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ImplicitRegion—Wolfram Language Documentation

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• See Also
  • ParametricRegion
  • RegionQ
  • DiscretizeRegion
  • MeshRegion
• Related Guides
  • Geometric Computation
  • Plane Geometry
  • Solid Geometry
  • Partial Differential Equations
  • Polygons
  • Polyhedra
  • 3D Printing

ImplicitRegion[cond,{x₁,…,x_n}]

represents a region in that satisfies the conditions cond.

ImplicitRegion[cond,{{x₁,a₁,b₁},…}]

represents a region in that satisfies the conditions cond as well as etc.

• The value of x_i in ImplicitRegion[cond,{x₁,…,x_n}] is taken to be localized, as in Block.
• ImplicitRegion[cond,{{x₁,a₁,b₁},…}] is equivalent to ImplicitRegion[cond∧a₁≤x₁≤b₁∧⋯,{x₁,…}].
• ImplicitRegion can be used in functions such as RegionDistance, Reduce, and Integrate.

A 0D region is a set of discrete points:

Find the points:

A purely 1D region is a union of (possibly unbounded) segments:

Find the segments:

A region with 1D and 0D components:

Find the components:

Compute the length:

A 0D region is a set of discrete points:

Find the points:

A purely 1D region is a curve:

Plot it:

A higher-degree algebraic curve:

Plot it:

A 2D region with two connected components:

Plot it:

A 2D region with lower-dimensional components:

Plot it:

A 0D region is a set of discrete points:

Find the points:

A purely 1D region is a curve; conic curves are intersections of a cone and a plane:

An ellipse:

A parabola:

A hyperbola:

A higher-degree algebraic curve:

Plot it:

The curve is an intersection of two surfaces:

A purely 2D region is a surface:

Plot it:

A higher-degree algebraic surface:

Plot it:

A purely 3D region is a solid:

Plot it:

A higher-degree algebraic solid:

Plot it:

A ball in :

Intersect with a 3D affine space and project on :

Plot it:

A torus embedded in :

Compute the area of the torus:

Embed the torus in using the mapping :

Plot it:

Implicitly described region:

Plot it:

Embedding dimension:

Geometric dimension:

Point membership test:

Plot it:

Get conditions for point membership:

Area and centroid:

Plot it:

Distance from a point:

Plot it:

Signed distance from a point:

Plot it:

Nearest point in the region:

Plot it:

Nearest points:

A trifolium is bounded:

Plot it:

Integrate over an implicitly defined region:

Optimize over an implicitly defined region:

Solve equations in an implicitly defined region:

Find the projection of an implicit region on the plane in :

Visualize the region:

Visualize the projection:

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