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  "body": "\n## Summary\n\nThe Zernike polynomials are an orthogonal family of polynomials on the closed unit disk.\nIn polar coordinates each one is a radial polynomial times $\\cos(m\\varphi)$ or $\\sin(m\\varphi)$.\nThey are the standard basis in optics for describing the aberrations of a wavefront through a circular lens or mirror.\nTheir radial parts are [[Jacobi polynomials|TRD7DSF2KR]] in disguise.\n\n## Explanations\n\n### L1 human\n\nThe light passing through a round lens arrives with small delays that vary across the disk of the lens.\nOptical engineers write this delay map as a sum of standard patterns on the disk, the Zernike polynomials.\nEach pattern has a name: $2\\rho^2 - 1$ is \"defocus\", a bowl shape, and $\\rho \\cos \\varphi$ is \"tilt\", a plane sloping in one direction.\nBecause the patterns are orthogonal, the size of each defect can be read off separately.\n",
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