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.
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In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. On a basic level, it is the path traced by a curve while rolling on another curve without slipping.
The Tusi couple (also known as Tusi's mechanism) is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and forth in linear motion along a diameter of the larger circle. The Tusi couple is a two-cusped hypocycloid.
The couple was first proposed by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in his 1247 Tahrir al-Majisti (Commentary on the Almagest) as a solution for the latitudinal motion of the inferior planets and later used extensively as a substitute for the equant introduced over a thousand years earlier in Ptolemy's Almagest.
Gerolamo Cardano (Italian: [dʒeˈrɔːlamo karˈdaːno]; also Girolamo or Geronimo; French: Jérôme Cardan; Latin: Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath whose interests and proficiencies ranged through those of mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, music theorist, writer, and gambler. He became one of the most influential mathematicians of the Renaissance and one of the key figures in the foundation of probability; he introduced the binomial coefficients and the binomial theorem in the Western world. He wrote more than 200 works on science.
Cardano partially invented and described several mechanical devices including the combination lock, the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate freely, and the Cardan shaft with universal joints, which allows the transmission of rotary motion at various angles and is used in vehicles to this day. He made significant contributions to hypocycloids - published in De proportionibus, in 1570. The generating circles of these hypocycloids, later named "Cardano circles" or "cardanic circles", were used for the construction of the first high-speed printing presses.