# Parallelepiped

Parallelepiped[p,{v1,,vk}]

represents a parallelepiped with origin p and directions vi.

# Details • Parallelepiped is also known as parallelogram, rhombohedron, and parallelotope.
• Parallelepiped represents , where the vectors vi have to be linearly independent.
• • Parallelepiped can be used as a geometric region and graphics primitive.
• Parallelepiped can be used in Graphics and Graphics3D.
• In graphics, the point p and vectors vi can be Scaled and Dynamic expressions.
• Graphics rendering is affected by directives such as FaceForm, EdgeForm, Opacity, and color.

# Examples

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## Basic Examples(3)

A Parallelepiped in 3D:

And in 2D:

Different styles applied to a parallelepiped:

Volume and centroid:

## Scope(16)

### Graphics(6)

#### Specification(2)

A parallelepiped in dimensions is specified by a base point and up to vectors:

A parallelepiped with specified origin and directions:

#### Styling(2)

Color directives specify the face color:

FaceForm and EdgeForm can be used to specify the styles of the faces and edges:

#### Coordinates(2)

In 2D and 3D, a parallelepiped can be specified with Scaled coordinates:

Use Offset coordinates:

### Regions(10)

Embedding dimension is the dimension of the space in which the parallelepiped lives:

Geometric dimension is the dimension of the shape itself:

Membership testing:

Conditions for membership:

Measure:

Centroid:

Distance from a point:

Visualize it:

Signed distance from a point:

Plot it:

Nearest point:

Visualize it:

A parallelepiped is bounded:

Compute a bounding box for the region:

Integrate over a Parallelepiped:

Optimize over it:

Solve equations over a Parallelepiped:

## Applications(3)

For a full-dimensional Parallelepiped, the measure is easily computed from the vectors:

The volume is equal to the absolute value of the determinant of the matrix :

For a lower-dimensional Parallelepiped, the square root of the Gram determinant is used:

The Gram determinant is the determinant of dotted with its Transpose:

Any full-dimensional Parallelepiped can tile space:

## Properties & Relations(5)

Parallelogram is the 2D full-dimensional case of Parallelepiped:

Rectangle is a 2D Parallelepiped with axis-aligned edges:

Cuboid is a 3D Parallelepiped with axis-aligned edges:

Any Parallelepiped is an AffineTransform of a Cuboid:

Hexahedron is a generalization of a 3D Parallelepiped: