# Limaçon
The **limaçon** (from the French for "snail", after Latin *limax*) is a plane algebraic curve of order 4, defined as the conchoid of a circle with respect to a point on its circumference; it is also called the limaçon of Pascal after Étienne Pascal, who treated it in the first half of the 17th century.[1](https://mathworld.wolfram.com/Limacon.html) • [2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on) In polar coordinates it is written
\[ r = b + a\cos\theta, \]
and its Cartesian form is \( (x^2+y^2-ax)^2 = l^2(x^2+y^2) \), where the two parameters control the shape: the ratio \( b/a \) determines whether the curve is a convex oval, a dimpled oval, a cardioid, or a curve with an inner loop.[2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on) • [1](https://mathworld.wolfram.com/Limacon.html)
| Key fact | Detail |
|---|---|
| Equation | Polar \( r = b + a\cos\theta \); Cartesian \( (x^2+y^2-ax)^2 = l^2(x^2+y^2) \); a quartic (order-4) algebraic curve[2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on) |
| Shape regimes | Inner loop for \( b < a \); cardioid at \( b = a \); dimpled for \( a < b < 2a \); convex for \( b \ge 2a \)[1](https://mathworld.wolfram.com/Limacon.html) |
| Area | \( S = \pi a^2/2 + \pi l^2 \), with the inner leaf counted double when a loop is present[2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on) |
| Arc length | Generally expressible through an elliptic integral of the second kind[1](https://mathworld.wolfram.com/Limacon.html) |
| Trisectrix case | At \( b = 2a \) the curve trisects an arbitrary angle[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lima%C3%A7on) |
| History | Drawn by Dürer in 1525 (as "Arachne"), treated by Étienne Pascal in 1630, named by Roberval in 1650[1](https://mathworld.wolfram.com/Limacon.html) • [4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml) |
| Modern use | Limaçon-shaped chaotic microlasers combine high quality factor with unidirectional emission[5](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.80.041807) |
## Definition and equation
The limaçon is the conchoid of a circle with respect to a point on its circumference: from a fixed point, each ray through the circle is marked off at a fixed distance beyond (or short of) the circle, and the locus of these marks is the curve.[1](https://mathworld.wolfram.com/Limacon.html) In the notation of the Encyclopedia of Mathematics, the base circle has diameter \( a \) and the conchoid offset is \( l \), giving the polar equation \( \rho = a\cos\phi + l \) and the rectangular equation \( (x^2+y^2-ax)^2 = l^2(x^2+y^2) \).[2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on) The 1911 Britannica gives the same curve as \( r = a + b\cos\theta \) with a mechanical interpretation: if the two extremities of a rod of length \( 2a \) move so that its midpoint traces a circle of diameter \( b \), each extremity traces a limaçon.[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lima%C3%A7on) The two conventions swap the names of the parameters, so ratios must be read carefully when comparing sources.
## Classification by shape
The single quantity \( b/a \) sorts all limaçons into four regimes.[1](https://mathworld.wolfram.com/Limacon.html)
- **Inner loop**, \( b < a \): the curve crosses itself at the pole, which is a node (a double point with two branches).[2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on)
- **Cardioid**, \( b = a \): the loop shrinks to a cusp at the origin, and the curve is the familiar heart-shaped cardioid.[2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on)
- **Dimpled**, \( a < b < 2a \): the curve is a single oval with a concave indentation near the pole.
