🫧 Parametric Surfaces — Torus, Klein Bottle & Möbius

🫧 What It Demonstrates

A parametric surface is defined by mapping two parameters (u, v) to 3D coordinates: x = f(u,v), y = g(u,v), z = h(u,v). This simulation renders five classic surfaces as interactive wireframe meshes:

• Torus — the doughnut shape with major radius R and minor radius r.
• Sphere — the unit sphere via latitude–longitude parametrisation.
• Klein Bottle — a non-orientable surface that has no inside or outside (immersed in 3D with self-intersection).
• Möbius Strip — a surface with only one side and one edge.
• Hyperboloid — a ruled surface of revolution, the shape of cooling towers and certain skyscrapers.

How to Use

• Choose a Surface from the dropdown.
• Adjust U/V segments to change mesh density (resolution).
• Rotation speed controls the auto-rotation of the object.
• Toggle Wireframe and Faces for different rendering.
• Drag on the canvas to manually rotate the view.

Did You Know?

The Klein bottle was first described by Felix Klein in 1882. In 4D, it has no self-intersections — only when immersed in 3D does it appear to pass through itself. The Möbius strip, discovered by August Möbius and Johann Listing in 1858, inspired M.C. Escher's famous "Möbius Strip II" showing ants marching in an endless loop. Cooling-tower hyperboloids are chosen for their structural strength — every point on the surface sits on a straight line, making construction with straight steel beams possible.