# Disk

Disk[{x,y},r]

represents a disk of radius r centered at {x,y}.

Disk[{x,y}]

gives a disk of radius 1.

Disk[{x,y},{rx,ry}]

gives an axis-aligned elliptical disk with semiaxes lengths rx and ry.

Disk[{x,y},,{θ1,θ2}]

gives a sector of a disk from angle θ1 to θ2.

# Details • Disk can be used as a geometric region and a graphics primitive.
• Disk[] is equivalent to Disk[{0,0}]. »
• • Disk represents the filled region .
• Angles are measured in radians counterclockwise from the positive x direction.
• Disk can be used in Graphics.
• In graphics, the points {xi,yi} can be Scaled, Offset, ImageScaled, and Dynamic expressions.
• Graphics rendering is affected by directives such as FaceForm, EdgeForm, and color.
• Disk can be used with symbolic points and quantities in GeometricScene.

# Background & Context

• Disk is a graphics and geometry primitive that represents a circular disk, elliptical disk or sector in the plane. In particular, Disk[{x,y},r] represents the disk of radius r in centered at {x,y}, Disk[{x,y},{rx,ry}] represents the axis-aligned filled ellipse in with center {x,y} and semiaxis lengths rx and ry, and Disk[{x,y},,{θ1,θ2}] represents the (potentially elliptical) sector centered at {x,y} ranging between angles θ1 and θ2 measured in radians counterclockwise from the positive axis. The shorthand form Disk[{x,y}] is equivalent to Disk[{x,y},1], while Disk[] autoevaluates to Disk[{0,0},1].
• Disk objects can be formatted by placing them inside a Graphics expression. The appearance of Disk objects in graphics can be modified by specifying edge and face directives EdgeForm and FaceForm, color directives such as Red, the transparency directive Opacity, and the style option Antialiasing.
• Disk may also serve as a region specification over which a computation should be performed. For example, Integrate[1,{x,y}Disk[{0,0},r]] and Area[Disk[{0,0},r]] both return the area of a disk of radius , and Perimeter[Disk[{x,y},r]] returns the perimeter .
• Disk is related to a number of other symbols. Circle represents the boundary of a disk, as can be computed using RegionBoundary[Disk[{x,y},r]]. Ball and Ellipsoid may be thought of as higher-dimensional analogs of disks. Annulus gives a region obtained by removing a small disk from the interior of a larger concentric disk. Disk[{x,y},r] may be alternately represented using Ball[{x,y},r], ImplicitRegion[(x-u)2+(y-v)2r2,{u,v}] or ParametricRegion[a{Cos[θ],Sin[θ]}-{x,y},{{θ,0,2π},{a,0,r}}]. Precomputed properties of the disk and its variants in standard position are available using LaminaData["entity","property"] or EntityValue[Entity["Lamina","entity"],"property"], where "entity" is one of "CircularSector", "Disk", "FilledEllipse", "FilledHalfEllipse", "HalfDisk", etc.

# Examples

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## Basic Examples(5)

A unit disk:

A disk sector:

An elliptical disk:

Differently styled unit disks:

Get the Area of a disk:

The area of an ellipse:

## Scope(22)

### Graphics(12)

#### Specification(6)

Specify centers:

A disk sector:

An elliptical disk:

An elliptical disk sector:

Short form for a unit disk at the origin:

#### Styling(2)

Color directives specify the face colors of disks:

FaceForm and EdgeForm can be used to specify the styles of the interiors and boundaries:

Boundaries of an elliptical disk sector:

#### Coordinate(4)

Use Offset coordinates:

Use Offset to specify the radii in printer's points:

### Regions(10)

Embedding dimension:

Geometric dimension:

Point membership test:

Get conditions for point membership:

Area:

Centroid:

Distance from a point:

The distance to the nearest point in the unit disk:

Signed distance from a point:

Signed distance to the unit disk:

Nearest point in the region:

Nearest points:

An ellipse is bounded:

Get its range:

Integrate over an ellipse:

Optimize over an ellipse:

Solve equations in an ellipse:

## Applications(11)

Create illusory contours:

Use a Disk to annotate a plot of a trig function:

Make a pie chart:

Archimedes' approximation of the circle area:

The square packing of disks:

The hexagonal packing of disks:

Simulation of elliptical gears:

An annulus is the RegionDifference of two disks with the same center:

Maximize the area of an ellipse of fixed perimeter:

As expected, the largest such ellipse is the circular disk:

Illustrate a function's radius of curvature:

By taking a RegionUnion of many disks, dilation of a mesh can be approximated:

Create disks of the dilation radius around the mesh boundary:

Then simply take the union of all disks plus the original mesh:

By removing a RegionUnion of many disks, erosion of a mesh can be approximated:

Create disks of the erosion radius around the mesh boundary:

Then subtract the union of the disks from the original mesh:

## Properties & Relations(9)

Use Rotate to get all possible elliptical disks:

The boundary of a disk defines a circle:

An implicit specification of a disk can be generated using RegionPlot:

A parametric specification of a disk can be generated using ParametricPlot:

Disk is a special case of Ball:

Disk is a special case of Ellipsoid:

ParametricRegion can represent any Disk:

ImplicitRegion can represent any Disk:

Disk is a norm ball for the Euclidean norm:

## Possible Issues(2)

Using Scaled radii will depend on the PlotRange:

Using ImageScaled sizes will depend on the ImageSize and AspectRatio:

## Neat Examples(4)

Random disk collections:

A family of disks:

Digital petals:

Yin and yang:

Introduced in 1991
(2.0)
|
Updated in 2014
(10.0)