I'm trying to construct a BoundaryMeshRegion from a set of 3D surfaces generated using ParametricPlot3D. Here's my workflow:
- I created several
ParametricPlot3Dsurfaces using equations of the form:
ParametricPlot3D[{fx, fy, fz}, {u, umin, umax}, {v, vmin, vmax}]
In this case, u represents the parameter for each surface. However, for my surfaces, the parameter u (called t in my code) differs in its range across some surfaces. Specifically:
- The
trange for two of the surfaces is the same. - For the other two surfaces, the
trange is different.
- I extracted points, lines, and polygons from the plots using
Cases. - Attempted to create a
BoundaryMeshRegionby combining the extracted data.
2D Grapics:
a = ParametricPlot[{2 Cos[t], 2 Sin[t]}, {t, 0, Pi}, PlotRange -> All];
b = ParametricPlot[{4 + 2 Cos[t], 2 Sin[t]}, {t, Pi, 2 Pi},
PlotRange -> All];
c = ParametricPlot[{8 + 2 Cos[t], 2 Sin[t]}, {t, 0, Pi}];
d = ParametricPlot[{4 + 6 Cos[t], 6 Sin[t]}, {t, Pi, 2 Pi},
PlotRange -> All];
3D Grapics:
However, I get the error:
BoundaryMeshRegion::coplnr: The vertices in the polygon [...] are not coplanar.
Here’s a simplified version of my code:
(* Parameters *)
wrap = 4 \[Pi];
P1 = 100;
P2 = 20;
P = P1 + ((P2 - P1)/wrap)*t;
L = 1/(2 Pi)*Integrate[P, {t, 0, \[Phi]r}];
(* Transformation Matrix *)
M = ({
{Cos[\[Phi]r], -Sin[\[Phi]r], 0, 0},
{Sin[\[Phi]r], Cos[\[Phi]r], 0, 0},
{0, 0, 1, L},
{0, 0, 0, 1}
});
(* Parametric Equations *)
{X[1], Y[1]} = {2 Cos[t], 2 Sin[t]} /. \[Phi]r -> wrap;
{X[2], Y[2]} = {4 + 2 Cos[t], 2 Sin[t]} /. \[Phi]r -> wrap;
{X[3], Y[3]} = {8 + 2 Cos[t], 2 Sin[t]} /. \[Phi]r -> wrap;
{X[4], Y[4]} = {4 + 6 Cos[t], 6 Sin[t]} /. \[Phi]r -> wrap;
(* Transformation *)
{abx, aby, abz, dm} = M . {X[1], Y[1], 0, 1};
{bcx, bcy, bcz, dm} = M . {X[2], Y[2], 0, 1};
{cdx, cdy, cdz, dm} = M . {X[3], Y[3], 0, 1};
{dex, dey, dez, dm} = M . {X[4], Y[4], 0, 1};
(* Parametric Plots *)
plot = {
ParametricPlot3D[
{abx, aby, abz} // Evaluate, {t, 0, Pi}, {\[Phi]r, 0, wrap},
MaxRecursion -> 0, PlotPoints -> {400, 400},
MeshFunctions -> {#4 &}, Mesh -> {{0, wrap}},
MeshStyle -> Directive@{Thick, Red}, Boxed -> False,
Axes -> False, Method -> {"BoundaryOffset" -> False}
],
ParametricPlot3D[
{bcx, bcy, bcz} // Evaluate, {t, Pi, 2 Pi}, {\[Phi]r, 0, wrap},
MaxRecursion -> 0, PlotPoints -> {400, 400},
MeshFunctions -> {#4 &}, Mesh -> {{0, wrap}},
MeshStyle -> Directive@{Thick, Red}, Boxed -> False,
Axes -> False, Method -> {"BoundaryOffset" -> False}
],
ParametricPlot3D[
{cdx, cdy, cdz} // Evaluate, {t, 0, Pi}, {\[Phi]r, 0, wrap},
MaxRecursion -> 0, PlotPoints -> {400, 400},
MeshFunctions -> {#4 &}, Mesh -> {{0, wrap}},
MeshStyle -> Directive@{Thick, Red}, Boxed -> False,
Axes -> False, Method -> {"BoundaryOffset" -> False}
],
ParametricPlot3D[
{dex, dey, dez} // Evaluate, {t, Pi, 2 Pi}, {\[Phi]r, 0, wrap},
MaxRecursion -> 0, PlotPoints -> {400, 400},
MeshFunctions -> {#4 &}, Mesh -> {{0, wrap}},
MeshStyle -> Directive@{Thick, Red}, Boxed -> False,
Axes -> False, Method -> {"BoundaryOffset" -> False}
]
};
(* Extract Points and Polygons *)
pts = Cases[plot, GraphicsComplex[pts_, rest__] :> pts, -1][[1]];
polys = Cases[plot, GraphicsGroup[data_] :> data, -1][[1, 1]];
lines = Cases[plot, _Line, -1];
(* Create MeshRegion *)
reg = BoundaryMeshRegion[pts,
Polygon[Join[lines /. Line[pts_] :> Rest@pts, First@First@polys]]]
My Questions:
- How can I address the non-coplanar vertex issue? Is there a reliable way to preprocess the data to ensure coplanarity?
- Does the fact that the
tranges differ between parametric equations affect how the points or polygons should be combined? - Should I use an alternative method (e.g.,
MeshRegion) instead ofBoundaryMeshRegion? - Is there a way to triangulate the extracted surface data to resolve this issue?
Thank you for your time and insights—any help would be greatly appreciated!
wrap? please post the full code about the 2D and 3D graphics. $\endgroup$