# Hyperplane

Hyperplane[n,p]

represents the hyperplane with normal n passing through the point p.

Hyperplane[n,c]

represents the hyperplane with normal n given by the points that satisfy .

# Details • Hyperplane can be used as a geometric region and a graphics primitive.
• Hyperplane[n] is equivalent to Hyperplane[n,0], a hyperplane through the origin.
• Hyperplane corresponds to an infinite line in and an infinite plane in .
• • Hyperplane represents the set or .
• Hyperplane is defined for nd, pd, and c.
• Hyperplane can be used in Graphics and Graphics3D.
• Hyperplane will be clipped by PlotRange when rendering.
• Graphics rendering is affected by directives such as Thickness, Dashing, Opacity, and color.
• Graphics3D rendering is affected by directives such as Opacity and color. FaceForm[front,back] can be used to specify different styles for the front and back, where front is defined to be in the direction of the normal n.

# Examples

open allclose all

## Basic Examples(3)

A Hyperplane in 2D:

And in 3D:

Different styles applied to a hyperplane region:

Determine if points belong to a given hyperplane region:

## Scope(15)

### Graphics(5)

#### Specification(2)

A hyperplane in 2D defined by a normal vector and a point:

The same hyperplane defined by a normal vector and a constant:

Define a hyperplane in 3D using a normal vector and a point:

Define the same hyperplane using a normal vector and a constant:

Hyperplanes varying in direction of the normal:

#### Styling(2)

Color directives specify the color of the hyperplane:

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 hyperplane has infinite measure and undefined centroid:

Distance from a point:

Signed distance from a point:

Nearest point in the region:

Nearest points:

A hyperplane is unbounded:

Find the region range:

In the axis-aligned case, it is bounded in some directions:

Integrate over a hyperplane:

Optimize over a hyperplane:

Solve equations over a hyperplane:

## Applications(11)

### Hyperplane Arrangements(7)

In a parallel arrangement of hyperplanes, all hyperplanes have the same normal n:

Orthogonal arrangements of hyperplanes:

Grids of hyperplanes:

Random arrangements of hyperplanes:

A pencil of hyperplanes is all hyperplanes through a point:

A sheaf of hyperplanes is all hyperplanes through a line:

A bundle of hyperplanes, where all pass through a common point:

### Tangent Planes(4)

A tangent plane to an implicitly defined curve in 2D is given by its normal at a point on the curve. Start by finding points on the curve:

Find tangent lines at each of the points:

Visualize the solution:

A tangent plane to an implicitly defined surface in 3D is also given by its normal and a point on the surface. Start by finding points on the surface:

Find tangent planes at each of the points:

Visualize the solution:

A tangent line for a parametric curve can be defined by its normal for some value of the parameter . Start by picking parameter values:

Find tangent lines for each parameter value:

Visualize the solution:

A tangent plane for a parametric surface can be defined by its normal for some value of the parameters and . Start by picking parameter values:

Find tangent planes at each of the points:

Visualize the solution:

## Properties & Relations(7)

Hyperplane is a special case of ConicHullRegion:

Hyperplane is a special case of AffineSpace:

InfiniteLine is a special case of Hyperplane:

InfinitePlane is a special case of Hyperplane:

ParametricRegion can represent any Hyperplane in :

In :

ImplicitRegion can represent any Hyperplane in :

In :

ClipPlanes for a given results in a graphic that does not render anything on the side of that is in the negative direction of the normal :

## Neat Examples(4)

A random collection of lines:

A random collection of planes:

Organized collection of lines:

Sweep a hyperplane around an axis: