# CoplanarPoints

CoplanarPoints[{p1,p2,p3,p4,,pn}]

tests whether the points p1,p2,p3,p4,,pn are coplanar.

# Details

• CoplanarPoints is also known as linearly dependent.
• Typically used to test whether a set of points lie on the same plane.
• CoplanarPoints[{p1,p2,p3,p4,,pn}] gives True if the points p4,,pn are on the plane passing through the points p1, p2 and p3.
• For coplanar points p1, p2, p3 and p4, the rank of the matrix {p2-p1,p3-p1,p4-p1} is less than or equal to 2.

# Examples

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## Basic Examples(2)

The points {0,0,0}, {1,1,-2}, {-1,2,-1}, {3,-4,1} are coplanar:

Plot the points:

Find the equation of the plane containing the points {0,0,0}, {1,1,-2} and {-1,2,-1}:

## Scope(4)

CoplanarPoints works with two-dimensional points:

Three-dimensional points:

n-dimensional points:

CoplanarPoints works with numerical coordinates:

Symbolic coordinates:

CoplanarPoints over a set of coordinates:

List of points:

Multi-points:

CoplanarPoints works for large sets:

## Applications(5)

### Basic Applications(4)

Find conditions for which two points lie on a plane passing through the origin:

Explicit instances:

2D points lie on the same plane:

Find the equation of a plane containing a set of points:

Draw coplanar points:

### Geometry(1)

A face of a polyhedron lies on a plane:

## Properties & Relations(5)

PositivelyOrientedPoints returns False for coplanar points:

NegativelyOrientedPoints returns False for coplanar points:

Collinear points are coplanar:

Use RegionMember to test whether points are coplanar:

Use InfinitePlane to draw a graphics image:

Wolfram Research (2020), CoplanarPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/CoplanarPoints.html.

#### Text

Wolfram Research (2020), CoplanarPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/CoplanarPoints.html.

#### CMS

Wolfram Language. 2020. "CoplanarPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CoplanarPoints.html.

#### APA

Wolfram Language. (2020). CoplanarPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CoplanarPoints.html

#### BibTeX

@misc{reference.wolfram_2024_coplanarpoints, author="Wolfram Research", title="{CoplanarPoints}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/CoplanarPoints.html}", note=[Accessed: 14-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_coplanarpoints, organization={Wolfram Research}, title={CoplanarPoints}, year={2020}, url={https://reference.wolfram.com/language/ref/CoplanarPoints.html}, note=[Accessed: 14-July-2024 ]}