Limaçon

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Construction of a limaçon

In mathematics, limaçons (pronounced with a soft c), also known as limaçons of Pascal, are heart-shaped mathematical curves. The cardioid is a special case, with a cusp.

They arise in polar coordinates in the form

$r = a + b \sin \theta \$

which in Cartesian coordinates is

$(x^2+y^2)^2 - (a^2+2by)(x^2+y^2) + b^2 y^2 = 0 \,$ .

Swapping x and y in the above equation also gives a limaçon, which in polar coordinates is

$r = a + b \cos \theta \$

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

The limaçon is a rational plane algebraic curve.

[edit] 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.