Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Hypocycloid
Summary
In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid created
by rolling a circle on a line.
The 2-cusped hypocycloid called Tusi couple was first described by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in Tahrir al-Majisti (Commentary on the Almagest). German painter and German Renaissance theorist Albrecht Dürer described epitrochoids in 1525, and later Roemer and Bernoulli concentrated on some specific hypocycloids, like the astroid, in 1674 and 1691, respectively.
If the smaller circle has radius r, and the larger circle has radius R = kr, then the
parametric equations for the curve can be given by either:
or:
If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not
differentiable). Specially for k = 2 the curve is a straight line and the circles are called Cardano circles. Girolamo Cardano was the first to describe these hypocycloids and their applications to high-speed printing.
If k is a rational number, say k = p/q expressed in simplest terms, then the curve has p cusps.
If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R − 2r.
Each hypocycloid (for any value of r) is a brachistochrone for the gravitational potential inside a homogeneous sphere of radius R.
The area enclosed by a hypocycloid is given by:
The arc length of a hypocycloid is given by:
Image:Hypocycloid-3.svg| k=3 → a [[deltoid curve|deltoid]]
Image:Hypocycloid-4.svg| k=4 → an [[astroid]]
Image:Hypocycloid-5.svg| k=5 → a pentoid
Image:Hypocycloid-6.svg| k=6 → an exoid
Image:Hypocycloid-2-1.svg| k=2.1 = 21/10
Image:Hypocycloid-3-8.svg| k=3.8 = 19/5
Image:Hypocycloid-5-5.svg| k=5.5 = 11/2
Image:Hypocycloid-7-2.svg| k=7.2 = 36/5
The hypocycloid is a special kind of hypotrochoid, which is a particular kind of roulette.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related lectures (1)
Elastic Collisions and Rotational Motion
Explores elastic collisions, rolling cylinders, and ballistic pendulums.
Related concepts (8)
Hypotrochoid
In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle. The parametric equations for a hypotrochoid are: where θ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because θ is not the polar angle). When measured in radian, θ takes values from 0 to (where LCM is least common multiple).
Cycloid
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve. The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve).
Epitrochoid
In geometry, an epitrochoid (ɛpᵻˈtrɒkɔɪd or ɛpᵻˈtroʊkɔɪd) is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle. The parametric equations for an epitrochoid are The parameter θ is geometrically the polar angle of the center of the exterior circle. (However, θ is not the polar angle of the point on the epitrochoid.) Special cases include the limaçon with R = r and the epicycloid with d = r.