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Curve studied by Dürer in 1525 and Rømer in 1674.
From the Greek trokhos: wheel.

Cartesian parametrization:  (where d is the distance from M to the centre of the circle).

The notion of trochoid refers to the curve described by a point linked to a disk with radius R rolling without slipping on a line (D); in other words, in it a roulette of a movement of a plane over a fixed plane the base of which is a line and the rolling curve of which is a circle.

For d < R, the curve is also called curtate cycloid and looks like a sinusoid, and it is one if the term  is neglected in x.

For d = R, we get the cycloid.

For d > R, the curve is also called prolate cycloid and can assume various shapes, with more and more double points as d increases.
The fact that the prolate cycloid has a loop is at the origin of the following paradox:
Show that, in a train, there always is a portion of mass that moves in the opposite direction of the train.
Answer: the bottom of the small edge of the wheels.
The bottom of the wheel of the train moves in the opposite direction of the train....
The trochoids can also be defined as the trajectories of a movement composed of a uniform linear motion and a circular motion, with complex parametrization:  ( ); they are cycloids if , curtate cycloids if , prolate cycloids if   (take  and d = r).
The ratio "translation speed over rotation speed" then entirely defines the trochoid, up to similarity, the elongation increasing with .

Opposite, two remarkable cases where the prolate cycloid is tangent to itself.
The ratios  are respectively 4.6.... and 7.8....

Concrete examples:

- you are walking regularly along a blackboard holding a chalk stick in your hand with a regular circular motion: you are tracing a trochoid, in general, prolate because you are moving slower than your hand is turning.

- reflectors on the spokes of the wheels of your bike describe curtate cycloids.

- the pedal of your bike describes, when you push down on it, a trochoid with ratio  where  is the radius of the chain wheel (dented wheel on the drive),  is the radius of the cogwheel (dented wheel on the back wheel),  is the "ratio", R is the radius of the back wheel, and r the length of the arm of the pedal. Since this number is always < 1, it is a curtate cycloid, the flattening increasing with the ratio.

- the impeller of a boat describes a curtate trochoid because the grip of the blades in the water is not perfect (the speed of the end of the blade is greater than the speed of the boat).


The projections of a circular helix on a fixed plane give all the possible shapes of trochoids, potentially scaled; in other words, the trochoids are, up to scaling, the views in parallel perspective, or the shades, of a spring (Montucla-Guillery theorem).


See also the epi- and hypotrochoid, the mascot curve , the Duporcq curve, and the minimal surface of Catalan.
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© Robert FERRÉOL 2017