS2NPC]], which prescribes values only, and Taylor polynomials, which prescribe derivatives at a single point.\nIf $N$ conditions are imposed in total, there is exactly one polynomial of degree at most $N - 1$ that satisfies them.\nIt can be computed with the Newton form using [[divided differences|N5ZHV4HWDC]] with repeated nodes, and it underlies cubic Hermite splines and Gaussian quadrature.\n\n## Explanations\n\n### L1 human\n\nOrdinary interpolation asks a curve to pass through given points.\nHermite interpolation also asks it to have given slopes there, and possibly given curvature and more.\nPassing through $(0, 0)$ and $(1, 1)$ with slope $0$ at both points gives four conditions, and the cubic $3x^2 - 2x^3$ meets all of them.\nThis curve leaves and arrives flat, which is why it is used to join pieces of curves smoothly.\nThe [[definition|DZQ4EW6907]] states the conditions.\n",
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