Kuen Surface
Shape · (x, y, z) = f(u, v, t)
(2 · cosh(v − 5 · π/4) · (cos(u) +(u − 2 · π) · sin(u))/(cosh(v − 5 · π/4)² +(u − 2 · π)²), (v − 5 · π/4) − 2 · sinh(v − 5 · π/4) · cosh(v − 5 · π/4)/(cosh(v − 5 · π/4)² +(u − 2 · π)²), 2 · cosh(v − 5 · π/4) · (sin(u) −(u − 2 · π) · cos(u))/(cosh(v − 5 · π/4)² +(u − 2 · π)²))
Open in the app The dials and keys named below are the app's.
What it draws
Theodor Kuen's surface of 1884, written with U = u − 2π and V = v − 5π/4 so that the opening dials cover |U| < 6.28 and |V| < 3.93. All three expressions share the denominator D = cosh²V + U², which never drops below 1, so the surface has no holes.
The first and last expressions place a point around the vertical axis, at distance 2 cosh V · √(1 + U²)/D. That is 2 at U = V = 0, the widest the surface gets, and it shrinks toward the axis as either grows. The middle expression is the height, V − 2 sinh V cosh V/D, between −2.04 and 2.04 here.
Curvature −1 everywhere
At every point the surface bends like a saddle, and by the same amount: its Gaussian curvature is exactly −1, the product of its two principal curvatures. The pseudosphere, a horn turned about an axis, has the same curvature.
Kuen's surface can be made from the pseudosphere by a Bäcklund transformation, which takes one surface of curvature −1 to another.
Why it has edges
David Hilbert proved in 1901 that no complete smooth surface of constant negative curvature fits in ordinary space. Every such surface must break somewhere. Here it breaks along the sharp creases the solid shows, where the surface folds back on itself and the saddle can no longer be smooth.
The substitution U → −U leaves the first two expressions alone and flips the sign of the third, so the surface is mirror symmetric across a vertical plane.
Try
- Turn Mesh down from its top stop to see the grid of u and v lines it is built on.
- Set Turns of u to 1. Only U < 0 is drawn: one mirror half of the surface.
- Set Span of v (×π) to 1. Only V from −3.93 to −0.79 is left, and the surface ends in an open edge at the top.