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HalfPlane
✖
HalfPlane

✖
HalfPlane[{p1,p2},w]

represents the half-plane bounded by the line through p1 and p2 and extended in the direction w.

✖
HalfPlane[p,v,w]

represents the half-plane bounded by the line through p along v and extended in the direction w.

Details

  • HalfPlane is also known as half-space in 2D.
  • HalfPlane can be used as a geometric region and graphics primitive.
  • HalfPlane represents a planar region or .
  • HalfPlane can be used in Graphics and Graphics3D.
  • HalfPlane will be clipped by PlotRange when rendering.
  • In graphics, the points p, pi and vector v can be Scaled and Dynamic expressions.
  • Graphics rendering is affected by directives such as FaceForm, EdgeForm, Opacity, and color.
  • FaceForm[front,back] can be used to specify different styles for the front and back in 3D. The front is defined by the right-hand rule and the direction from {p1,w,p2} or {p,v,w}.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

A HalfPlane in 2D:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-e7un64
Out[1]=1

And in 3D:

In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-e7ewm
Out[2]=2

Different styles applied to a half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-ha30gf
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-db0xcy
Out[2]=2

The Area of a half-plane is infinite:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-bfbp1
Out[1]=1

Determine if points belong to a given half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-jndyz9
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-dgs4g1
Out[2]=2

Scope  (18)Survey of the scope of standard use cases

Graphics  (8)

Specification  (3)

Define the upper half-plane using a point and two vectors:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-b5woeh
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-vjyj3
Out[2]=2

Or using two points and a single vector:

In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-b49gsf
In[4]:=4
✖
https://wolfram.com/xid/0en1pqlz3-g94d9
Out[4]=4

Define a half-plane in 3D using a point and two vectors:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-eemyer
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-cokj9k
Out[2]=2

Or using two points and a single vector:

In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-bmzjqp
In[4]:=4
✖
https://wolfram.com/xid/0en1pqlz3-e0zase
Out[4]=4

A half-plane with symbolic parameters:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-nlkqsx
Out[1]=1

Styling  (2)

Color directives specify the color of the half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-i29k3n
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-e04uby
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-h1pi0h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-4hfyj
Out[2]=2

Coordinates  (3)

Specify coordinates by fractions of the plot range:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-e3u77e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-yimo99
Out[2]=2

Specify scaled offsets from the ordinary coordinates in 2D:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-f8i2ip
Out[1]=1

Points and vectors can be Dynamic:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-9cjx7z
Out[1]=1

Regions  (10)

Embedding dimension is the dimension of the coordinates:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-11a0x6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-hn6tsf
Out[2]=2

Geometric dimension is the dimension of the region itself:

In[3]:=3
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https://wolfram.com/xid/0en1pqlz3-mji3x4
Out[3]=3

Membership testing:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-qtw6nw
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-l0qk5q
Out[2]=2

Get conditions for membership:

In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-pvff7t
Out[3]=3

Half-planes have infinite measure and undefined centroid:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-zsl8fn
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-0x62cm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-c4n99m
Out[3]=3

Distance from a point to a half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-beic9u
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-4wmd2m
Out[2]=2

Visualizing it:

In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-7xgx3
Out[3]=3

Signed distance to a half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-b7ypyd
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-z3n36v
Out[2]=2

Plotting it:

In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-vddrjy
Out[3]=3

Nearest point:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-8ujy2g
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-k26agk
Out[2]=2

Visualize it:

In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-7ej43v
In[4]:=4
✖
https://wolfram.com/xid/0en1pqlz3-n8fd3
Out[4]=4

A half-plane is unbounded:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-2qbzkc
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-pjepv6
Out[2]=2

Find the region range:

In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-o5laas
Out[3]=3

Integrate over a half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-udk0sy
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-f4zxih
Out[2]=2

Optimize over a half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-0156ri
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-5qcc45
Out[2]=2

Solve equations over a half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-x3q5bp
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-6jvs5i
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

Define regions that occupy two adjacent quadrants:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-67fjfp
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-p5o7j0
Out[2]=2

Partition space in a BubbleChart:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-1fm7gd
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-euq1kf

Combine the graphics:

In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-r8p9a
Out[3]=3

Find the intersection points of a sphere, a half-plane, and a surface defined by :

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-qlnp21
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-h539x7
Out[2]=2

Visualize intersection points:

In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-ozql0o
In[4]:=4
✖
https://wolfram.com/xid/0en1pqlz3-0jr7hl
In[5]:=5
✖
https://wolfram.com/xid/0en1pqlz3-0hkhnu
In[6]:=6
✖
https://wolfram.com/xid/0en1pqlz3-lnw1r2
In[7]:=7
✖
https://wolfram.com/xid/0en1pqlz3-3hs7ij
Out[7]=7

Properties & Relations  (4)Properties of the function, and connections to other functions

Any HalfPlane can be represented by ConicHullRegion:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-d88ckb
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-f123x
In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-k6jos
Out[3]=3

ImplicitRegion can be used to represent any HalfPlane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-eon2fb
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-bp9ufs
In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-dr0t97
Out[3]=3

ParametricRegion can be used to represent any HalfPlane:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-fiy33a
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-gb7v1
In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-dmt09h
Out[3]=3

Any InfinitePlane can be represented as a union of two half-planes:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-3addd
In[2]:=2
✖
https://wolfram.com/xid/0en1pqlz3-d9shn8
In[3]:=3
✖
https://wolfram.com/xid/0en1pqlz3-jth8x6
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

A collection of random half-planes:

In[1]:=1
✖
https://wolfram.com/xid/0en1pqlz3-erlfyn
Out[1]=1
Wolfram Research (2014), HalfPlane, Wolfram Language function, https://reference.wolfram.com/language/ref/HalfPlane.html (updated 2016).
✖
Wolfram Research (2014), HalfPlane, Wolfram Language function, https://reference.wolfram.com/language/ref/HalfPlane.html (updated 2016).

Text

Wolfram Research (2014), HalfPlane, Wolfram Language function, https://reference.wolfram.com/language/ref/HalfPlane.html (updated 2016).

✖
Wolfram Research (2014), HalfPlane, Wolfram Language function, https://reference.wolfram.com/language/ref/HalfPlane.html (updated 2016).

CMS

Wolfram Language. 2014. "HalfPlane." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/HalfPlane.html.

✖
Wolfram Language. 2014. "HalfPlane." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/HalfPlane.html.

APA

Wolfram Language. (2014). HalfPlane. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HalfPlane.html

✖
Wolfram Language. (2014). HalfPlane. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HalfPlane.html

BibTeX

@misc{reference.wolfram_2025_halfplane, author="Wolfram Research", title="{HalfPlane}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/HalfPlane.html}", note=[Accessed: 01-April-2025 ]}

✖
@misc{reference.wolfram_2025_halfplane, author="Wolfram Research", title="{HalfPlane}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/HalfPlane.html}", note=[Accessed: 01-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_halfplane, organization={Wolfram Research}, title={HalfPlane}, year={2016}, url={https://reference.wolfram.com/language/ref/HalfPlane.html}, note=[Accessed: 01-April-2025 ]}

✖
@online{reference.wolfram_2025_halfplane, organization={Wolfram Research}, title={HalfPlane}, year={2016}, url={https://reference.wolfram.com/language/ref/HalfPlane.html}, note=[Accessed: 01-April-2025 ]}