How to plot a quarter disk with specified height?

Question

Kind of important: the solution posted does work, and answers the question in that sense. However, it is very slow, so a faster option would be much appreciated.

I'm trying to create the following image in Mathematica.

I apologize for my terrible paint skills. The idea is that I have a certain vector, say $({0,-\sqrt{0.25},\sqrt{0.5},-\sqrt0.25})$ (it's a unit vector, apart from that the components can take any real value). Now, what I want is to plot a disk, where each quarter section corresponds to a value of the vector. So in the image I drew, the first component is 0, so that quarter is just a quarter circle in the z = 0 plane. Then the second component would have height $\sqrt{0.25}$ in the second quarter section of the disk, in the negative z direction, et cetera.

Now, to be honest, I'm not sure where to start. I first thought I could possible use ParametricPlot3D, for example like

ParametricPlot3D[{ Cos[u], Sin[u], v}, {u, 0, 2 Pi/4}, {v, -0.5, 0}, 
 Mesh -> None] 

However, this only draws the outer shell, and I'm not sure to fill it up. Nor am I sure that I can combine this with another 3 commands in order to get a full disk, but perhaps that is not too difficult. In any case, I was wondering if someone had an idea of how to do this, and if I'm taking the right approach.

To extend a little on this, I tried combining them like this but it doesn't seem to work

p1 = ParametricPlot3D[{ Cos[u], Sin[u], v}, {u, 0, 2 Pi/4}, {v, -0.5, 
    0}, Mesh -> None, PlotStyle -> Opacity[.5]];
p2 = ParametricPlot3D[{ Cos[u], Sin[u], v}, {u, 2 Pi/4, 2 Pi/2}, {v, 
    0, Sqrt[0.5]}, Mesh -> None, PlotStyle -> LightBlue];
Show[p1, p2]

I can however do it in 2D, with circles. If only here was an easy way to plot sections of cylinders.

p1 = Graphics[{Blue, Disk[{0, 0}, 0, {0, 2 Pi/4}]}];
p2 = Graphics[{Red, Disk[{0, 0}, Sqrt[0.5], {2 Pi/4, 2 Pi/2}]}];
p3 = Graphics[{Blue, Disk[{0, 0}, 0.5, {Pi, 3/2*Pi}]}];
p4 = Graphics[{Red, Disk[{0, 0}, Sqrt[0.5], {3/2*Pi, 2 Pi/1}]}];
Show[p1, p2, p3, p4]

Answer 1

You can create a rather fast function to create a section of a cylinder of a given height combining some ParametricPlot3D using the following code:

cylinderSection[h_, \[Theta]0_, \[Theta]1_, 
  OptionsPattern[plotStyle -> Green]] := With[{r = 1},
  Show[
   ParametricPlot3D[
    {{\[Rho] Cos[\[Theta]], \[Rho] Sin[\[Theta]], 0},
     {\[Rho] Cos[\[Theta]], \[Rho] Sin[\[Theta]], h}},
    {\[Rho], 0, r},
    {\[Theta], \[Theta]0, \[Theta]1},
    Mesh -> None,
    PlotStyle -> OptionValue@plotStyle
    ],
   ParametricPlot3D[
    {Cos[\[Theta]], Sin[\[Theta]], z},
    {\[Theta], \[Theta]0, \[Theta]1},
    {z, 0, h},
    Mesh -> None,
    PlotStyle -> OptionValue@plotStyle
    ],
   ParametricPlot3D[
    {{t Cos[\[Theta]0], t Sin[\[Theta]0], z},
     {t Cos[\[Theta]1], t Sin[\[Theta]1], z}},
    {t, 0, 1},
    {z, 0, h},
    Mesh -> None,
    PlotStyle -> OptionValue@plotStyle
    ],
   PlotRange -> All,
   RotationAction -> "Clip",
   ImageSize -> Large
   ]
  ]

This is called in exactly the same way as the solution with RegionPlot3D (see below):

Show[
 cylinderSection[0.001, 0, Pi/2, plotStyle -> Blue],
 cylinderSection[-Sqrt[0.25], Pi/2, Pi, plotStyle -> Green],
 cylinderSection[Sqrt[0.5], -Pi, -Pi/2, plotStyle -> Orange],
 cylinderSection[-Sqrt[0.25], -Pi/2, 0, plotStyle -> Red]
 ]

And this is the output:

You want the whole thing generated by only giving as input heights, angles and colors? Do it with the function makePie:

makePie[hh_List, angles_List, plotStyles_List] := Show[
  Append[
   Table[
    cylinderSection[hh[[i]] /. (0 -> 0.001), angles[[i]], 
     angles[[i + 1]], plotStyle -> plotStyles[[i]]],
    {i, Length@hh - 1}
    ],
   cylinderSection[hh[[Length@hh]] /. (0 -> 0.001), 
    angles[[Length@hh]], 2 Pi, plotStyle -> plotStyles[[Length@hh]]]
   ]
  ]

