Centered trochoid - AbsoluteAstronomy.com

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a centered trochoid is the roulette

Roulette (curve)

In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....

formed by a circle rolling along another circle. That is, the path traced by a point attached to a circle as the circle rolls without slipping along a fixed circle. The term encompasses both epitrochoid

Epitrochoid

An epitrochoid 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 a distance d from the center of the exterior circle....

and hypotrochoid

Hypotrochoid

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 center of this curve is defined to be the center of the fixed circle.

Alternatively, a centered trochoid can be defined as the path traced by the sum of two vectors, each moving at a uniform speed in a circle. Specifically, a centered trochoid is a curve that can be parameterized in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

or in the Cartesian plane by

where

is rational then the curve is closed and algebraic. Otherwise the curve winds around the origin an infinite number of times, and is dense in the annulus

Annulus (mathematics)

In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...

with outer radius

and inner radius

Terminology

Most authors use epitrochoid to mean a roulette of a circle rolling around the outside of another circle, hypotrochoid to mean a roulette of a circle rolling around the inside of another circle, and trochoid to mean a roulette of a circle rolling along a line. However, some authors (for example http://www.monmouth.com/~chenrich/Trochoids/Trochoids.html following F. Morley

Frank Morley

Frank Morley was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry...

) use "trochoid" to mean a roulette of a circle rolling along another circle, though this is inconsistent with the more common terminology. The term Centered trochoid as adopted by http://www.mathcurve.com/courbes2d/trochoid/trochoidale.shtml combines epitrochoid and hypotrochoid into a single concept to streamline mathematical exposition and remains consistent with the existing standard.

The term Trochoidal curve describes epitrochoids, hypotrochoids, and trochoids (see http://www.mathcurve.com/courbes2d/trochoid/trochoidale.shtml). A trochoidal curve can be defined as the path traced by the sum of two vectors, each moving at a uniform speed in a circle or in a straight line (but not both moving in a line).

In the parametric equations given above, the curve is an epitrochoid if

and

have the same sign, and a hypotrochoid if they have opposite signs.

Dual generation

Let a circle of radius

be rolled on a circle of radius

, and a point

is attached to the rolling circle. The fixed curve can be parameterized as

and the rolling curve can be parameterized as either

depending on whether the parameterization traverses the circle in the same direction or in the opposite direction as the parameterization of the fixed curve. In either case we may use

where

. Let

be attached to the rolling circle at

. Then, applying the formula for the roulette

Roulette

Roulette is a casino game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even....

, the point traces out a curve given by:

This is the parameterization given above with

.

Conversely, given

, and

, the curve

can be reparameterized as

and the equations

can be solved for

and

to get

The curve

remains the same if the indexes
1 and 2 are reversed but the resulting values of

and

, in general, do not. This produces the Dual generation theorem which states that, with the exception of the special case discussed below, any centered trochoid can be generated in two essentially different ways as the roulette of a circle rolling on another circle.

Cardioid

The cardioid

Cardioid

A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp...

is parameterized by

. Take

to get

. The circles both have radius 1 and, since c < 0, the rolling circle is rolling around the outside of the fixed circle. The point p is 1 unit from the center of the rolling so it lies on its circumference. This is the usual definition of the cardioid. We may also parameterize the curve as

, so we may also take

to get

In this case the fixed circle has radius 1, the rolling circle has radius 2, and, since c > 0, the rolling circle revolves around the fixed circle in the fashion of a hula hoop

Hula hoop

A hula hoop is a toy hoop that is twirled around the waist, limbs or neck.Although the exact origins of hula hoops are unknown, children and adults around the world have played with hoops, twirling, rolling and throwing them throughout history...

. This produces an essentially different definition of the same curve.

Ellipse

then we obtain the parametric curve

, or

. If

, this is the equation of an ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

with axes

and

. Evaluating

, and

as before; either

. This gives two different ways of generating an ellipse, both of which involve a circle rolling inside a circle with twice the diameter.

Straight line

If additionally, next to

, then

in both cases and the two ways of generating the curve are the same. In this case the curve is simply

or a segment of the x-axis.

Likewise, if

, then

. The circle is symmetric about the origin, so both of these give the same pair of circles. In this case the curve is simply

: a segment of the y-axis.

So the case

is an exception (in fact the only exception) to the dual generation theorem stated above. This degenerate case, in which the curve is a straight-line segment, underlies the Tusi-couple

Tusi-couple

The Tusi-couple 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...

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.