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InscribedBall

InscribedBall

InscribedBall[{p₁,p₂,…}]

gives the largest ball that lies inside the convex hull of the points p₁,p₂,….

Details

InscribedBall is also known as plug gauge circle or maximum inscribed circle.
InscribedBall gives the Ball of largest measure (arc length, area, etc.) that can be inscribed in the convex hull of a set of points.

Examples

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Basic Examples (2)

A 2D inscribed ball from points:

Wolfram Language code: pts = RandomReal[{-1, 1}, {50, 2}];

Wolfram Language code: ℛ = InscribedBall[pts]

The region is the largest ball that lies inside the convex hull of the points:

Wolfram Language code: Region[ℛ, Epilog -> {Opacity[.5], ConvexHullRegion[pts], Black, Point[pts]}]

A 3D inscribed ball from points:

Wolfram Language code: pts = RandomReal[{-1, 1}, {50, 3}];

Wolfram Language code: ℛ = InscribedBall[pts]

The region is the largest ball that lies inside the convex hull of the points:

Wolfram Language code: Show[{Region[ℛ], Region[Style[ConvexHullRegion[pts], Opacity[.5]]]}]

Scope (1)

Create a 1D inscribed ball from a set of points:

Wolfram Language code: pts = RandomReal[{-1, 1}, {50, 1}];

Wolfram Language code: ℛ = InscribedBall[pts]

A 2D inscribed ball:

Wolfram Language code: pts = RandomReal[{-1, 1}, {50, 2}];

Wolfram Language code: ℛ = InscribedBall[pts]

A 3D inscribed ball:

Wolfram Language code: pts = RandomReal[{-1, 1}, {50, 3}];

Wolfram Language code: ℛ = InscribedBall[pts]

Applications (1)

Inscribed balls inside a Voronoi mesh:

Wolfram Language code: ℛ = VoronoiMesh[RandomReal[{-1, 1}, {100, 3}]]

Wolfram Language code: InscribedBall[PolyhedronCoordinates[#]]& /@ MeshPrimitives[ℛ, 3];

Wolfram Language code:

Show[{MeshRegion[ℛ, PlotTheme -> "Lines", MeshCellShapeFunction -> {1 -> (Tube[#1, .04]&)}, BaseStyle -> Hue[0.6, 0.77, 0.85]], Graphics3D[{Purple, %}]}, BaseStyle -> {Specularity[White, 25]}]

Properties & Relations (3)

InscribedBall is the largest Ball that lies inside the convex hull of points:

Wolfram Language code: pts = {{0, 0}, {1, 0}, {2 / 3, 1 / 2}, {0, 1}};

Wolfram Language code: Graphics[{GrayLevel[.5, .5], ConvexHullRegion[pts], Red, Point[pts], Blue, InscribedBall[pts]}]

Use CircumscribedBall to get the smallest Ball that encloses the points:

Wolfram Language code: pts = {{0, 0}, {1, 0}, {2 / 3, 1 / 2}, {0, 1}};

Wolfram Language code: Graphics[{ GrayLevel[.5, .5], CircumscribedBall[pts], Red, Point[pts]}]

Use Insphere to get the Sphere that can be inscribed in the simplex defined by points:

Wolfram Language code: pts = {{0, 0}, {1, 0}, {0, 1}};

Wolfram Language code: Graphics[{GrayLevel[.5, .5], Simplex[pts], Red, Point[pts], Blue, Insphere[pts]}]

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