Misplaced Pages

#52947

36-531: 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

72-406: A curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps. Consider a smooth real-valued function of two variables , say f ( x , y ) where x and y are real numbers . So f is a function from the plane to the line. The space of all such smooth functions is acted upon by

108-524: A diameter of the larger circle. The Tusi couple is a 2-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 . The translation of

144-442: A final non-divisibility condition (giving type exactly A 4 ). To see where these extra divisibility conditions come from, assume that f has a degenerate quadratic part L and that L divides the cubic terms. It follows that the third order taylor series of f is given by L 2 ± L Q , {\displaystyle L^{2}\pm LQ,} where Q is quadratic in x and y . We can complete

180-409: A hypocycloid is given by: s = 8 ( k − 1 ) k R = 8 ( k − 1 ) r {\displaystyle s={\frac {8(k-1)}{k}}R=8(k-1)r} The hypocycloid is a special kind of hypotrochoid , which is a particular kind of roulette . A hypocycloid with three cusps is known as a deltoid . A hypocycloid curve with four cusps

216-654: A hypocycloid with pole at the center of the hypocycloid is a rose curve . The isoptic of a hypocycloid is a hypocycloid. Curves similar to hypocycloids can be drawn with the Spirograph toy. Specifically, the Spirograph can draw hypotrochoids and epitrochoids . The Pittsburgh Steelers ' logo, which is based on the Steelmark , includes three astroids (hypocycloids of four cusps ). In his weekly NFL.com column "Tuesday Morning Quarterback," Gregg Easterbrook often refers to

252-585: Is quartic (order four) in x 1 and y 1 . The divisibility condition for type A ≥4 is that x 1 divides P 1 . If x 1 does not divide P 1 then we have type exactly A 3 (the zero-level-set here is a tacnode ). If x 1 divides P 1 we complete the square on x 1 2 + P 1 {\displaystyle x_{1}^{2}+P_{1}} and change coordinates so that we have x 2 2 + P 2 {\displaystyle x_{2}^{2}+P_{2}} where P 2

288-461: Is quintic (order five) in x 2 and y 2 . If x 2 does not divide P 2 then we have exactly type A 4 , i.e. the zero-level-set will be a rhamphoid cusp. Cusps appear naturally when projecting into a plane a smooth curve in three-dimensional Euclidean space . In general, such a projection is a curve whose singularities are self-crossing points and ordinary cusps. Self-crossing points appear when two different points of

324-562: Is a power series of order k (degree of the nonzero term of the lowest degree) larger than m . The number m is sometimes called the order or the multiplicity of the cusp, and is equal to the degree of the nonzero part of lowest degree of F . In some contexts, the definition of a cusp is restricted to the case of cusps of order two—that is, the case where m = 2 . The definitions for plane curves and implicitly-defined curves have been generalized by René Thom and Vladimir Arnold to curves defined by differentiable functions :

360-601: Is a non-negative integer. A function f is said to be of type ⁠ A k ± {\displaystyle A_{k}^{\pm }} ⁠ if it lies in the orbit of x 2 ± y k + 1 , {\displaystyle x^{2}\pm y^{k+1},} i.e. there exists a diffeomorphic change of coordinate in source and target which takes f into one of these forms. These simple forms x 2 ± y k + 1 {\displaystyle x^{2}\pm y^{k+1}} are said to give normal forms for

396-420: 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 Tusi Couple. Nasir al-Din al-Tusi 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

SECTION 10

#1732869692053

432-557: Is given in the figure. A cusp is thus a type of singular point of a curve . For a plane curve defined by an analytic , parametric equation a cusp is a point where both derivatives of f and g are zero, and the directional derivative , in the direction of the tangent , changes sign (the direction of the tangent is the direction of the slope lim ( g ′ ( t ) / f ′ ( t ) ) {\displaystyle \lim(g'(t)/f'(t))} ). Cusps are local singularities in

468-449: Is known as an astroid . The hypocycloid with two "cusps" is a degenerate but still very interesting case, known as the Tusi couple . Any hypocycloid with an integral value of k , and thus k cusps, can move snugly inside another hypocycloid with k +1 cusps, such that the points of the smaller hypocycloid will always be in contact with the larger. This motion looks like 'rolling', though it

