Limaçon


In geometry, a limaçon or limacon, also known as a limaçon of Pascal, is defined as a roulette 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, 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 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.

Equations

The equation of a limaçon in polar coordinates has the form
This can be converted to Cartesian coordinates by multiplying by r, and substituting and 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:
If we were to shift horizontally by, i.e.,
we would, by changing the location of the origin, convert to the usual form of the equation of a centered trochoid. Note the change of independent variable at this point to make it clear that we are no longer using the default polar coordinate parameterization.

Special cases

In the special case, the polar equation is
or
making it a member of the sinusoidal spiral family of curves. This curve is the cardioid.
In the special case, 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

When, 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, the area bounded by the curve is convex, and when, the curve has an indentation bounded by two inflection points. At, the point is a point of 0 curvature.
As is decreased relative to, the indentation becomes more pronounced until, at, the curve becomes a cardioid, and the indentation becomes a cusp. For, the cusp expands to an inner loop, and the curve crosses itself at the origin. As 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. When this counts the area enclosed by the inner loop twice. In this case the curve crosses the origin at angles, the area enclosed by the inner loop is
the area enclosed by the outer loop is
and the area between the loops is

Relation to other curves