Chop Sticks, Compounded Radii, Fountain Pens and Fuckwits

What is the curve (track) called, that an identical ball can roll down, compared to a straight track, with the same distance from the start to the end point, with the same drop, but the curved track is faster?

The curve you are looking for is called the Brachistochrone curve. The name is derived from the Greek words brachistos (shortest) and chronos (time). It is the path of fastest descent for a body moving under the influence of gravity, between two points that are not vertically aligned. Key Characteristics: The Shape: The curve is an inverted cycloid—the path traced by a point on the rim of a rolling wheel. The Paradox: Even though the straight-line path (a linear slope) is the shortest distance, the Brachistochrone is the fastest time. The Reason: On a curved track, the ball drops more steeply at the beginning. This converts potential energy into kinetic energy (speed) much faster than the straight track. Even though the ball travels a longer distance on the curve, the higher average velocity results in a shorter travel time.

https://en.wikipedia.org/wiki/....Brachistochrone_curv

So not only are my chopsticks made to a compounding radii, they are also faster.

I was actually thinking of this - https://en.wikipedia.org/wiki/Catenary