Brachistochrone comparison

One release, three paths

Compare a frictionless point bead released from rest at A=(0,0) on each of three fixed, guided tracks. The coordinate y points downward. B=(πH/2,H) is the same endpoint for all tracks. Gravity is constant, g=9.81 m/s². Conservation of energy gives v=√(2gy); mass cancels. This is a point-particle model with a two-sided guide, not a rolling ball or a contact-force calculation for a one-sided ramp.

Cycloid and straight line

The half-cycloid is x=(H/2)(θ−sinθ), y=(H/2)(1−cosθ), with 0≤θ≤π. Its travel time is T_c=π√(H/(2g)). The angle grows linearly with time, even though the bead starts with zero speed. The cycloid is the brachistochrone for these endpoints.

Let X=πH/2 and D=√(X²+H²). The straight line has travel time T_s=D√(2/(gH)); its position fraction is (t/T_s)². The displayed extra time is 100(T/T_c−1), with the cycloid as the reference.

A genuine circle

A circle can join A and B only if R≥D/2. No input is silently enlarged. Define h=√(R²−D²/4), C_x=X/2+hH/D, C_y=H/2−hX/D, and α=2atan2(D/2,h). The selected smaller arc around C is: x=C_x(1−cosu)−C_ysinu and y=C_xsinu+C_y(1−cosu), for 0≤u≤α. It starts downward and its true radius is R.

This arc may first move left of A or pass below B and rise again. These are permitted track shapes; the diagram includes the complete arc at equal x/y scale. Its travel time is the integral of R du/√(2gy(u)). The substitution u=αq² removes the release singularity. Controlled quadrature and a monotone time interpolation provide the circular bead position without an artificial starting speed.

Controls and interpretation

H accepts continuous values from 0.8 to 4 m. R must be positive, at most 6 m, and at least half the distance AB. Curves and arrival times update immediately. Run starts the race; Step advances the common clock by 0.04 s. Reset returns all beads to A and keeps the inputs. Editing stops the old race. Invalid or unfinished entries remain saved but show no stale result. Reloading or returning to the tool prepares a new comparison at rest.

Markers remain at B after arrival; the displayed speed is the arrival speed, without a stop or impact model. Time fraction is elapsed time divided by each track’s travel time, not distance travelled. Hiding the tool pauses playback; returning does not catch up elapsed wall time. A long display frame advances at most 0.25 s of model time. Replay starts the same race again.

Sources

Rutgers: the brachistochrone problem uses a normalized gravity convention; this tool uses the explicit physical g above. See also Liberzon: the brachistochrone.