# AffineHalfSpace

AffineHalfSpace[{p1,,pk+1},w]

represents AffineSpace[{p1,,pk+1}] extended in the direction w.

AffineHalfSpace[p,{v1,,vk},w]

represents AffineSpace[p,{v1,,vk}] extended in the direction w.

# Details

• AffineHalfSpace can be used as a geometric region and graphics primitive.
• AffineHalfSpace represents the region or . The dimension is if the pi are affinely independent or the vi are linearly independent.
• AffineHalfSpace can be used in Graphics and Graphics3D.
• AffineHalfSpace will be clipped by PlotRange when rendering.
• Graphics rendering is affected by directives such as Opacity and color as well as:
•  Thickness,Dashing 1-dimensional () FaceForm 2-dimensional ()
• For a two-dimensional AffineSpace, FaceForm[front,back] can be used to specify different styles for the front and back, where the front is defined to be in the direction of the normal Cross[v1,v2] or Cross[p2-p1,p3-p1], depending on which input form is used.

# Examples

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## Basic Examples(3)

An AffineHalfSpace in 2D:

And in 3D:

Different styles applied to an affine half-space region:

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

## Scope(18)

### Graphics(8)

#### Specification(3)

Define an affine half-space in 3D using two points and a directional vector:

Define the same affine half-space using a point, a tangent vector, and a directional vector:

An affine half-space in 3D defined by a single point and directional vector:

Affine half-spaces varying in tangent vector:

Affine half-spaces varying in directional vector:

#### Styling(2)

Color directives specify the color of the affine half-space:

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

#### Coordinates(3)

Specify coordinates by fractions of the plot range:

Specify scaled offsets from the ordinary coordinates:

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:

An affine half-space has infinite measure and undefined centroid:

Distance from a point:

Signed distance from a point:

Nearest point in the region:

Nearest points:

An affine half-space is unbounded:

Find the region range:

Integrate over an affine half-space:

Optimize over an affine half-space:

Solve equations over an affine half-space:

## Applications(2)

Visualize the upper half-plane:

The lower half-plane:

The left half-plane:

The right half-plane:

Visualize the upper half-space:

The lower half-space:

The left half-space:

The right half-space:

The front half-space:

The back half-space:

## Properties & Relations(6)

HalfLine is a special case of AffineHalfSpace:

HalfPlane is a special case of AffineHalfSpace:

HalfSpace is a special case of AffineHalfSpace:

AffineHalfSpace is a special case of ConicHullRegion:

ParametricRegion can represent any AffineHalfSpace in :

In :

ImplicitRegion can represent any AffineHalfSpacein :

In :

## Neat Examples(1)

A collection of random half-spaces in :

In :

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_affinehalfspace, author="Wolfram Research", title="{AffineHalfSpace}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/AffineHalfSpace.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_affinehalfspace, organization={Wolfram Research}, title={AffineHalfSpace}, year={2015}, url={https://reference.wolfram.com/language/ref/AffineHalfSpace.html}, note=[Accessed: 14-September-2024 ]}