2
$\begingroup$

Edit: This question has very useful answers, but none address how to work with spherical coordinates.

I want to plot a 2D disk in 3D space, angled perpendicular to a given direction. That direction is given by $\theta$ and $\phi$ in spherical coordinates (where $\theta$ is polar and $\phi$ is azimuthal cause I'm a physicist, sorry math people!).

Here's four approaches I've tried that nearly work but not quite:

  1. ContourPlot3D in cartesian coordinates

    ContourPlot3D[{x + y + z == 1}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]

    This gives an inclined plane, but I can't figure out how to put it at the desired angles $\theta$ and $\phi$. I also wouldn't know how to make this a disk with a finite radius.

    enter image description here

  2. ContourPlot3D in spherical coordinates, by converting x+y+z==1 into spherical

    ContourPlot3D[{r*Sin[θ]*Cos[ϕ] + r*Sin[θ]*Sin[ϕ] 
     + r*Cos[θ] == 1},
{r, 0, 1}, {θ, 0, Pi}, {ϕ, 0, 2*Pi}, AxesLabel -> Automatic]

    This uses {r,$\theta$,$\phi$} as the plot coordinates, rather than plotting in proper cartesian space.

    enter image description here

  3. ParametricPlot3D to make a disk, then rotate it

    ParametricPlot3D[{r*Sin[Pi/2]*Cos[ϕ],r*Sin[Pi/2]*Sin[ϕ], r*Cos[Pi/2]},
  {r, 0, 1}, {ϕ, 0, 2*Pi}, 
 PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}, AxesLabel -> {x, y, z}]

    This can produce a disk, but I can't figure out how to angle it. Here I set $\theta$=$\pi$/2 to make the disk.

    enter image description here

  4. Use Disk with Graphics3D and Rotate

    This would work if Disk could be used as a 3D object! Argh!

That's all my ideas. If you see how to fix one of these, or have another, better idea, please advise!

Improve this question
$\endgroup$
2
  • $\begingroup$ Have a look at this answer by Taiki. $\endgroup$ – Henrik Schumacher Feb 16 '18 at 6:53
  • 2
    $\begingroup$ Some of the techniques used to create a 3D circle are certainly relevant for creating a 3D disk, but I don't think that makes this question a duplicate, especially since what I consider the best answer to that question uses BSplineCurve, and a BSplineCurve is not capable of creating a disk. I just created a resource function Disk3D based on BSplineSurface that can create a 3D disk, so I think it makes sense to reopen this question. $\endgroup$ – Carl Woll Dec 16 '19 at 22:41
3
$\begingroup$

First, define the normal in spherical coordinates:

n[theta_, phi_] := {Sin[theta]*Cos[phi], Sin[theta]*Sin[phi], Cos[theta]}

Then multiply your (syntax-corrected) ParametricPlot3D with RotationMatrix and wrap it in Manipulate:

Manipulate[
  Show[
     ParametricPlot3D[
       RotationMatrix[{{0, 0, 1}, n[theta, phi]}].  
         {r*Sin[Pi/2]*Cos[Phi], r*Sin[Pi/2] Sin[Phi], r*Cos[Pi/2]}
         , {r, 0, 1}, {Phi, 0, 2*Pi}
         , PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}
         , AxesLabel -> {x, y, z}]
      , Graphics3D[{Red, Arrow[{{0, 0, 0}, n[theta, phi]}]}]]
  , {theta, 0, 2 Pi}, {phi, 0, Pi}]

enter image description here

Improve this answer
$\endgroup$
1
  • $\begingroup$ This is beautiful, thank you. RotationMatrix was the missing piece to my idea #3 above. Small note: typically theta goes 0->Pi and phi goes 0->2Pi. $\endgroup$ – Max Feb 17 '18 at 0:04
3
$\begingroup$

Given the normal vector

n={Cos[\[Theta]] Cos[\[CurlyPhi]], Cos[\[Theta]] Sin[\[CurlyPhi]],Sin[\[Theta]]}

the plane orthogonal to n is given by 

e={Sin[\[CurlyPhi]], -Cos[\[CurlyPhi]], 0}
f={-Cos[\[CurlyPhi]] Sin[\[Theta]], -Sin[\[Theta]] Sin[\[CurlyPhi]],Cos[\[Theta]]}

The 3D-disk can be plotted using 

Block[{\[Theta] = 50 \[Degree], \[CurlyPhi] = 10 \[Degree] },
Show[{Graphics3D[{Point[{0, 0, 0}], Opacity[.2], Sphere[]}], 
ParametricPlot3D [r (Cos[t] e + Sin[t] f), {r, 0, 1}, {t, 0, 2 Pi}]}, Boxed-> False]
]

enter image description here

Improve this answer
$\endgroup$
3
  • $\begingroup$ Nothing wrong with you answer, but @anderstood built upon one of my solutions which made it easier for me to work with. Small feedback: aren't your n and e defined the same? $\endgroup$ – Max Feb 17 '18 at 0:10
  • $\begingroup$ Sorry, cut &paste error: I edited my answer. $\endgroup$ – Ulrich Neumann Feb 17 '18 at 14:00
  • $\begingroup$ ... n!=e: By the way I didn't agree, that using a RotationMatrix is easier than the evaluation of two orthogonal vectors... $\endgroup$ – Ulrich Neumann Feb 17 '18 at 14:07
3
$\begingroup$

The resource function Disk3D generates a BSplineSurface representation of a 3D disk. For example, a 3D disk with symmetry axis in the direction {1, 1, 1}:

Graphics3D @ ResourceFunction["Disk3D"][{0,0,0}, 1, {1, 1, 1}]

enter image description here

A 3D elliptical disk in the plane spanned by {1, 1, 0} and {0, 1, 1} with the semimajor axis in the direction {1, 1, 0}:

Graphics3D[{
    ResourceFunction["Disk3D"][{0, 0, 0}, {3, 1}, {{1, 1, 0}, {0, 1, 1}}], 
    Red, Arrow[{{0,0,0}, {1,1,0}}],
    Blue, Arrow[{{0,0,0}, {0,1,1}}]
}]

enter image description here

A graphic with many disks:

Graphics3D[{
    LightGreen,
    MapThread[
        ResourceFunction["Disk3D"],
        {RandomReal[10, {300, 3}], RandomReal[3, 300], RandomReal[1, {300, 3}]}
    ]
}]

enter image description here

An example of the BSplineSurface primitive:

ResourceFunction["Disk3D"][{0,0,0}]

BSplineSurface[{{{-1, 0, 0}, {-1, -1, 0}, {0, -1, 0}}, {{-1, 1, 0}, {1/9, 0, 0}, {1, -1, 0}}, {{0, 1, 0}, {1, 1, 0}, {1, 0, 0}}}, SplineKnots -> {{0, 0, 0, 1, 1, 2}, {0, 0, 0, 1, 1, 2}}, SplineWeights -> {{1, 1/Sqrt[2], 1}, {1/Sqrt[2], 1, 1/Sqrt[2]}, {1, 1/Sqrt[2], 1}}]

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.