# "Filled in" ParametricPlot3D? **URL:** \ **Category:** Questions\ **Tags:** geometry, graphics-and-visualization\ **Created:** [November 23, 2017, 8:55pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570 "2017-11-23T20:55:37Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![davidjlynch20](https://avatars.discourse-cdn.com/v4/letter/d/58956e/32.png) [@davidjlynch20](https://community.wolfram.com/u/davidjlynch20)\ **Post date:** [November 23, 2017, 8:55pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/1 "2017-11-23T20:55:37Z")
Hello all! I’m currently finding my way through Mathematica (novice here), and am having issues “filling in” a Parametric3DPlot, as shown below: ``` ParametricPlot3D[{(1 + Cos[(8 t)]) Cos[t], (1 + Cos[8 t]) Sin[t], u}, {t, 0, 2 Pi}, {u, 0, 1}, Mesh -> None, PlotStyle -> Opacity[.9]] ``` While these “petals” somewhat work, how do I fill them in, to make them a solid shape? Thanks! ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [November 24, 2017, 7:12am UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/2 "2017-11-24T07:12:21Z")
Here is a way: ``` f[t_, u_, r_] = {r (1 + Cos[(8 t)]) Cos[t], r (1 + Cos[8 t]) Sin[t], u}; Show[ParametricPlot3D[f[t, u, 1], {t, 0, 2 Pi}, {u, 0, 1}, Mesh -> None, PlotStyle -> Opacity[.9], PlotPoints -> {40, 2}], ParametricPlot3D[{f[t, 1, r], f[t, -1, r]}, {t, 0, 2 Pi}, {r, 0, 1}, Mesh -> None, PlotStyle -> Opacity[.9], PlotPoints -> {40, 2}]] ``` ---
**Author:** ![mroge02](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/mroge02/32/28775_2.png) [@mroge02](https://community.wolfram.com/u/mroge02)\ **Post date:** [November 24, 2017, 3:47pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/3 "2017-11-24T15:47:07Z")
The following uses the vertices of the top and bottom edges, where `u` is approximately one of `{0, 1}`. It is only approximate because `ParametricPlot3D` uses open sampling. The advantage to using the actually discrete edges of the graphics is that the lids “fit tightly.” ``` ParametricPlot3D[{(1 + Cos[(8 t)]) Cos[t], (1 + Cos[8 t]) Sin[t], u}, {t, 0, 2 Pi}, {u, 0, 1}, PlotPoints -> {120, 5}, Mesh -> None, PlotStyle -> Opacity[0.9], BoundaryStyle -> Opacity[0.9]] /. GraphicsComplex[pts_, g_, opts_] :> GraphicsComplex[ pts, (* array of coordinates of points in the graphics *) g /. Line[idcs_] :> { (* points are specified by their indices in the coordinate array pts *) EdgeForm[], Polygon[ SplitBy[ Select[ idcs, Min@Abs[pts[[#, 3]] - {0, 1}] < 1*^-6 &], (* select indices at the boundary edges {0, 1} *) Nearest[{0, 1}, pts[[#, 3]]] &], (* split the indices according to bottom, top {0, 1} *) VertexNormals -> None] (* the polygons are virtually flat and should not inherit the VertexNormals assigned to their points on the original surface (the sides) *) }, opts] ``` Alternatively, one can construct a cylinder given a base polygon. ``` base = First@Cases[ ParametricPlot[{(1 + Cos[(8 t)]) Cos[t], (1 + Cos[8 t]) Sin[t]}, {t, 0, 2 Pi}], Line[p_] :> p, Infinity]; RegionProduct[Polygon[base], Line[{{0.}, {1.}}]] ``` Or just construct the base directly with `Table`: ``` base = Most@ Table[{(1 + Cos[(8 t)]) Cos[t], (1 + Cos[8 t]) Sin[t]}, {t, 0., 2 Pi, 2 Pi/8/100}]; (* 100 yields a sufficient sampling density *) ``` ---
**Author:** ![davidjlynch20](https://avatars.discourse-cdn.com/v4/letter/d/58956e/32.png) [@davidjlynch20](https://community.wolfram.com/u/davidjlynch20)\ **Post date:** [November 24, 2017, 6:13pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/4 "2017-11-24T18:13:10Z")
This is so helpful, thank you all. One more question: I am trying to combine this with another ParametricPlot3D, but whenever I try the command Show[{figure1,figure2,figure3}, PlotRange-\>All], it returns that it cannot combine them. Any help with this part? ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [November 24, 2017, 6:29pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/5 "2017-11-24T18:29:54Z")
