Is it possible in Opengl to specify bezier curves as the edges of figures (rotated, textured, shaded, gradient, etc. - like any other polygon)?
No.
Legacy OpenGL has evaluators (glEvalMesh etc). Modern OpenGL (4.0+) has tessellation shaders. But neither of these allow you to directly fill a region bounded by Bézier curves. You either have to tessellate the interior into triangles, or tessellate the boundary into line segments then fill it as a triangle fan (using stencilling if it isn’t convex).
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