# HalfSpace

HalfSpace[n,p]

represents the half-space of points such that .

HalfSpace[n,c]

represents the half-space of points such that .

# Examples

open allclose all

## Basic Examples(3)

A HalfSpace in 2D:

And in 3D:

Different styles applied to a half-space region:

Determine if points belong to a given half-space region:

## Scope(15)

### Graphics(5)

#### Specification(2)

A half-space in 2D defined by a normal vector and a point:

The same half-space defined by a normal vector and a constant:

Define a half-space in 3D using a normal vector and a point:

Define the same half-space using a normal vector and a constant:

Half-spaces varying in direction of the normal:

#### Styling(2)

Color directives specify the color of the half-space:

FaceForm and EdgeForm can be used to specify the styles of the faces and edges:

#### Coordinates(1)

Points and vectors can be Dynamic:

### Regions(10)

Embedding dimension is the dimension of the coordinates:

Geometric dimension is the dimension of the region itself:

Point membership test:

Get the conditions for membership:

A half-space has infinite measure and undefined centroid:

Distance from a point:

Signed distance from a point:

Nearest point in the region:

Nearest points:

A half-space is unbounded:

Find the region range:

In the axis-aligned case:

Integrate over a half-space:

Optimize over a half-space:

Solve equations over a half-space:

## Applications(5)

Visualize 2D half-planes:

The upper half-plane:

The lower half-plane:

The left half-plane:

The right half-plane:

Visualize 3D half-spaces:

The upper half-space:

The lower half-space:

The left half-space:

The right half-space:

The front half-space:

The back half-space:

Partition space in a BubbleChart:

Combine the graphics:

Any convex polygon in 2D can be represented as an intersection of half-spaces:

Any convex polyhedron in 3D can be represented as an intersection of half-spaces:

## Properties & Relations(7)

ClipPlanes, for a given , results in a graphic that does not render anything within the :

HalfSpace is a special case of ConicHullRegion:

HalfSpace is a special case of AffineHalfSpace:

HalfLine is a special case of HalfSpace:

HalfPlane is a special case of HalfSpace:

ImplicitRegion can represent any HalfSpace in :

In :

In :

ParametricRegion can represent any HalfSpace in :

In :

In :

## Neat Examples(1)

A collection of random half-spaces in :

In :

Wolfram Research (2015), HalfSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/HalfSpace.html.

#### Text

Wolfram Research (2015), HalfSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/HalfSpace.html.

#### CMS

Wolfram Language. 2015. "HalfSpace." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HalfSpace.html.

#### APA

Wolfram Language. (2015). HalfSpace. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HalfSpace.html

#### BibTeX

@misc{reference.wolfram_2024_halfspace, author="Wolfram Research", title="{HalfSpace}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/HalfSpace.html}", note=[Accessed: 08-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_halfspace, organization={Wolfram Research}, title={HalfSpace}, year={2015}, url={https://reference.wolfram.com/language/ref/HalfSpace.html}, note=[Accessed: 08-August-2024 ]}