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Ball
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Ball

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Ball[p]

represents the unit ball centered at the point p.

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Ball[p,r]

represents the ball of radius r centered at the point p.

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Ball[{p1,p2,…},r]

represents a collection of balls of radius r.

Details and Options

  • Ball is also known as center interval, disk, ball, and hyperball.
  • Ball can be used as a geometric region and a graphics primitive.
  • Ball[] is equivalent to Ball[{0,0,0}].
  • Ball[n] for integer n is equivalent to Ball[{0,…,0}], a unit ball in .
  • Ball represents a filled ball {x|TemplateBox[{{x, -, p}}, Norm]<=r}. The region is dimensional for point p of length .
  • Ball allows p to be any point in and r any positive real number.
  • Ball can be used in Graphics and Graphics3D.
  • In graphics, the points p, pi and radii r can be Scaled and Dynamic expressions.
  • Graphics rendering is affected by directives such as FaceForm, EdgeForm, Specularity, Opacity, and color.
  • Ball[{p1,p2,…},{r1,r2,…}] represents a collection of spheres with centers pi and radii ri.

Background & Context

  • Ball is a graphics and geometry primitive that represents a ball in -dimensional space. In particular, Ball[p,r] represents a (filled-in) ball {x:TemplateBox[{{x, -, p}}, Norm]<=r} in TemplateBox[{}, Reals]^n with center p and radius r, where r may be any non-negative real number and p can have any positive length . The shorthand form Ball[p] is equivalent to Ball[p,1] and Ball[n] is equivalent to Ball[ConstantArray[0, n],1], while Ball[] autoevaluates to Ball[{0,0,0}].
  • Collections of ball objects (multi-balls) of common radius may be efficiently represented using Ball[{p1,…,pk},r] and balls of varying radii represented using Ball[{p1,…,pk},{r1,…,rk}].
  • Ball objects can be visually formatted in two and three dimensions using Graphics and Graphics3D, respectively. The appearance of Ball objects in graphics can be modified by specifying the edge directive EdgeForm (in 2D) or face directive FaceForm (in 3D), color directives such as Red, the transparency and specularity directives Opacity and Specularity, and the style option Antialiasing.
  • Ball may also serve as a region specification over which a computation should be performed. For example, Integrate[1,{x,y,z}∈Ball[{0,0,0},r]] and Volume[Ball[{0,0,0},r]] both return the volume of a 3D ball of radius .
  • Ball is related to a number of other symbols. Sphere represents the boundary of a ball, as can be computed using RegionBoundary[Ball[{x,y,z},r]]. Ball is a generalization of Interval and Disk to arbitrary dimension, and Ellipsoid is a generalization of Ball in the sense that Ball[{p1,…,pk},1] is equivalent to Ellipsoid[{p1,…,pk},ConstantArray[1,k]] for all . SphericalShell gives a filled shell obtained by removing a small ball from the interior of a larger concentric ball. Ball objects in 3D may be represented as ImplicitRegion[(x-u)2+(y-v)2+(z-w)2≤r2,{u,v,w}] or ParametricRegion[a{Cos[θ]Sin[ϕ],Sin[θ]Sin[ϕ],Cos[ϕ]}-{x,y,z},{{θ,0,2π},{ϕ,0,π},{a,0,r}]. Precomputed properties of the 3D ball and its variants in standard position are available using SolidData["entity", "property"] or EntityValue[Entity["Ball","entity"],"property"], where "entity" is one of "Ball" or "HalfBall".

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

A unit ball in 3D:

In[1]:=1
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https://wolfram.com/xid/0giv1q-zi21kv
Out[1]=1

In 2D:

In[2]:=2
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https://wolfram.com/xid/0giv1q-jsgjlp
Out[2]=2

Volume and centroid:

In[1]:=1
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https://wolfram.com/xid/0giv1q-eft1lp
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0giv1q-cwz9t
Out[2]=2

Scope  (22)Survey of the scope of standard use cases

Graphics  (12)

Specification  (4)

The default is a unit ball at the origin in 3D:

In[1]:=1
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https://wolfram.com/xid/0giv1q-ggb1ww
Out[1]=1

Unit balls in different dimensions:

In[1]:=1
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https://wolfram.com/xid/0giv1q-4hpwzh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0giv1q-snmrge
Out[2]=2