You can also use the derived geometric region functions: ``` RegionProduct[ BoundaryDiscretizeGraphics@ ParametricPlot[{(1 + Cos[(8 t)]) Cos[t], (1 + Cos[8 t]) Sin[t]}, {t, Pi/8, 2 Pi + Pi/8}], DiscretizeRegion@Line[{{0}, {1}}]] ``` ---
**Author:** ![davidjlynch20](https://avatars.discourse-cdn.com/v4/letter/d/58956e/32.png) [@davidjlynch20](https://community.wolfram.com/u/davidjlynch20)\ **Post date:** [November 24, 2017, 8:07pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/6 "2017-11-24T20:07:02Z")
[@Gianluca Gorni](http://community.wolfram.com/web/gianlucagorni) thank you so much! One last question (sorry!): how do I change the color of this graphic? I’m confused as to where I would put it in the coding. 🙂 ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [November 24, 2017, 8:28pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/7 "2017-11-24T20:28:29Z")
Right now I cannot check with Mathematica, but you may look up `MeshPrimitives`, that converts a Region to `Graphics3D`, to which you can give color styling. ---
**Author:** ![davidjlynch20](https://avatars.discourse-cdn.com/v4/letter/d/58956e/32.png) [@davidjlynch20](https://community.wolfram.com/u/davidjlynch20)\ **Post date:** [November 24, 2017, 9:00pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/8 "2017-11-24T21:00:03Z")
[@Gianluca Gorni](http://community.wolfram.com/web/gianlucagorni) Hm… can’t seem to exactly find the right resources to convert the code you gave me into a MeshFunction. Perhaps when you get the chance you could help me. Thanks! ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [November 25, 2017, 3:00pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/9 "2017-11-25T15:00:12Z")
Here is an example: ``` reg = RegionProduct[ BoundaryDiscretizeGraphics@ ParametricPlot[{(1 + Cos[(8 t)]) Cos[t], (1 + Cos[8 t]) Sin[ t]}, {t, Pi/8, 2 Pi + Pi/8}], DiscretizeRegion@Line[{{0}, {1}}]]; Graphics3D[{Red, EdgeForm[], MeshPrimitives[reg, 2]}] ``` ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [November 26, 2017, 9:08pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/10 "2017-11-26T21:08:43Z")
This is a ligher version, with only the boundary shown: ``` reg = RegionBoundary@ RegionProduct[ BoundaryDiscretizeGraphics@ ParametricPlot[{(1 + Cos[(8 t)]) Cos[t], (1 + Cos[8 t]) Sin[ t]}, {t, Pi/8, 2 Pi + Pi/8}], DiscretizeRegion@Line[{{0}, {1}}]]; Graphics3D[{Red, EdgeForm[], Opacity[.5], MeshPrimitives[reg, 2]}] ``` ---
**Author:** ![anon21378091](https://avatars.discourse-cdn.com/v4/letter/a/bcef8e/32.png) [@anon21378091](https://community.wolfram.com/u/anon21378091)\ **Post date:** [November 27, 2017, 2:16pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/11 "2017-11-27T14:16:24Z")
Very interesting solutions! Michael, thanks, but where do you get and what is the meaning of: ``` Line[{{0}, {1}}] ? ``` I cannot find in the Line docs any such syntax for Line[]. Also, ``` Graphics[Line[{{0}, {1}}]] ``` gives the error: "Array {{0}, {1}} has the wrong dimensions for a graphics coordinate list. Am I missing something? ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [November 27, 2017, 3:40pm UTC](https://community.wolfram.com/t/filled-in-parametricplot3d/11570/12 "2017-11-27T15:40:19Z")
The special syntax `Line[{{0}, {1}}]` is not used for 2D graphics, but for geometric regions. It means the line in 1D space, connecting the two zero-dimensional points `{0}` and `{1}`.