Compute Nearest Points 

For a point and a region, the nearest point in that region to the given point can be computed. The nearest point is given by ArgMin[, ∈ℛ], where is the given point and is the region.

Some regions have simple closed-form representation for the nearest point.

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X RegionNearest[Disk[], {x, y}]

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X RegionNearest[Rectangle[], {x, y}]

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Others are more involved.

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X RegionNearest[Triangle[{{0, 0}, {1, 0}, {0, 1}}], {x, y}]

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Visualize the different case where there is a different nearest projector function.

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X piecewiseConditions[HoldPattern[Piecewise[cases_, default_]]] := Module[{cl = Last /@ cases}, Append[cl, And @@ (Not /@ cl)]]

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X conds = piecewiseConditions[RegionNearest[Triangle[], {x, y}]];

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X RegionPlot[conds, {x, -1, 2}, {y, -1, 2}, ImageSize -> Medium]

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Similarly for a standard tetrahedron.

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X RegionNearest[Tetrahedron[], {x, y, z}]

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X conds = piecewiseConditions[RegionNearest[Tetrahedron[], {x, y, z}]];

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X Show[Table[ RegionPlot3D[conds[[i]], {x, -1, 2}, {y, -1, 2}, {z, -1, 2}, Mesh -> None, PlotStyle -> Directive[Opacity[0.4], ColorData[95, i]], BoundaryStyle -> Gray, PlotPoints -> 35], {i, Length[conds]}], ImageSize -> Medium, Lighting -> "Neutral"]

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