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

Examples

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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 TemplateBox[{{f, (, {x, ,, y}, )}, {{, {x, ,, y}, }}}, Grad] 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 TemplateBox[{{f, (, {x, ,, y, ,, z}, )}, {{, {x, ,, y, ,, z}, }}}, Grad] 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:

Introduced in 2015
 (10.2)