Finding Nearest Points 

Compute the nearest point in any dimension.

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X RegionNearest[Cylinder[], {1, 2, 3}]

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X RegionNearest[Simplex[5], {1, 2, 3, 4, 5}]

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Compute formulas for the nearest point on a region in any dimension.

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X RegionNearest[Ball[4], {w, x, y, z}]

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Create nearest functions that can be efficiently reapplied to multiple points.

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X nf = RegionNearest[Disk[{0, 0}, {2, 1}]]

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show complete Wolfram Language inputhide input

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X pts = Table[3 {Cos[k 2 \[Pi]/16], Sin[k 2 \[Pi]/16]}, {k, 0., 15}]; nst = nf[pts];

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X Legended[Graphics[{{LightBlue, EdgeForm[Darker@LightBlue], \[ScriptCapitalR]}, PointSize[Medium], {Thin, Gray, Line[Transpose[{pts, nst}]]}, {Red, Point[pts]}, {Blue, Point[nst]}}], PointLegend[{Red, Blue}, {"start", "nearest"}]]

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