SphericalShell

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`SphericalShell`

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SphericalShell[c,{r_inner,r_outer}]

represents a filled spherical shell centered at c with inner radius r_inner and outer radius r_outer.

Details and Options

SphericalShell can be used as a geometric region and a graphics primitive.
SphericalShell[] is equivalent to SphericalShell[{0,0,0},{1/2,1}].
SphericalShell[r] is equivalent to SphericalShell[{0,0,0},{r/2,r}].
SphericalShell[{r_inner,r_outer}] is equivalent to SphericalShell[{0,0,0},{r_inner,r_outer}].

SphericalShell represents a filled spherical shell ${x|r_(inner)<=TemplateBox[{{x, -, c}}, Norm]<=r_(outer)}$ . The region is dimensional for point c of length .
SphericalShell allows c to be any point in and .
SphericalShell can be used in Graphics and Graphics3D.
In graphics, the point c and radius r can be Dynamic expressions.
Graphics rendering is affected by directives such as FaceForm, EdgeForm, Specularity, Opacity, and color.

Examples

open allclose all

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

The standard spherical shell at the origin:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-du6svg

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-c7o

Out[2]=2

Volume and centroid:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-7wedhk

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-kxw6nj

Out[2]=2

Scope (20)Survey of the scope of standard use cases

Graphics (10)

Specification (5)

The standard spherical shell:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-btz1q2

Out[1]=1

Spherical shells with different outer radii:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-l0hn2

Out[1]=1

Spherical shells with different inner radii:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-hky91r

Out[1]=1

Short form for a spherical shell with radii at the origin:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-37iz6

Out[1]=1

Short form for a spherical shell with radii at the origin:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-fwdxat

Out[1]=1

Styling (4)

Colored spherical shells:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-cdxdbh

Out[1]=1

Different properties can be specified for the front and back of faces using FaceForm:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-vb6wx

Out[1]=1

Spherical shells with different specular exponents:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-d1tyd0

Out[1]=1

White spherical shell that glows red:

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-ca9zt

Out[2]=2

Opacity specifies the face opacity:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-v6995

Out[1]=1

Coordinates (1)

Points can be Dynamic:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-6gl1zz

Out[1]=1

Regions (10)

Embedding dimension is the dimension of the space in which the spherical shell lives:

In[3]:=3

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https://wolfram.com/xid/0n4frng4azh-y220

Out[3]=3

Geometric dimension is the dimension of the shape itself:

In[4]:=4

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https://wolfram.com/xid/0n4frng4azh-bx9tom

Out[4]=4

Membership testing:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-c7lq97

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-f70gib

Out[2]=2

Get conditions for point membership:

In[3]:=3

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https://wolfram.com/xid/0n4frng4azh-inebf9

Out[3]=3

Volume:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-se0twe

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-e06l44

Out[2]=2

Centroid:

In[3]:=3

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https://wolfram.com/xid/0n4frng4azh-gwq4b4

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0n4frng4azh-oknxhk

Out[4]=4

Distance from a point:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-jcsb4b

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-8aexdg

Out[2]=2

The equidistance contours for a spherical shell:

In[3]:=3

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https://wolfram.com/xid/0n4frng4azh-ew8anh

Out[3]=3

Signed distance from a point:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-kjgbyj

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-zognbt

Out[2]=2

Nearest point in the region:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-d7g53y

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-mtue

Out[2]=2

Nearest points to an enclosing sphere:

In[3]:=3

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https://wolfram.com/xid/0n4frng4azh-e29k5d

In[4]:=4

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https://wolfram.com/xid/0n4frng4azh-5ksoo8

In[5]:=5

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https://wolfram.com/xid/0n4frng4azh-uv1cfm

Out[5]=5

A spherical shell is bounded:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-ypd96t

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-po0eks

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0n4frng4azh-dym4fu

In[4]:=4

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https://wolfram.com/xid/0n4frng4azh-i3tfrr

Out[4]=4

Find its range:

In[5]:=5

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https://wolfram.com/xid/0n4frng4azh-l3exhn

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0n4frng4azh-bx1r15

Out[6]=6

Integrate over a spherical shell region:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-fivgav

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-banwkr

Out[2]=2

Optimize over a spherical shell region:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-nf9ton

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-hyz4dq

Out[2]=2

Solve equations in a spherical shell region:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-bnrw6

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-lhnpf3

Out[2]=2

Applications (5)Sample problems that can be solved with this function

A standard Ping-Pong ball, a spherical shell, has a mass of 2.7 grams and an outer radius of 40 millimeters:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-e5g8rs

It is made of celluloid, which has a density of . Knowing that , find the inner radius of a Ping-Pong ball:

In[3]:=3

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https://wolfram.com/xid/0n4frng4azh-cfx2ja

In[4]:=4

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https://wolfram.com/xid/0n4frng4azh-bpcki9

Out[4]=4

The thickness is the difference between the inner and outer radius:

In[5]:=5

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https://wolfram.com/xid/0n4frng4azh-bzw9ov

Out[5]=5

It was once believed that the universe was a system of concentric celestial spheres. At the center of the universe was Earth, and surrounding it were the spherical shells for the planets and the Sun. There was disagreement about the exact order of the planets and the Sun. Plato and Ptolemy ordered the celestial spheres differently:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-boeyc9

Visualize and compare the two philosophers' universes using a set of labeled spherical shells:

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-ebwncv

In[3]:=3

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https://wolfram.com/xid/0n4frng4azh-dwhmr0

Out[3]=3

A pearl is formed, layer by layer, by a mollusk. The pearl's optical properties come from light reflecting off many translucent layers, as opposed to only an opaque surface layer. The difference that translucent layers make can be seen by creating two sets of nested spherical shells that differ in opacity:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-noxhyp

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-togky

Out[2]=2

Golf balls have multiple layers, the outer, dimpled shell, and one or more inner layers. Model the outer layer as the RegionDifference between a spherical shell and a collection of balls. Get a distribution for the dimples using the MeshCoordinates of a sphere's mesh:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-h5iw2r

Discretize the region over an interval to get a "cut-away" view of half of it:

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-dtsfms

Finally, make a ball for the inner core of the golf ball, offset from the center to get a visual separation of layers:

In[3]:=3

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https://wolfram.com/xid/0n4frng4azh-blftrz

In[4]:=4

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https://wolfram.com/xid/0n4frng4azh-cftdxt

Out[4]=4

A balloon can be approximated as a spherical shell. Suppose the balloon is made of 10 units of material. Find out how the thickness depends on the outer radius of the balloon:

In[1]:=1

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https://wolfram.com/xid/0n4frng4azh-cryyaq

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0n4frng4azh-cgfnxp

Out[2]=2

Suppose the material is such that under the given conditions it will break if it has thickness less than . Find the largest possible outer radius balloon given that constraint: