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  "body": "\n## Summary\n\nA projective conic is the set of points of a projective plane over a field where a quadratic form in three variables vanishes.\nIt is the projective version of the [[conic section|00HJCMC2CD]].\nEllipses, parabolas and hyperbolas become the same kind of curve once points at infinity are added.\nNon-degenerate conics are the simplest curves after lines, and in finite planes they are the model examples of [[ovals|7B03TB64N9]].\n\n## Explanations\n\n### L1 human\n\nAn ellipse, a parabola and a hyperbola look different in the plane.\nIn the projective plane, a parabola gains one point at infinity and a hyperbola gains two, and all three become closed curves of the same type.\nIn homogeneous coordinates $[x : y : z]$ each one is given by an equation of degree $2$, such as $xz = y^2$.\nThis curve contains the points $[1 : t : t^2]$ for every $t$, and also the point $[0 : 0 : 1]$.\n",
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