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Wikipedia / Limaçon

In geometry, a limaçon or limacon (), also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that 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.

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 4.

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. The curve was named by Gilles de Roberval when he used it as an example for finding tangent lines, deriving the word from the latin limax: snail.

Equations

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

This can be converted to Cartesian coordinates by multiplying by r to obtain

Applying the parametric form of the polar to Cartesian conversion, we also have

while setting

yields this parameterization as a curve in the complex plane:

, i.e.,

  1. ,

.

Special cases

, the polar equation is

or

making it a member of the sinusoidal spiral family of curves. This curve is the cardioid.

, the centered trochoid form of the equation becomes

or, in polar coordinates,

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

Form

, 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.

, the curve has an indentation bounded by two inflection points is a point of 0 curvature.

, the curve becomes a cardioid, and the indentation becomes a cusp 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 inner loop is

the area enclosed by the outer loop is

and the area between the loops is

The circumference of the limaçon is given by a complete elliptic integral of the second kind:

Relation to other curves

  • is a limaçon.

    A pedal of a circle.

    The inverse is

  • which is the equation of a conic section 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 a 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.

    A particular special case of a Cartesian oval is a limaçon.

See also

References

Further reading