How do engineers and designers 3D print real-world functional containers like coffee mugs, flower vases, cereal bowls, or solid chess pieces?

Instead of tracing a 1D path in space, we can use one of the most powerful techniques in 3D geometry: a surface of revolution (also known as lathing).

Imagine drawing a 2D cross-section of a cup wall on paper—graphing its radius $r$ at every vertical height $z$. If you take that 2D curve and spin it $360^\circ$ around the central vertical axis, it sweeps out a symmetrical, solid 3D object!

Our function revolve3d(r, z, thickness) turns your 2D profile into a 3D-printable model:

• Hollow Containers (Cups, Vases, Bowls): When you specify a wall thickness (like thickness = 2.2 mm), revolve3d automatically generates an outer wall, a smooth top rim, a hollow interior cavity, and a flat, solid base floor that sits securely on the print bed!
  
  
• Solid Objects (Chess Pawns, Spinning Tops, Domes): If you set thickness = 0 (or omit it), revolve3d revolves your profile into a 100% solid body of revolution.

Here is how to call revolve3d:

Move the mouse over a dotted box for more information.

• Watertight Solid Base: For hollow vessels (thickness > 0), our 3D engine automatically closes the bottom with a solid base of thickness $t$ resting flat on the printer bed at $Z = 0$, connects the outer wall to the inner cavity with a smooth rim, and ensures the entire model is 100% watertight (manifold).

• Mathematical Profiling: You can create any shape you like by calculating $r$ as a function of height $z$ in a simple loop: straight tapers ($r = a + b \cdot z$), smooth power curves ($r = a + b \cdot z^p$), or wavy fluted bodies ($r = a + b \cdot \sin(c \cdot z)$).

• Interactive 3D Viewer: Click and drag with your mouse to rotate the model, use your scroll wheel to zoom, and right-drag to pan. Toggle Spin to watch your cup rotate, or Wire to inspect its triangular polygon mesh.

• Ready for Your Slicer: Click the Download STL button below the output to save your .stl file, which you can load directly into Cura, PrusaSlicer, Bambu Studio, or Tinkercad to slice and print!