Alain's Curve

What it draws

The formula is a curve of the plane written in polar form. Put x = r·cos(θ) and y = r·sin(θ) into (x² − y²)² = a²x² − b²y² and it becomes r²·cos(2θ)² = a²·cos(θ)² − b²·sin(θ)², which is exactly what is on the plate. Here b = 1.2, since 1.44 = 1.2², and a = 0.7 + 0.3·sin(t), breathing between 0.4 and 1.0 every 2π ≈ 6.28 seconds at Speed 1.

The square root is what shapes the picture. It is real only where a²·cos(θ)² ≥ b²·sin(θ)², that is within arctan(a/b) of the horizontal. That wedge is 18.4° wide when a is at its smallest and 39.8° at its largest. Everywhere else the formula has no value and a gap is left, so two thirds of the angles are empty at rest.

The two loops

Straight ahead, at θ = 0, the formula gives r = a. At the edge of the wedge the quantity under the root falls to zero, so r returns to the origin. Each lobe is a closed loop that leaves the centre, reaches out and comes back, and the two of them cross at the centre in a figure of eight. As a grows the wedge widens and both loops swell.

The denominator sets the limit. cos(2θ) vanishes at 45°, and as long as a stays below b the wedge stops short of it and r stays finite. Let a reach b and the two meet: the loops tear open along the diagonals and run off to infinity. That is why the breathing is set to keep a under 1.2. The curve is a rational quartic, and it is catalogued as Alain's curve after Alain Juhel, who proposed its study.

Try

• Hold the breath with √(0.49·cos(θ)² − 1.44·sin(θ)²) / cos(2θ), which is a fixed at 0.7.
• Narrow the wedge by raising b: √((0.7 + 0.3·sin(t))²·cos(θ)² − 4·sin(θ)²) / cos(2θ) puts b = 2, and the lobes thin to slivers.
• Bring a up to meet b with √((0.7 + 0.5·sin(t))²·cos(θ)² − 1.44·sin(θ)²) / cos(2θ): near the top of each breath the loops burst open and the scale-to-fit gives up on them.
• Top in the deck: the figure of eight seen flat, with the empty wedges above and below.

Read more

• Alain's Curve
• Quartic plane curve
• Lemniscate
• Polar coordinate system