Definitions

pascal limaçon

Limaçon

[lim-uh-son]
In geometry, limaçons (pronounced with a soft c), also known as limaçons of Pascal, are heart-shaped mathematical curves. A limaçon is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius (see the diagram on the right). Thus, they part of the family of curves called centered trochoids; more specifically they are epitrochoids. The cardioid is the special case where the point generating the roulette lies on the rolling circle and the resulting curve has a cusp.

The term derives from the Latin word limax which means "snail".

The limaçon is a rational plane algebraic curve.

History

Formal research on limaçons is attributed to Étienne Pascal, father of Blaise Pascal. However investigations began earlier by the German Renaissance artist, Albrecht Dürer. Dürer's Underweysung der Messung (Instruction in Measurement), contains specific geometric methods for producing limaçons.

In 1982, Jane Grossman discovered that in addition to the three limacons listed in the CRC Tables, there is it fact a fourth type, which she called the dimpled limacon.

Equations

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

r = b + a cos theta

which in Cartesian coordinates is

(x^2+y^2-ax)^2=b^2(x^2+y^2). ,

Parametrically, this becomes

x = {aover 2} + b cos theta + {aover 2} cos 2theta,, y = b sin theta + {aover 2} sin 2theta.

In the complex plane this takes the form

z = {aover 2} + b e^{itheta} + {aover 2} e^{2itheta}.

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

z = b e^{it} + {aover 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 + cos theta) = 2bcos^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} + e^{2it}) = b e^{3itover 2} (e^{itover 2} + e^{-itover 2}) = 2b cos {tover 2} e^{3itover 2} ,

or, in polar coordinates,

r = 2bcos{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 a 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 + {{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 + {{a^2}over 2})arccos {b over a} - {3over 2} b sqrt {{a^2} - {b^2}}, the area enclosed by the outer loop is (b^2 + {{a^2}over 2})(pi - arccos {b over a}) + {3over 2} b sqrt {{a^2} - {b^2}}, and the area between the loops is (b^2 + {{a^2}over 2})(pi - 2arccos {b over a}) + 3 b sqrt {{a^2} - {b^2}}.

Relation to other curves

  • Let P be a point and C be a circle whose center is not P. Then the envelope of those circles whose center lies on C and that pass through P is a limaçon.
  • A pedal of a circle is a limaçon. In fact, the pedal with respect to the origin of the circle with radius b and center (a,0) has polar equation r = b + a cos theta
  • The inverse with respect to the unit circle of r = b + a cos theta is r = {1 over {b + a cos theta}} which is the equation of a conic section with eccentricity a/b and focus at the origin. Thus a limaçon can be defined as the inverse of a conic where the center of inversion is one of the foci. If the conic is parabola then the inverse will be a cardioid, if the conic is a hyperbola then the corresponding limaçon will have an inner loop, and if the conic is an ellipse then the corresponding limaçon will have no loop.
  • The conchoid of a circle with respect to a point on the circle is a limaçon.

References

Additional reading

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