Balls with different positions and radii:

In[1]:=1
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https://wolfram.com/xid/0giv1q-rnl2wv
Out[1]=1

Multiple balls with equal radii:

In[1]:=1
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https://wolfram.com/xid/0giv1q-ppozsb
Out[1]=1

Styling  (4)

Balls with different specular exponents:

In[1]:=1
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https://wolfram.com/xid/0giv1q-78t7nc
Out[1]=1

Black ball that glows red:

In[1]:=1
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https://wolfram.com/xid/0giv1q-tdbv9f
Out[1]=1

Opacity specifies the face opacity:

In[1]:=1
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https://wolfram.com/xid/0giv1q-v6r7w7
Out[1]=1

2D styling:

In[1]:=1
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https://wolfram.com/xid/0giv1q-q2gss6
Out[1]=1

Coordinates  (4)

Specify coordinates by fractions of the plot range:

In[1]:=1
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https://wolfram.com/xid/0giv1q-z39q1r
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0giv1q-nxagkd
Out[2]=2

Specify radius by fractions of the plot range:

In[1]:=1
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https://wolfram.com/xid/0giv1q-421ein
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0giv1q-stpkgg
Out[2]=2

Specify scaled offsets from the ordinary coordinates:

In[1]:=1
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https://wolfram.com/xid/0giv1q-u898xy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0giv1q-nn6a0g
Out[2]=2

Points can be Dynamic:

In[1]:=1
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https://wolfram.com/xid/0giv1q-6gl1zz
Out[1]=1

Regions  (10)

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

In[1]:=1
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https://wolfram.com/xid/0giv1q-y220
Out[1]=1

Geometric dimension is the dimension of the shape itself:

In[2]:=2
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https://wolfram.com/xid/0giv1q-bx9tom
Out[2]=2

Membership testing:

In[1]:=1
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https://wolfram.com/xid/0giv1q-6e5svh
In[2]:=2
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https://wolfram.com/xid/0giv1q-3pjv4f
Out[2]=2

Get conditions for point membership:

In[3]:=3
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https://wolfram.com/xid/0giv1q-2p4iz
Out[3]=3

Volume:

In[1]:=1
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https://wolfram.com/xid/0giv1q-se0twe
In[2]:=2
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https://wolfram.com/xid/0giv1q-e06l44
Out[2]=2

Centroid:

In[3]:=3
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https://wolfram.com/xid/0giv1q-gwq4b4
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0giv1q-oknxhk
Out[4]=4

Distance from a point:

In[1]:=1
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https://wolfram.com/xid/0giv1q-8idvu6
In[2]:=2
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https://wolfram.com/xid/0giv1q-lkl9xy
Out[2]=2

The distance to the nearest point for a 2D ball:

In[3]:=3
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https://wolfram.com/xid/0giv1q-da2zys
Out[3]=3

The equidistance contours for a 3D ball:

In[4]:=4
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https://wolfram.com/xid/0giv1q-ew8anh
Out[4]=4

Signed distance from a point:

In[1]:=1
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https://wolfram.com/xid/0giv1q-bfhwbm
In[2]:=2
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https://wolfram.com/xid/0giv1q-0f4ou3
Out[2]=2

Signed distance to a 2D ball:

In[3]:=3
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https://wolfram.com/xid/0giv1q-g7ezsg
Out[3]=3

Nearest point in the region:

In[1]:=1
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https://wolfram.com/xid/0giv1q-d7g53y
In[2]:=2
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https://wolfram.com/xid/0giv1q-mtue
Out[2]=2

Nearest points to an enclosing sphere:

In[3]:=3
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https://wolfram.com/xid/0giv1q-e29k5d
In[4]:=4
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https://wolfram.com/xid/0giv1q-5ksoo8
In[5]:=5
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https://wolfram.com/xid/0giv1q-uv1cfm
Out[5]=5

A ball is bounded:

In[1]:=1
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https://wolfram.com/xid/0giv1q-dym4fu
In[2]:=2
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https://wolfram.com/xid/0giv1q-i3tfrr
Out[2]=2

Find its range:

In[3]:=3
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https://wolfram.com/xid/0giv1q-l3exhn
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0giv1q-bx1r15
Out[4]=4

Integrate over a ball region:

