The Brachistochrone curve (from Greek brachistos, “shortest,” and chronos, “time”) is the path between two points that allows an object to descend under gravity in the shortest possible time. First posed as a challenge by Johann Bernoulli in 1696, this problem spurred breakthroughs in mathematics and physics. The solution—unexpectedly a cycloid rather than a straight line or circular arc—reveals how optimal paths often defy intuition. Bernoulli, Newton, Leibniz, and others derived the curve using early calculus techniques, marking a pivotal moment in the development of calculus of variations, a field dedicated to optimizing functionals like time or energy.
The brachistochrone’s parametric equations are:
x=r(θ−sinθ)
y=r(1−cosθ)
Where ( r ) defines the cycloid’s scale, and ( theta θ) is the rolling angle of the generating circle.
It’s a concave curve rather than a straight line, and it minimizes the time it takes for an object to travel from point A to point B under the influence of gravity alone. This shape allows the ball to gain speed quickly and then maintain that speed over a longer distance, resulting in the shortest travel time.

In Figure 1, the black sloped line is a stright line connecting point A at top left to the point B at bottom right. The blue curve line is a brachistochrone path. If two balls of the same mass (black ball along the straight line, and red ball along the brachistochrone curve) were released at the same time at point A, the red ball would arrive at point B first. In fact, quicker than any other curve or path under the sole influence of gravity. The time it would take for the red ball to reach destination point B will be shorter than the time it would take the black ball.
But How?
Though the curve’s path may be longer than a straight line, its steeper initial drop allows the object to accelerate faster, balancing increased speed against total distance. This optimization leverages the principle of least action, where the object naturally follows the path minimizing travel time. Remarkably, the brachistochrone is also a tautochrone—a curve where objects released from any point reach the bottom simultaneously.
That’s right! That means if several balls were to let go from different locations along the curve, all of them will reach the destination point at the same time! This seems impossible at first, but it makes sense when you realize the fact that the object closest to the destination has less distance to travel but also has less kinetic energy, therefore, speed than another object let go from farther point, which picks up enough kientic energy (therefore, speed) to make up for its increased distance from the destination.

Figure 2 is a Python animation that shows the relationship between the rolling circle and resulting cycloid. The red ball starts at height 2r (potential energy) and accelerates down the concave curve. Speed increases as the ball approaches lowest point. The brachistochrone curve cycloid is shown with a blue line. The ball follows the curve. A radius line (green dash) showing the connection between the circle and ball.
Cycloid parametric equations:
x=r(θ−sinθ)
y=r(1−cosθ)
Here’s another Python animation of a ball rolling down another brachistochrone curve (Figure 3).

Remember, at the starting point, the ball has maximum potential energy due to its height (potential energy in this case only depends on height*).
As the ball rolls down the curve, its potential energy is converted into kinetic energy. Kinetic energy gives it the horizontal propulsion or speed more than making up the longer path along the curve than a straight line.
Cool! But What’s its Use?
Beyond its mathematical elegance, the brachistochrone has practical applications. Engineers use its principles in roller coaster design to optimize thrill and safety, while physicists apply variational methods derived from this problem to fields like optics (Fermat’s principle) and quantum mechanics. It remains a cornerstone example of how abstract mathematics can unlock nature’s hidden efficiencies—a timeless lesson in the interplay between theory and reality.
Fun Fact
The concept of optimizing paths and trajectories is indeed applicable to rocket launches. So, by turning the brachistochrone curve upside down, at least in theory, rockets could achieve the most efficient ascent. However, its direct application to rocket launches isn’t practical due to the different forces and constraints involved in space travel. Instead, rockets utilize a Gravity Turn. That trajectory is used by rockets and satellites to efficiently reach orbit (including SpaceX, Blue Origin, Virgin Galactic).
The animation scripts were written in Python using matplotlib library for plotting and animation, and numpy library for math calculations. If you’re interested in the code for both of the animations, contact me.
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