You can now generate the above pie with the following:

makePie[{0, -Sqrt[0.25], Sqrt[0.5], -Sqrt[0.25]}, {0, Pi/2, Pi, 
  3 Pi/2}, {Blue, Green, Orange, Red}]

But you can use it to generate pies with an arbitrary number of partitions, not just four. For fun, you can try this:

With[{n = RandomInteger@{3, 20}}, 
 makePie[RandomReal[{-2, 2}, n], Sort@RandomReal[{0, 2 Pi}, n], 
  RandomColor@n]]

The (slow) solution I originally found, using RegionPlot3D:

First create a function drawing an arbitrary section of a cylinder with

cylinderSection[h_, \[Theta]1_, \[Theta]2_, 
  OptionsPattern[plotStyle -> Red]] := RegionPlot3D[
  x^2 + y^2 < 1 &&
   \[Theta]1 < ArcTan[x, y] < \[Theta]2,
  {x, -1, 1},
  {y, -1, 1},
  {z, 0, h},
  Mesh -> None,
  PlotRange -> All,
  RotationAction -> "Clip",
  PlotStyle -> OptionValue@plotStyle,
  PlotPoints -> 100,
  ImageSize -> Large
  ]

Then combine it with Show to obtain the desired result.

Show[
 cylinderSection[0.001, 0, Pi/2, plotStyle -> Blue],
 cylinderSection[-Sqrt[0.25], Pi/2, Pi, plotStyle -> Green],
 cylinderSection[Sqrt[0.5], -Pi, -Pi/2, plotStyle -> Orange],
 cylinderSection[-Sqrt[0.25], -Pi/2, 0, plotStyle -> Red]
 ]

Unfortunately, this has the problem of being rather slow, so possibly something better with only Graphics3D can be realized.

Answer 2

seg[{a_, b_}, {g_, h_}, col_, op_] := 
 Show[ParametricPlot3D[{Cos[t], Sin[t], z}, {t, a, b}, {z, g, h}, 
   Mesh -> False, PlotStyle -> {col, Opacity[op]}, 
   BoundaryStyle -> {Black, Thick}], 
  ParametricPlot3D[{u Cos[t], u Sin[t], g}, {t, a, b}, {u, 0, 1}, 
   Mesh -> False, PlotStyle -> {col, Opacity[op]}], 
  ParametricPlot3D[{u Cos[t], u Sin[t], h}, {t, a, b}, {u, 0, 1}, 
   Mesh -> False, PlotStyle -> {col, Opacity[op]}], 
  Graphics3D[{col, Opacity[op], EdgeForm[{Black, Thick}], 
      Polygon[{{0, 0, g}, {0, 0, h}, {Cos[#], Sin[#], h}, {Cos[#], 
         Sin[#], g}}]} & /@ {a, b}]]

Visualizing:

Manipulate[
 Show[seg[{0 + a, Pi/2}, {0, 1}, Red, opacity], 
  seg[{-Pi/2, Pi + a}, {-1, 0}, Blue, opacity],
  seg[{Pi + a, 3 Pi/2}, {-.01, 0}, Gray, opacity],
  PlotRange -> Table[{-2, 2}, {3}], Boxed -> False, Axes -> False, 
  Background -> White], {opacity, 0.3, 1}, {{a, 0}, -1, 1}]

Answer 3

ChartElementData["CylindricalSector3D"]

The built-in function ChartElementData["CylindricalSector3D"] takes two or more arguments. The first argument, {{angle0_, angle1_}, {radius0_, radius1_}, {height0_, height1_}} specifies the angles, radii and heights needed to specify a cylindrical sector. The second argument needs to be specified but can be anything for our application. We use this function to generate the graphics primitives in the following function:

cylindricalSectors[styles : {__}, specs : {__}, opts : OptionsPattern[]] := 
 Graphics3D[{#, EdgeForm[], ChartElementData["CylindricalSector3D"][#2, {}]} & @@@ 
   Transpose[{styles, specs}], opts, Axes -> False, Boxed -> False, Lighting -> "Neutral"]

Examples:

cylindricalSectors[{Red}, {{{Pi/2, Pi}, {0, 1}, {0, 1/2}}},  Axes -> True]

specs = {{{0, Pi/2}, {0, 2}, {0, 1}}, {{Pi/2, Pi}, {0, 2}, {0,  1}},
 {{Pi, 3 Pi/2}, {0, 2}, {0, 1}}, {{3 Pi/2, 2 Pi}, {0, 2}, {1, 1}}, 
 {{Pi/2, Pi}, {0, 2}, {1, 2}}};
colors = ColorData[54, "ColorList"][[;; 5]];

cylindricalSectors[colors, specs]

random = Transpose[{Sort /@ RandomReal[{0, 2 Pi}, {7, 2}],
     {0, #} & /@ RandomReal[{1, 3}, 7], Sort /@ RandomReal[{0, 10}, {7, 2}]}];
colors = ColorData["Rainbow"] /@ Range[1/7, 1, 1/7];
cylindricalSectors[Opacity[.7, #] & /@ colors, random]