# ImplicitRegion

ImplicitRegion[cond,{x1,,xn}]

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.

# Details # Examples

open allclose all

## Basic Examples(1)

Specify a disk as an ImplicitRegion:

A circle:

## Scope(27)

### Regions in 1D(3)

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:

### Regions in 2D(5)

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:

### Regions in 3D(7)

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:

### Regions in D(2)

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:

### Properties(10)

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:

## Applications(1)

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

Visualize the region:

Visualize the projection: