Saddle (elliptic)
z = (x²/9 − y²/4) · (1 + 0.25 · sin(t)) +(x²/9 + y²/4 < 1 ? 0 : 0/0)
Open in the app The dials and keys named below are the app's.
What it draws
The height is x²/9 − y²/4: a rise along the x axis, a fall along the y axis, the plainest saddle there is. The two divisors set how fast each happens. The sheet reaches +1 at x = ±3 and −1 at y = ±2.
The second half of the formula is not a shape but a cut. Where x²/9 + y²/4 is under 1 it adds nothing. Everywhere else it evaluates 0/0, which is undefined, so no height is drawn there and the sheet is trimmed to the ellipse with semi-axes 3 and 2. The factor 1 + 0.25·sin t runs between 0.75 and 1.25 over 2π ≈ 6.28 seconds at Speed 1. It never reaches zero, so the saddle deepens and eases but never flattens or turns over.
The rim
Walk the boundary as x = 3·cos u, y = 2·sin u, and the height along it is cos²u − sin²u = cos(2u). That is up at the two ends of the long axis and down at the two ends of the short one, twice around in one circuit. That is the potato crisp, and it is why a cut of this kind is called a saddle at all.
Inside the rim the sheet is doubly ruled. Factor it: x²/9 − y²/4 = (x/3 − y/2)·(x/3 + y/2). Hold either bracket fixed and the height varies in a straight line, so through every point of the surface run two straight lines that lie in it, and the curved sheet can be built out of straight members. Félix Candela made a career of the form. His thin concrete shells in Mexico in the 1950s, among them the Los Manantiales restaurant at Xochimilco of 1958, are hyperbolic paraboloids poured on formwork of straight boards.
Try
- Front in the deck, looking along y: the rise alone, a shallow parabola opening upward.
- Drop the cut, (x²/9 − y²/4)·(1 + 0.25·sin t), and the sheet fills the square corner to corner.
- Change the cut to a disc, (x²/9 − y²/4)·(1 + 0.25·sin t) + (x² + y² < 4 ? 0 : 0/0): a round crisp of radius 2.
- Let the breathing reach zero, (x²/9 − y²/4)·(1 + sin t) + (x²/9 + y²/4 < 1 ? 0 : 0/0): once a cycle the whole thing lies flat.
- Top in the deck shows the cut for what it is, an ellipse 6 across and 4 deep.