# InfinitePlane

InfinitePlane[{p1,p2,p3}]

represents the plane passing through the points p1, p2, and p3.

InfinitePlane[p,{v1,v2}]

represents the plane passing through the point p in the directions v1 and v2.

# Details • InfinitePlane is also known as plane or hyperplane.
• InfinitePlane can be used as a geometric region and graphics primitive.
• • InfinitePlane represents a plane or .
• Hyperplane[n,p] is an alternative representation using a normal n in 3D.
• InfinitePlane can be used in Graphics and Graphics3D.
• InfinitePlane will be clipped by PlotRange when rendering.
• In graphics, the points p, pi and vector v can be Dynamic expressions.
• Graphics rendering is affected by directives such as FaceForm, 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 of the points pi or the vectors vi.
• InfinitePlane can be used with functions such as RegionMeasure, RegionCentroid, etc.

# Examples

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## Basic Examples(3)

An InfinitePlane in 3D:

Different styles applied to an infinite plane:

Determine if points belong to a given infinite plane:

## Scope(17)

### Graphics(7)

#### Specification(2)

Define an infinite plane in 3D using three points:

Define the same plane using a single point and two tangent vectors:

An infinite plane varying in direction:

#### Styling(2)

Color directives specify the color of the infinite 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:

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:

An infinite plane has infinite measure and undefined centroid:

Distance from a point:

Signed distance from a point:

Nearest point in the region:

Nearest points:

An infinite plane is unbounded:

Find the region range:

Integrate over an infinite plane:

Optimize over an infinite plane:

Solve equations over an infinite plane:

## Applications(7)

Find the plane in which a triangle is embedded:

InfinitePlane can use the same parametrization as Triangle:

Find the plane in which a polygon is embedded:

To find the plane, take the first three points (or any three points not on a line):

The tangent plane to a parametric surface f[u,v] is given by InfinitePlane[f[u,v],{uf[u,v],vf[u,v]}]. Find the tangent plane to the parametric surface :

Find the tangent plane to the surface :

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

Visualize intersection points:

Partition space in a BubbleChart:

Combine the graphics:

Visualize a reflection plane:

Define a reflection plane:

Define a ReflectionTransform using a point on the plane and its normal vector:

Visualize the reflection of a unit cube about the plane:

## Properties & Relations(6)

InfinitePlane[{p1,p2,p3}] is equivalent to InfinitePlane[p1,{p2-p1,p3-p1}]:

InfinitePlane[p,{v1,v2}] is equivalent to Hyperplane[Cross[v1,v2],p] in 3D:

ParametricRegion can represent any InfinitePlane:

ImplicitRegion can represent any InfinitePlane:

InfinitePlane is a special case of ConicHullRegion:

Any InfinitePlane can be represented as a union of two HalfPlane regions:

## Neat Examples(2)

A random collection of planes:

Sweep an infinite plane around an axis: