Viviani's Curve
(1 + cos(u), sin(u), 2 · sin(u/2))
Open in the app The dials and keys named below are the app's.
What it draws
None of the three expressions uses t, so the figure is fixed and playing changes nothing on screen. The second expression, sin(u), is the height, which stays within ±1. Two facts about the point pin it down, and both hold exactly. Its distance from the origin is always 2, so the curve lies on a sphere of radius 2.
The first two expressions alone satisfy (x − 1)² + y² = 1, a circle of radius 1 through the origin. The curve therefore lies as well on the cylinder standing on that circle, whose axis runs into depth. Viviani's curve is exactly where those two surfaces meet: a cylinder half as wide as the sphere, touching it from inside at the single point (2, 0, 0). The half angle u/2 in the third expression is why Turns of u opens at 2. After u = 2π the point is back at that touching point having drawn only one of the two lobes, and the second turn draws the other.
The three shadows
Each flat view of the curve is a different classical figure. Looking down the depth axis, along the cylinder, the two lobes fall on top of each other and the shadow is the circle of radius 1. Looking down the vertical, the shadow is the parabola x = 2 − z²/2. Looking along the remaining horizontal axis it is a figure eight, y² = z²·(1 − z²/4), the lemniscate of Gerono.
History
Vincenzo Viviani, the last pupil of Galileo, published the curve in 1692 as a challenge to the mathematicians of Europe. Cut four such windows from a hemispherical dome, he asked, so that the vault left standing has an area that can be squared exactly, with no π in it. Answers came quickly, Leibniz and Jakob Bernoulli among them, and the shape has carried Viviani's name since.
Try
- Drop Turns of u to 1: only half the curve is drawn, an open arc that stops at the point where the cylinder touches the sphere.
- Side: the shadow is the figure eight, the lemniscate of Gerono.
- Top: the shadow is the parabola, the flattest of the three views.
- Turn on trace: with nothing else moving, the pen is the only motion, and it draws one lobe before the other.
- Change the third expression to 2·sin(u/2)·cos(t). The curve now flattens onto its circle and swells back out every 2π ≈ 6.28 seconds, and it is Viviani's curve only at the two extremes.