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  "body": "\n## Definition\n\nThe spherical coordinate map is $S \\colon [0, \\infty) \\times [0, \\pi] \\times \\mathbb{R} \\to \\mathbb{R}^3$,\n$$S(\\rho, \\varphi, \\theta) = (\\rho \\sin \\varphi \\cos \\theta, \\ \\rho \\sin \\varphi \\sin \\theta, \\ \\rho \\cos \\varphi).$$\nA triple $(\\rho, \\varphi, \\theta)$ with $S(\\rho, \\varphi, \\theta) = (x, y, z)$ is a triple of spherical coordinates of $(x, y, z)$.\nHere $\\rho = \\sqrt{x^2 + y^2 + z^2}$ is the distance to the origin, $\\varphi$ is the polar angle from the positive $z$-axis, and $\\theta$ is the azimuth.\n\n## Remarks\n\nThe pair $(\\rho \\sin \\varphi, \\theta)$ is a pair of [[polar coordinates|0RJZV05MJA]] of $(x, y)$, and $\\cos \\varphi = z / \\rho$ for $\\rho \u003e 0$.\nThe coordinates are unique for points off the $z$-axis if $\\rho \u003e 0$, $0 \u003c \\varphi \u003c \\pi$ and $\\theta \\in [0, 2\\pi)$.\nThe sphere of radius $a$ is $\\rho = a$, and $\\varphi = \\pi / 4$ is a cone around the positive $z$-axis.\nThis is a common convention in mathematics. Physics texts often write $(r, \\theta, \\varphi)$, with $\\theta$ the polar angle and $\\varphi$ the azimuth.\n\n## Explanations\n\n### L1 human\n\nTo locate a point in space, give its distance from the origin, how far it tilts away from the north pole, and its longitude.\nThis is like latitude and longitude on the Earth, with the polar angle measured from the north pole.\n",
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