In[1]:=1
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https://wolfram.com/xid/0giv1q-fivgav
In[2]:=2
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https://wolfram.com/xid/0giv1q-banwkr
Out[2]=2

Optimize over a ball region:

In[1]:=1
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https://wolfram.com/xid/0giv1q-nf9ton
In[2]:=2
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https://wolfram.com/xid/0giv1q-hyz4dq
Out[2]=2

Solve equations in a ball region:

In[1]:=1
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https://wolfram.com/xid/0giv1q-bnrw6
In[2]:=2
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https://wolfram.com/xid/0giv1q-c7lkth
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

Find the minimum surface area for a ball with volume :

In[1]:=1
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https://wolfram.com/xid/0giv1q-opgf2q
In[2]:=2
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https://wolfram.com/xid/0giv1q-bnifa8
Out[2]=2

Total mass for a ball region with density given by :

In[1]:=1
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https://wolfram.com/xid/0giv1q-h2ora
In[2]:=2
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https://wolfram.com/xid/0giv1q-p48pf6
Out[2]=2

Find the mass of caffeine in a ball with a radius of 3 centimeters:

In[1]:=1
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https://wolfram.com/xid/0giv1q-u48x65

Density of caffeine:

In[2]:=2
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https://wolfram.com/xid/0giv1q-6bgb0m
Out[2]=2

Volume of ball:

In[3]:=3
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https://wolfram.com/xid/0giv1q-6fqwu0
Out[3]=3

Mass of caffeine in the ball:

In[4]:=4
✖
https://wolfram.com/xid/0giv1q-8she50
Out[4]=4

Properties & Relations  (5)Properties of the function, and connections to other functions

Disk is a special case of Ball:

In[1]:=1
✖
https://wolfram.com/xid/0giv1q-f599a7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0giv1q-r6dlf
Out[2]=2

Sphere is the RegionBoundary of Ball:

In[1]:=1
✖
https://wolfram.com/xid/0giv1q-h6x47m
Out[1]=1

Ellipsoid is a generalization of Ball:

In[1]:=1
✖
https://wolfram.com/xid/0giv1q-bgriyc
In[2]:=2
✖
https://wolfram.com/xid/0giv1q-yms35e
Out[2]=2

ImplicitRegion can represent any Ball:

In[1]:=1
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https://wolfram.com/xid/0giv1q-hw5ae3
In[2]:=2
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https://wolfram.com/xid/0giv1q-ki4s56
Out[2]=2

Ball is a norm ball for the Euclidean norm:

In[1]:=1
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https://wolfram.com/xid/0giv1q-mysp6a
In[2]:=2
✖
https://wolfram.com/xid/0giv1q-b56hh2
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Random ball collections:

In[4]:=4
✖
https://wolfram.com/xid/0giv1q-coltip
Out[4]=4
Wolfram Research (2014), Ball, Wolfram Language function, https://reference.wolfram.com/language/ref/Ball.html.
✖
Wolfram Research (2014), Ball, Wolfram Language function, https://reference.wolfram.com/language/ref/Ball.html.

Text

Wolfram Research (2014), Ball, Wolfram Language function, https://reference.wolfram.com/language/ref/Ball.html.

✖
Wolfram Research (2014), Ball, Wolfram Language function, https://reference.wolfram.com/language/ref/Ball.html.

CMS

Wolfram Language. 2014. "Ball." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Ball.html.

✖
Wolfram Language. 2014. "Ball." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Ball.html.

APA

Wolfram Language. (2014). Ball. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Ball.html

✖
Wolfram Language. (2014). Ball. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Ball.html

BibTeX

@misc{reference.wolfram_2025_ball, author="Wolfram Research", title="{Ball}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Ball.html}", note=[Accessed: 10-July-2025 ]}

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@misc{reference.wolfram_2025_ball, author="Wolfram Research", title="{Ball}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Ball.html}", note=[Accessed: 10-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_ball, organization={Wolfram Research}, title={Ball}, year={2014}, url={https://reference.wolfram.com/language/ref/Ball.html}, note=[Accessed: 10-July-2025 ]}

✖
@online{reference.wolfram_2025_ball, organization={Wolfram Research}, title={Ball}, year={2014}, url={https://reference.wolfram.com/language/ref/Ball.html}, note=[Accessed: 10-July-2025 ]}