# RegionEqual

RegionEqual[reg1,reg2]

returns True if the regions reg1 and reg2 are equal.

RegionEqual[reg1,reg2,reg3,]

returns True if the regions reg1, reg2, reg3, are all equal.

# Details and Options • The regions reg1 and reg2 are equal if all points in reg1 are points in reg2 and vice versa.
• • If all regi are parameter-free regions, i.e. ConstantRegionQ[regi] is True, the regions are point sets, and typically True or False is returned.
• If some regi depend on parameters, i.e. ConstantRegionQ[regi] is False, then regi represents a family of regions, and RegionEqual will attempt to compute conditions on parameters such that the regions are equal.
• The following options can be given:
•  Assumptions \$Assumptions assumptions to make about parameters GenerateConditions False whether to generate conditions on parameters

# Examples

open allclose all

## Basic Examples(2)

Test whether two regions are equal:

Generate conditions for which two regions are equal:

## Scope(16)

### Basic Uses(4)

Test for equality:

Show two regions are not equal:

Find conditions that make regions equal:

Compare three regions at once:

### Basic Regions(4)

Regions in including Line and Interval:

Ball:

Regions in including Point:

Line:

Disk, Ball and Ellipsoid:

Regions in including Point:

Line:

Cuboid and Hexahedron:

Ball and Ellipsoid:

Regions in including Cuboid and Parallelepiped in :

Ellipsoid and Ball in :

### Formula Regions(4)

Implicit regions:

Parametric regions:

Compare two formula regions:

Nonconstant formula regions:

### Mesh Regions(3)

Compare MeshRegion in :

In :

In :

Compare BoundaryMeshRegion in :

In :

In :

Compare MeshRegion with BoundaryMeshRegion in :

In :

### Derived Regions(1)

Compare BooleanRegion:

## Options(2)

### Assumptions(1)

Find all radii that describe the unit circle:

### GenerateConditions(1)

Find when the unit disk is equal to an implicitly described annulus:

Show the conditions for which the result is valid:

Explicitly allow for degenerate cases:

## Applications(6)

Find when two lamina are equal:

Create the unit disk by covering the hole of an annulus with a disk. Find all possible radii:

Show an instance of this:

Create the unit disk by covering the hole of an annulus with a square. Find all possible side lengths:

Show an instance of this:

Find all ways to express the unit rectangle in terms of Parallelogram:

Verify the results:

Express an implicitly defined ellipse through Ellipsoid:

Find a value for each parameter:

Visualize both representations:

Demonstrate De Morgan's law, :

## Properties & Relations(4)

Equivalent regions have all points in common:

RegionEqual can be expressed with RegionWithin:

For nonempty regions, RegionDisjoint returns False when RegionEqual returns True:

Use FindInstance to find points that lie in one region but not the other:

Use RandomPoint to find a uniform sampling of points that lie in one region but not the other:

Use Reduce to find where two regions differ: