Limaçon

In geometry, a limaçon or limacon ( /ˈlɪməsɒn/), also known as a limaçon of Pascal, is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp.

The term derives from the Latin word limax, which means "snail". Depending on the position of the point generating the curve, it may have inner and outer loops (giving the family its name), it may be heart-shaped, or it may be oval.

A limaçon is a bicircular rational plane algebraic curve of degree 2.

Contents

History

The earliest formal research on limaçons is generally attributed to Étienne Pascal, father of Blaise Pascal. However, some insightful investigations regarding them had been undertaken earlier by the German Renaissance artist Albrecht Dürer. Dürer's Underweysung der Messung (Instruction in Measurement) contains specific geometric methods for producing limaçons.

Equations

The equation (up to translation and rotation) of a limaçon in polar coordinates has the form

r = b %2B a \cos \theta \ .

This can be converted to Cartesian coordinates by multiplying by r (thus introducing a point at the origin which in some cases is spurious), and substituting r^2 = x^2%2By^2 and r \, \cos \theta = x to obtain[1]

(x^2%2By^2-ax)^2=b^2(x^2%2By^2). \,

Parametrically, this becomes

x = {a\over 2} %2B b \cos \theta %2B {a\over 2} \cos 2\theta,\, y = b \sin \theta %2B {a\over 2} \sin 2\theta.

In the complex plane this takes the form

z = {a\over 2} %2B b e^{i\theta} %2B {a\over 2} e^{2i\theta}.

If we shift this horizontally by a/2 we obtain the equation in the usual form for a centered trochoid:

z = b e^{it} %2B {a\over 2} e^{2it}.

This is the equation obtained when the center of the curve (as a centered trochoid) is taken to be the origin.

Special cases

In the special case a = b, the polar equation is r = b(1 %2B \cos \theta) = 2b\cos^2 {\theta \over 2} or r^{1 \over 2} = (2b)^{1 \over 2} \cos {\theta \over 2} making it a member of Sinusoidal spiral family of curves. This curve is the Cardioid

In the special case a = 2b the centered trochoid form of the equation becomes

z = b (e^{it} %2B e^{2it}) = b e^{3it\over 2} (e^{it\over 2} %2B e^{-it\over 2}) = 2b \cos {t\over 2} e^{3it\over 2} ,

or, in polar coordinates,

r = 2b\cos{\theta \over 3}

making it a member of the rose family of curves. This curve is a trisectrix, and is sometimes called the limaçon trisectrix.

Form

When b > a the limaçon is a simple closed curve. However, the origin satisfies the Cartesian equation given above so the graph of this equation has an acnode or isolated point.

When b > 2a the area bounded by the curve is convex and when a < b < 2a the curve has an indentation bounded by two inflection points. At b = 2a the point (-a, 0) is a point of 0 curvature.

As b is decreased relative to a, the indentation becomes more pronounced until, at b = a, the cardioid, it becomes a cusp. For 0 < b < a, the cusp expands to an inner loop and the curve crosses itself at the origin. As b approaches 0 the loop fills up the outer curve and, in the limit, the limaçon becomes a circle traversed twice.

Measurement

The area enclosed by the limaçon is (b^2 %2B {{a^2}\over 2})\pi. When b < a this counts the area enclosed by the inner loop twice. In this case the curve crosses the origin at angles \pi \pm \arccos {b \over a}, the area enclosed by the inner loop is (b^2 %2B {{a^2}\over 2})\arccos {b \over a} - {3\over 2} b \sqrt {{a^2} - {b^2}}, the area enclosed by the outer loop is (b^2 %2B {{a^2}\over 2})(\pi - \arccos {b \over a}) %2B {3\over 2} b \sqrt {{a^2} - {b^2}}, and the area between the loops is (b^2 %2B {{a^2}\over 2})(\pi - 2\arccos {b \over a}) %2B 3 b \sqrt {{a^2} - {b^2}}.

Relation to other curves

References

  1. ^ J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 113–118. ISBN 0-486-60288-5. 
  2. ^ O'Connor, John J.; Robertson, Edmund F., "Cartesian Oval", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Curves/Cartesian.html .

Additional reading