# Create region bounded by parametric curves

How can I define a region that is bounded by a set of parametric curves?

E.g. three parametric curves - which are cubic Bezier curves - define the boundary of a shape:

cubicbez[a_, b_, c_, d_, t_] := (1 - t)^3*a + 3*(1 - t)^2*t*b +
3*(1 - t)*t^2*c + t^3*d

bezierRegion[{a_, b_, c_, d_}] :=
ParametricRegion[cubicbez[a, b, c, d, t], {{t, 0, 1}}]

regions = {bezierRegion[{{5, 5}, {10, 13}, {18, 4}, {20, 30}}],
bezierRegion[{{20, 30}, {18, 40}, {17, 40}, {15, 35}}],
bezierRegion[{{15, 35}, {25, 25}, {5, 20}, {5, 5}}]};

Show[Region /@ regions, Frame -> True] Is there a way to get the inside of that shape as a Mathematica region?

Edit

• Another way is just use
Show[Region /@ regions, Frame -> False] // BoundaryDiscretizeGraphics

BezierCurve[{{5, 5}, {10, 13}, {18, 4}, {20, 30}, {18, 40}, {17,
40}, {15, 35}, {25, 25}, {5, 20}, {5,
5}}] // BoundaryDiscretizeGraphics

fig = Graphics@
FilledCurve@
BezierCurve[{{5, 5}, {10, 13}, {18, 4}, {20, 30}, {18, 40}, {17,
40}, {15, 35}, {25, 25}, {5, 20}, {5, 5}}];
reg = BoundaryDiscretizeGraphics[fig] An alternative approach: You can use the coordinate data to define BezierFunctions and use them to construct a polygon which can be used as is as a region or discretized using BoundaryDiscretizeRegion:

bzfuncs = {BezierFunction[{{5, 5}, {10, 13}, {18, 4}, {20, 30}}],
BezierFunction[{{20, 30}, {18, 40}, {17, 40}, {15, 35}}],
BezierFunction[{{15, 35}, {25, 25}, {5, 20}, {5, 5}}]};

poly = Polygon[Join @@ (Map[#]@Subdivide & /@ bzfuncs)];

RegionQ @ poly

True

Graphics[{EdgeForm[Gray], FaceForm[LightBlue], poly}] BoundaryDiscretizeRegion @ poly 