Limaçon

In geometry, a limaçon or limacon /ˈlɪməsɒn/, 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.

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.

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.

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 (thus introducing a point at the origin which in some cases is spurious), and substituting

to obtain[1] Applying the parametric form of the polar to Cartesian conversion, we also have[2] 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

, the polar equation is or making it a member of the sinusoidal spiral family of curves.

, the centered trochoid form of the equation becomes or, in polar coordinates, making it a member of the rose family of curves.

, the curve has an indentation bounded by two inflection points.

, the cusp expands to an inner loop, and the curve crosses itself at the origin.

approaches 0, the loop fills up the outer curve and, in the limit, the limaçon becomes a circle traversed 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 The circumference of the limaçon is given by a complete elliptic integral of the second kind: