ImplicitRegion

✖
`ImplicitRegion`

✖

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.

Details

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.

Examples

open allclose all

Basic Examples (1)Summary of the most common use cases

Specify a disk as an ImplicitRegion:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-f8sguu

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-xa3xmw

Out[2]=2

A circle:

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-pco1p

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-f9arvb

Out[4]=4

Scope (27)Survey of the scope of standard use cases

Regions in 1D (3)

A 0D region is a set of discrete points:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-bisuod

Out[1]=1

Find the points:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-d1bfx3

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-f1bi9j

Out[1]=1

Find the segments:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-j0p2zw

Out[2]=2

A region with 1D and 0D components:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-lcgie6

Out[1]=1

Find the components:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-km1m2a

Out[2]=2

Compute the length:

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-ip9qd8

Out[3]=3

Regions in 2D (5)

A 0D region is a set of discrete points:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-gdp0hn

Out[1]=1

Find the points:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-e7esag

Out[2]=2

A purely 1D region is a curve:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-e6kfp

Out[1]=1

Plot it:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-b0c4z4

Out[2]=2

A higher-degree algebraic curve:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-glu07x

Plot it:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-377lc

Out[2]=2

A 2D region with two connected components:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-d721hv

Plot it:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-ia60c6

Out[2]=2

A 2D region with lower-dimensional components:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-bms3j

Plot it:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-fhkrpq

Out[2]=2

Regions in 3D (7)

A 0D region is a set of discrete points:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-rs4v3

Out[1]=1

Find the points:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-b8v81v

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-o1uqgm

An ellipse:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-j4314

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-b0dcw

Out[3]=3

A parabola:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-dxf1ql

In[5]:=5

✖

https://wolfram.com/xid/05for8hl66-k2e1qz

Out[5]=5

A hyperbola:

In[6]:=6

✖

https://wolfram.com/xid/05for8hl66-bawy7j

In[7]:=7

✖

https://wolfram.com/xid/05for8hl66-ckiusj

Out[7]=7

A higher-degree algebraic curve:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-gn5mya

Plot it:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-dzexie

Out[2]=2

The curve is an intersection of two surfaces:

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-cmm8xn

Out[3]=3

A purely 2D region is a surface:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-bb74z

Plot it:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-f7hzus

Out[2]=2

A higher-degree algebraic surface:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-bn6tbz

Plot it:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-sj7ss

Out[2]=2

A purely 3D region is a solid:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-fwdbm

Plot it:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-t9m7g

Out[2]=2

A higher-degree algebraic solid:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-cevz56

Plot it:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-d1jkkb

Out[2]=2

Regions in D (2)

A ball in :

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-irko

Intersect with a 3D affine space and project on :

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-fgj3wr

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-epgn3b

Out[3]=3

Plot it:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-n2svzm

Out[4]=4

A torus embedded in :

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-b9nrxc

Compute the area of the torus:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-kf587e

Out[2]=2

Embed the torus in using the mapping :

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-f50512

Out[3]=3

Plot it:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-famjo0

Out[4]=4

Properties (10)

Implicitly described region:

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-be4ii0

Out[3]=3

Plot it:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-esb2mb

Out[4]=4

Embedding dimension:

In[5]:=5

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https://wolfram.com/xid/05for8hl66-y220

Out[5]=5

Geometric dimension:

In[6]:=6

✖

https://wolfram.com/xid/05for8hl66-bx9tom

Out[6]=6

Point membership test:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-lxvzyg

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-x1qns

Out[2]=2

Plot it:

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-mv61rz

Out[3]=3

Get conditions for point membership:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-x0pt

Out[4]=4

Area and centroid:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-b1sb39

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-z6qch

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-dxlbau

Out[3]=3

Plot it:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-dlwkyz

Out[4]=4

Distance from a point:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-oc6hy

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-bjikjq

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-bruj1e

Out[3]=3

Plot it:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-e1s5s1

Out[4]=4

Signed distance from a point:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-cybvpc

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-bm4ed

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-b7y6q0

Out[3]=3

Plot it:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-cu3ekg

Out[4]=4

Nearest point in the region:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-btejnv

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-et4yza

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-bzhv7w

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-c0d30r

Out[4]=4

Plot it:

In[5]:=5

✖

https://wolfram.com/xid/05for8hl66-fwzoi9

Out[5]=5

Nearest points:

In[6]:=6

✖

https://wolfram.com/xid/05for8hl66-eoyt0e

In[7]:=7

✖

https://wolfram.com/xid/05for8hl66-fq1ssz

Out[7]=7

A trifolium is bounded:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-e1slmz

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-b5d6xy

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-ma7dh0

Out[3]=3

Plot it:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-d4963l

Out[4]=4

Integrate over an implicitly defined region:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-fivgav

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-banwkr

Out[2]=2

Optimize over an implicitly defined region:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-nf9ton

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-hyz4dq

Out[2]=2

Solve equations in an implicitly defined region:

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-bnrw6

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-c7lkth

Out[2]=2

Applications (1)Sample problems that can be solved with this function

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

In[1]:=1

✖

https://wolfram.com/xid/05for8hl66-i6pul9

Visualize the region:

In[2]:=2

✖

https://wolfram.com/xid/05for8hl66-gloawf

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/05for8hl66-qmif60

Visualize the projection:

In[4]:=4

✖

https://wolfram.com/xid/05for8hl66-4x6ich

Out[4]=4

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