# 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

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## Basic Examples(4)

A HalfPlane in 2D:

And in 3D:

Different styles applied to a half-plane:

The Area of a half-plane is infinite:

Determine if points belong to a given half-plane:

## Scope(18)

### Graphics(8)

#### Specification(3)

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

Or using two points and a single vector:

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

Or using two points and a single vector:

A half-plane with symbolic parameters:

#### Styling(2)

Color directives specify the color of the half-plane:

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 in 2D:

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:

Membership testing:

Get conditions for membership:

Half-planes have infinite measure and undefined centroid:

Distance from a point to a half-plane:

Visualizing it:

Signed distance to a half-plane:

Plotting it:

Nearest point:

Visualize it:

A half-plane is unbounded:

Find the region range:

Integrate over a half-plane:

Optimize over a half-plane:

Solve equations over a half-plane:

## Applications(3)

Partition space in a BubbleChart:

Combine the graphics:

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

Visualize intersection points:

## Properties & Relations(4)

Any HalfPlane can be represented by ConicHullRegion:

ImplicitRegion can be used to represent any HalfPlane:

ParametricRegion can be used to represent any HalfPlane:

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

## Neat Examples(1)

A collection of random half-planes: