# Ellipsoid

Ellipsoid[p,{r1,}]

represents an axis-aligned ellipsoid centered at the point p and with semiaxes lengths ri.

Ellipsoid[p,Σ]

represents an ellipsoid centered at p and weight matrix Σ.

# Details • Ellipsoid is also known as center interval, ellipse, and hyperellipsoid.
• Ellipsoid can be used as a geometric region and a graphics primitive.
• • Ellipsoid represents the axis-aligned filled ellipsoid or general ellipsoid .
• Ellipsoid allows p to be any point in , ri any positive real numbers, and Σ any real symmetric positive definite matrix.
• Ellipsoid can be used in Graphics and Graphics3D.
• In graphics, the points p, pi, and radii ri can be Scaled and Dynamic expressions.
• Graphics rendering is affected by directives such as FaceForm, Specularity, Opacity, and color.

# Examples

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## Basic Examples(2)

An axis-aligned ellipsoid in 3D:

In 2D:

Measure and centroid:

## Scope(20)

### Graphics(10)

#### Specification(4)

An axis-aligned ellipsoid in 3D:

In 2D:

A general ellipsoid in 3D:

In 2D:

#### Styling(4)

Balls with different specular exponents:

Black ball that glows red:

Opacity specifies the face opacity:

2D styling:

#### Coordinates(2)

Specify coordinates by fractions of the plot range:

Specify scaled offsets from the ordinary coordinates:

### Regions(10)

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

Geometric dimension is the dimension of the shape itself:

Membership testing:

Get conditions for point membership:

Volume:

Centroid:

Distance from a point:

The distance to the nearest point for an ellipse:

Signed distance from a point:

Signed distance to an ellipse:

Nearest point in the region:

Nearest points to an enclosing sphere:

An ellipsoid is bounded:

Find its range:

Integrate over an ellipsoid region:

Optimize over an ellipsoid region:

Solve equations in an ellipsoid region:

## Applications(4)

A spheroid is an ellipsoid with two equal axes:

Compute its volume:

Total mass for an ellipsoid region with density given by :

Find the mass of methanol in an Ellipsoid:

Density of methanol:

Volume of ellipsoid:

Mass of methanol in the ellipsoid:

Find a bounding Ellipsoid to a region's bounding box:

Compute the bounding box:

Compute a bounding ellipsoid to the bounding box:

Compute the difference in Volume of the bounding solids:

Visualize bounding surfaces:

## Properties & Relations(4)

Disk is a special case of Ellipsoid:

Ball is a special case of Ellipsoid:

Ellipsoid is a generalization of Ball:

ImplicitRegion can represent any Ellipsoid:

## Neat Examples(2)

Random ellipsoid collections:

Sweep an ellipsoid around an axis: