HalfSpace

HalfSpace[n,p]

represents the half-space of points such that .

HalfSpace[n,c]

represents the half-space of points such that .

Details

Examples

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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: 30-December-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: 30-December-2024 ]}