# Parallelogram

Parallelogram[p,{v1,v2}]

represents a parallelogram with origin p and directions v1 and v2.

# Details • Parallelogram is also known as rhomboid and rhombus.
• Parallelogram represents , where the vectors vi have to be linearly independent.
• • is equivalent to Parallelogram[{0,0},{{1,0},{1,1}}].
• CanonicalizePolygon can be used to convert a parallelogram to an explicit Polygon object.
• Parallelogram can be used as a geometric region and graphics primitive.
• Parallelogram can be used in Graphics.
• 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 standard parallelogram:

Different styles applied to a parallelogram:

Compute the Area of a parallelogram:

Centroid:

## Scope(16)

### Graphics(6)

#### Specification(2)

A standard parallelogram:

A parallelogram 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 interior and boundary:

#### Coordinates(2)

Use Scaled coordinates:

Use Offset coordinates:

### Regions(10)

Embedding dimension is the dimension of the space in which the vertices exist:

Geometric dimension is the dimensionality of the region itself:

Point membership test:

Get conditions for point membership:

Measure and centroid:

Distance from a point to a parallelogram:

Visualizing:

Signed distance to a parallelogram:

Visualizing:

Nearest point:

Visualizing:

A parallelogram is bounded and convex:

Compute a bounding box:

Integrate over a parallelogram:

Optimize over a parallelogram:

Solve equations in a parallelogram:

## Applications(5)

A rhombus is a parallelogram in which all edges are the same length:

Visualize:

A parallelogram with sides that form right angles is a rectangle:

Visualize:

Any rectangle can easily be converted to a parallelogram:

The area of a parallelogram can easily be computed from the direction vectors:

Simply treat the vectors as a matrix and take the absolute value of the determinant:

Compare with Area:

A Parallelogram can tile the plane:

## Properties & Relations(6)

Rectangle is a special case of Parallelogram:

Polygon is a generalization of Parallelogram:

Parallelepiped generalizes Parallelogram to any dimension:

ImplicitRegion can represent any parallelogram:

ParametricRegion can represent any parallelogram:

A parallelogram can be represented as the union of two triangles: