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  "body": "\n## Summary\n\nHomogeneous coordinates describe a point of the [[projective space|ZT9A6JK8QF]] $\\mathbb{P}^n$ by $n + 1$ numbers, not all zero, taken up to a common nonzero factor.\nThey turn points at infinity into ordinary tuples, so that projective geometry can be done with linear algebra.\nProjective transformations become matrices, and curves become zero sets of homogeneous polynomials.\nThey are standard in computer graphics, where translations and perspective become matrix products.\n\n## Explanations\n\n### L1 human\n\nA point $(x, y)$ of the plane gets three coordinates $[x : y : 1]$, and any multiple such as $[2x : 2y : 2]$ names the same point.\nTo go back, divide by the last coordinate.\nTriples with last coordinate $0$, such as $[1 : 1 : 0]$, cannot be divided back: they are the points at infinity, one for each direction of lines.\nSo the lines $y = x$ and $y = x + 1$, which are parallel, meet at $[1 : 1 : 0]$.\n",
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