504-491: Is not technically rolling in the sense of classical mechanics, since it involves slipping. Hypocycloid shapes can be related to special unitary groups , denoted SU( k ), which consist of k × k unitary matrices with determinant 1. For example, the allowed values of the sum of diagonal entries for a matrix in SU(3), are precisely the points in the complex plane lying inside a hypocycloid of three cusps (a deltoid). Likewise, summing

540-487: The Taylor expansion of F are a power of a linear polynomial ; however, not all singular points that have this property are cusps. The theory of Puiseux series implies that, if F is an analytic function (for example a polynomial ), a linear change of coordinates allows the curve to be parametrized , in a neighborhood of the cusp, as where a is a real number , m is a positive even integer , and S ( t )

576-454: The group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target . This action splits the whole function space up into equivalence classes , i.e. orbits of the group action . One such family of equivalence classes is denoted by ⁠ A k ± , {\displaystyle A_{k}^{\pm },} ⁠ where k

612-532: The Steelers as the Hypocycloids. Chilean soccer team CD Huachipato based their crest on the Steelers' logo, and as such features hypocycloids. The first Drew Carey season of The Price Is Right ' s set features astroids on the three main doors, giant price tag, and the turntable area. The astroids on the doors and turntable were removed when the show switched to high definition broadcasts starting in 2008, and only

648-873: The Tusi couple can be interpreted as a rolling curve where the rotation of the inner circle satisfies a no-slip condition as its tangent point moves along the fixed outer circle. The term "Tusi couple" is a modern one, coined by Edward Stewart Kennedy in 1966. It is one of several late Islamic astronomical devices bearing a striking similarity to models in Nicolaus Copernicus 's De revolutionibus , including his Mercury model and his theory of trepidation . Historians suspect that Copernicus or another European author had access to an Arabic astronomical text, but an exact chain of transmission has not yet been identified, The 16th century scientist and traveler Guillaume Postel has been suggested as one possible facilitator. Since

684-435: The Tusi couple was developed within an astronomical context, later mathematicians and engineers developed similar versions of what came to be called hypocycloid straight-line mechanisms. The mathematician Gerolamo Cardano designed a system known as Cardan's movement (also known as a Cardan gear ). Nineteenth-century engineers James White, Matthew Murray , as well as later designers, developed practical applications of

720-717: The Tusi-couple are still extant in Italy. Another possibility is that he encountered the manuscript of the "Straightening of the Curves" (Sefer Meyasher 'Aqov) while studying in Italy. There are other sources for this mathematical model for converting circular motions to reciprocating linear motion. It is found in Proclus's Commentary on the First Book of Euclid and the concept was known in Paris by

756-628: The Tusi-couple was used by Copernicus in his reformulation of mathematical astronomy, there is a growing consensus that he became aware of this idea in some way. It has been suggested that the idea of the Tusi couple may have arrived in Europe leaving few manuscript traces, since it could have occurred without the translation of any Arabic text into Latin. One possible route of transmission may have been through Byzantine science ; Gregory Chioniades translated some of al-Tusi's works from Arabic into Byzantine Greek . Several Byzantine Greek manuscripts containing

SECTION 20

#1732869692053

792-401: The copy of Tusi's original description of his geometrical model alludes to at least one inversion of the model to be seen in the diagrams: The description is not coherent and appears to arbitrarily combine features of several both possible and impossible inversions of the geometric model. Algebraically, the model can be expressed with complex numbers as Other commentators have observed that

828-723: 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 − 2 r . 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: A = ( k − 1 ) ( k − 2 ) k 2 π R 2 = ( k − 1 ) ( k − 2 ) π r 2 {\displaystyle A={\frac {(k-1)(k-2)}{k^{2}}}\pi R^{2}=(k-1)(k-2)\pi r^{2}} The arc length of

864-500: The curves have the same projection. Ordinary cusps appear when the tangent to the curve is parallel to the direction of projection (that is when the tangent projects on a single point). More complicated singularities occur when several phenomena occur simultaneously. For example, rhamphoid cusps occur for inflection points (and for undulation points ) for which the tangent is parallel to the direction of projection. In many cases, and typically in computer vision and computer graphics ,

900-432: The diagonal entries of SU(4) matrices gives points inside an astroid, and so on. Thanks to this result, one can use the fact that SU( k ) fits inside SU( k+1 ) as a subgroup to prove that an epicycloid with k cusps moves snugly inside one with k +1 cusps. The evolute of a hypocycloid is an enlarged version of the hypocycloid itself, while the involute of a hypocycloid is a reduced copy of itself. The pedal of

