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  "body": "\n## Summary\n\nConformal geometric algebra is the [[geometric algebra|N8H460W3E2]] of $\\R^{n+1,1}$, used as a model of Euclidean space $\\R^n$.\nIn it, points, spheres and planes are all represented by vectors.\nA point $x \\in \\R^n$ is embedded as a null vector $X$, and the inner product of two embedded points is $-\\tfrac{1}{2}$ times their squared distance.\nSpheres and planes are the vectors orthogonal to the points that lie on them.\nAll Möbius transformations of $\\R^n \\cup \\set{\\infty}$ act by conjugation with elements of the algebra.\nIt is used in computer graphics and robotics, where rigid motions, intersections and tangencies become algebraic operations.\n\n## Explanations\n\n### L1 human\n\nThe trick is to add two extra dimensions to space, one that squares to $+1$ and one that squares to $-1$.\nIn this bigger space a point becomes a vector of length zero, and the distance between two points is read from their inner product.\nA sphere is then also a single vector, and asking whether a point lies on it is just asking whether an inner product is zero.\nRotations, translations, scalings and inversions in spheres all become the same kind of algebraic move.\n",
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