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AffineHalfSpace

AffineHalfSpace[{p₁,…,p_k+1},w]

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

AffineHalfSpace[p,{v₁,…,v_k},w]

represents AffineSpace[p,{v₁,…,v_k}] 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 p_i are affinely independent or the v_i 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 :

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