936-399: The giant price tag prop still features them today. Tusi couple 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

972-447: The hypocycloid straight-line mechanism. A practical and mechanically simple version of the Tusi couple, which avoids the use of an external rim gear, was developed in 2021 by John Goodman in order to provide linear motion. It uses 3 standard spur gears. A rotating (blue) arm is mounted on a central shaft, to which a fixed (yellow) gear is mounted. A (red) idler gear on the arm meshes with the fixed gear. A third (green) gear meshes with

1008-406: The idler. The third gear has half the number of teeth of the fixed gear. An (orange) arm is fixed to the third gear. If the length of the arm equals the distance between the fixed and outer gears = d, the arm will describe a straight line of throw = 2d. An advantage of this design is that, if standard modulus gears that do not provide the required throw, the idler gear does not have to be colinear with

1044-470: The middle of the 14th Century. In his questiones on the Sphere (written before 1362), Nicole Oresme described how to combine circular motions to produce a reciprocating linear motion of a planet along the radius of its epicycle. Oresme's description is unclear and it is not certain whether this represents an independent invention or an attempt to come to grips with a poorly understood Arabic text. Although

1080-436: The other two gears A property of the Tusi couple is that points on the inner circle that are not on the circumference trace ellipses . These ellipses, and the straight line traced by the classic Tusi couple, are special cases of hypotrochoids . Cusp (singularity) In mathematics , a cusp , sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example

1116-412: The sense that they involve only one value of the parameter t , in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point. For a curve defined by an implicit equation which is smooth , cusps are points where the terms of lowest degree of

Hypocycloid - Misplaced Pages Continue

1152-1458: 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: x ( θ ) = ( R − r ) cos ⁡ θ + r cos ⁡ ( R − r r θ ) y ( θ ) = ( R − r ) sin ⁡ θ − r sin ⁡ ( R − r r θ ) {\displaystyle {\begin{aligned}&x(\theta )=(R-r)\cos \theta +r\cos \left({\frac {R-r}{r}}\theta \right)\\&y(\theta )=(R-r)\sin \theta -r\sin \left({\frac {R-r}{r}}\theta \right)\end{aligned}}} or: x ( θ ) = r ( k − 1 ) cos ⁡ θ + r cos ⁡ ( ( k − 1 ) θ ) y ( θ ) = r ( k − 1 ) sin ⁡ θ − r sin ⁡ ( ( k − 1 ) θ ) {\displaystyle {\begin{aligned}&x(\theta )=r(k-1)\cos \theta +r\cos \left((k-1)\theta \right)\\&y(\theta )=r(k-1)\sin \theta -r\sin \left((k-1)\theta \right)\end{aligned}}} If k

1188-422: The source takes x 2 + y k + 1 {\displaystyle x^{2}+y^{k+1}} to x 2 − y 2 n + 1 . {\displaystyle x^{2}-y^{2n+1}.} So we can drop the ± from ⁠ A 2 n ± {\displaystyle A_{2n}^{\pm }} ⁠ notation. The cusps are then given by

1224-663: The square to show that L 2 ± L Q = ( L ± Q / 2 ) 2 − Q 4 / 4. {\displaystyle L^{2}\pm LQ=(L\pm Q/2)^{2}-Q^{4}/4.} We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with linearly independent linear parts) so that ( L ± Q / 2 ) 2 − Q 4 / 4 → x 1 2 + P 1 {\displaystyle (L\pm Q/2)^{2}-Q^{4}/4\to x_{1}^{2}+P_{1}} where P 1

1260-561: The type ⁠ A k ± {\displaystyle A_{k}^{\pm }} ⁠ -singularities. Notice that the ⁠ A 2 n + {\displaystyle A_{2n}^{+}} ⁠ are the same as the ⁠ A 2 n − {\displaystyle A_{2n}^{-}} ⁠ since the diffeomorphic change of coordinate ⁠ ( x , y ) → ( x , − y ) {\displaystyle (x,y)\to (x,-y)} ⁠ in

1296-434: The zero-level-sets of the representatives of the ⁠ A 2 n {\displaystyle A_{2n}} ⁠ equivalence classes, where n ≥ 1 is an integer. For a type A 4 -singularity we need f to have a degenerate quadratic part (this gives type A ≥2 ), that L does divide the cubic terms (this gives type A ≥3 ), another divisibility condition (giving type A ≥4 ), and

#52947