HOME

TheInfoList



OR:

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a trochoid () is a
roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...
curve formed by a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
rolling along a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
. It is the
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the point is on the circle, the trochoid is called ''common'' (also known as a
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 cu ...
); if the point is inside the circle, the trochoid is ''curtate''; and if the point is outside the circle, the trochoid is ''prolate''. The word "trochoid" was coined by Gilles de Roberval.


Basic description

As a circle of radius rolls without slipping along a line , the center moves parallel to , and every other point in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let .
Parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
s of the trochoid for which is the -axis are :\begin & x = a\theta - b \sin \theta \\ & y = a - b \cos \theta \end where is the variable angle through which the circle rolls.


Curtate, common, prolate

If lies inside the circle (), on its circumference (), or outside (), the trochoid is described as being curtate ("contracted"), common, or prolate ("extended"), respectively. A curtate trochoid is traced by a pedal (relative to the ground) when a normally geared bicycle is pedaled along a straight line. A prolate trochoid is traced by the tip of a paddle (relative to the water's surface) when a boat is driven with constant velocity by paddle wheels; this curve contains loops. A common trochoid, also called a
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 cu ...
, has
cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...
s at the points where touches the line .


General description

A more general approach would define a trochoid as the locus of a point (x,y)
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
ing at a constant rate around an axis located at (x',y'), :x=x'+r_1\cos(\omega_1 t+\phi_1),\ y=y'+r_1\sin(\omega_1 t+\phi_1),\ r_1>0, which axis is being translated in the ''x-y''-plane at a constant rate in ''either'' a straight line, :\begin x'=x_0+v_ t,\ y'=y_0+v_ t\\ \therefore x = x_0+r_1\cos(\omega_1 t+\phi_1)+v_ t,\ y = y_0+r_1 \sin(\omega_1 t+\phi_1)+v_ t,\\ \end or a circular path (another orbit) around (x_0,y_0) (the hypotrochoid/ epitrochoid case), :\begin x' = x_0+r_2\cos(\omega_2 t+\phi_2),\ y' = y_0+r_2\sin(\omega_2 t+\phi_2),\ r_2\ge 0\\ \therefore x = x_0+r_1\cos(\omega_1 t+\phi_1)+r_2\cos(\omega_2 t+\phi_2),\ y = y_0+r_1 \sin(\omega_1 t+\phi_1)+r_2\sin(\omega_2 t+\phi_2),\\ \end The ratio of the rates of motion and whether the moving axis translates in a straight or circular path determines the shape of the trochoid. In the case of a straight path, one full rotation coincides with one period of a periodic (repeating) locus. In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions, \omega_1/\omega_2, is a rational number, say p/q, where p & q are
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
, in which case, one period consists of p orbits around the moving axis and q orbits of the moving axis around the point (x_0,y_0). The special cases of the epicycloid and hypocycloid, generated by tracing the locus of a point on the perimeter of a circle of radius r_1 while it is rolled on the perimeter of a stationary circle of radius R, have the following properties: :\begin \text&\omega_1/\omega_2&=p/q=r_2/r_1=R/r_1+1,\ , p-q, \text\\ \text&\omega_1/\omega_2&=p/q=-r_2/r_1=-(R/r_1-1),\ , p-q, =, p, +, q, \text \end where r_2 is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with "cusps" replaced by either "radial maxima" or "radial minima".


See also

*
Aristotle's wheel paradox Aristotle's wheel paradox is a paradox or problem appearing in the Greek work ''Mechanica'', traditionally attributed to Aristotle. It states as follows: A wheel is depicted in two-dimensional space as two circles. Its larger, outer circle is tan ...
* Brachistochrone * Cyclogon *
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 cu ...
* Epitrochoid * Hypotrochoid * List of periodic functions * Roulette (curve) * Spirograph * Trochoidal wave


References

{{reflist


External links


Online experiments with the Trochoid using JSXGraph
Roulettes (curve)