# How to generate a 2D region from a 1D region boundary?

For example:

L1 = ParametricRegion[{2 \[Theta] Cos[\[Theta]],
2 \[Theta] Sin[\[Theta]]}, {{\[Theta], 0, 2 Pi}}];

L2 = Line[{{0, 0}, {2*2 Pi, 0}}];

L=Region[RegionUnion[L1, L2]]


How to generate a 2D Region R from the 1D Region Boundary L?

The following code can generate such Region. But is there a simpler and more common way?

Region[ParametricRegion[{2 \[Theta]*t* Cos[\[Theta]],
2 \[Theta]*t* Sin[\[Theta]]}, {{\[Theta], 0, 2 \[Pi]}, {t, 0, 1}}]]


How to obtain a 2D Region without knowing the boundary function expression and only knowing the 1D boundary line region?

l1 = ParametricPlot[{2 θ Cos[θ],
2 θ Sin[θ]}, {θ, 0, 2 Pi}];
l2 = Graphics[Line[{{0, 0}, {2*2 Pi, 0}}]];
Show[l1, l2] // BoundaryDiscretizeGraphics


L1 = ParametricRegion[{2 θ Cos[θ],
2 θ Sin[θ]}, {{θ, 0, 2 Pi}}];
L2 = Line[{{0, 0}, {2*2 Pi, 0}}];
Show[Region /@ {L1, L2}] // BoundaryDiscretizeGraphics

L1 = ParametricRegion[{2 θ Cos[θ],
2 θ Sin[θ]}, {{θ, 0, 2 Pi}}];

L2 = Line[{{0, 0}, {2*2 Pi, 0}}];

L = Region[RegionUnion[L1, L2]]
Show[L] // BoundaryDiscretizeGraphics

• Thank you very much!@cvgmt Feb 2, 2022 at 9:20
• R = Show[Region /@ {L1, L2}] // BoundaryDiscretizeGraphics Feb 2, 2022 at 9:21