- **Convex**, \( b \ge 2a \): the indentation disappears and the curve is a convex oval; at \( b = 2a \) exactly it is the trisectrix.[1](https://mathworld.wolfram.com/Limacon.html)
Mathcurve classifies the same family by eccentricity \( e \): \( e < 1 \) gives elliptic limaçons (convex, flat-sided, or bean-shaped, with an isolated point at the pole in the acnodal case), \( e = 1 \) the cuspidal cardioid, \( e > 1 \) hyperbolic limaçons with a loop, and \( e = 2 \) the trisectrix limaçon, which is also a rose.[4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml) The pedal-point view matches: with the pole outside the generating circle the limaçon is hyperbolic, with the pole on the circumference it is the cardioid, and with the pole inside it is elliptical; limiting cases are a double circle (pole at infinity, ratio infinite), the trisectrix (ratio 2), and a circle (ratio 0).[6](http://www.scielo.org.co/pdf/dyna/v88n216/2346-2183-dyna-88-216-196.pdf)
## Geometric constructions and transformations
The limaçon appears under many constructions.[4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml)
- **Pedal of a circle.** Using the traditional 90° pedal construction, the pedal curve of the circle \( r = a\cos\theta \) with respect to any fixed pedal point is the limaçon \( r = b + a\cos\theta \).[6](http://www.scielo.org.co/pdf/dyna/v88n216/2346-2183-dyna-88-216-196.pdf) The trisectrix limaçon is in turn the pedal curve of a cardioid with respect to the center of its conchoidal circle.[7](https://mathworld.wolfram.com/LimaconTrisectrix.html)
- **Envelope of circles.** Fix a point and draw the family of circles with centers on a given circle, all passing through the fixed point; their envelope is a limaçon, and if the fixed point lies on the circumference the envelope is a cardioid.[1](https://mathworld.wolfram.com/Limacon.html) Equivalently, the limaçon is the envelope of circles with diameters running from the fixed point to points of the base circle.[8](https://www.maplesoft.com/applications/Preview.aspx?id=4872)
- **Epitrochoid.** It is the epitrochoid generated when one circle rolls around an equal fixed circle.[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lima%C3%A7on) • [4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml)
- **Inversion of a conic.** With respect to a focus, the limaçon is the inverse of a central conic: an ellipse for the elliptic cases and a hyperbola for the looping cases.[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lima%C3%A7on) • [4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml)
- **Cissoid and anallagmatic property.** The limaçon is the cissoid of two circles and is anallagmatic, meaning it is its own inverse with respect to a suitable circle.[4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml) • [1](https://mathworld.wolfram.com/Limacon.html)
- **Caustics and mechanics.** The evolutes of limaçons are the caustics by reflection of a circle illuminated by a source at finite distance, and limaçons appear in the Bélidor drawbridge system.[4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml)
## By the numbers
**Area.** The area bounded by the limaçon is
\[ S = \frac{\pi a^2}{2} + \pi l^2, \]
where \( a \) is the diameter of the base circle and \( l \) the conchoid offset; when the curve has an inner loop (\( a > l \)), the inner leaf is counted twice in this formula.[2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on) The 1911 Britannica states the same result as \( (b^2 + a^2/2)\pi \) in its own parameter convention.[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lima%C3%A7on)
**Arc length.** For a general limaçon, the perimeter is not expressible in elementary functions: it is expressed through an elliptic integral of the second kind, and the full curve length is a complete elliptic integral of the second kind.[1](https://mathworld.wolfram.com/Limacon.html) • [3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lima%C3%A7on)
**Curvature and the dimple.** For the normalized curve \( r(t) = l + \sin t \), the sign of the curvature is the sign of \( l^2 + 3l\sin t + 2 \). The equation \( l^2 + 3l\sin t + 2 = 0 \) has real solutions exactly when \( 1 \le |l| \le 2 \); for \( 1 < |l| < 2 \), the curvature changes sign at two values of \( t \), and it is this change of sign that accounts for the dimple.[9](https://faculty.kfupm.edu.sa/math/hassanaz/teaching%20articles/limacon.pdf) The same analysis shows the curve is non-singular whenever \( |l| \ne 1 \), the singular case being the cardioid's cusp.[9](https://faculty.kfupm.edu.sa/math/hassanaz/teaching%20articles/limacon.pdf)
## How it compares with related curves
The cardioid is exactly the special case \( b = a \), so every limaçon family contains it as the boundary between looping and dimpled forms.[1](https://mathworld.wolfram.com/Limacon.html) The family sits inside larger classes of curves: limaçons are rational bicircular quartics, and each is a special case of a complete Cartesian oval, with the hyperbolic limaçon the union of two Cartesian ovals and the elliptic limaçon a Cartesian oval together with an isolated point.[4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml) • [2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on) The name "trisectrix" is shared: the limaçon with \( b = 2a \) is a trisectrix, but it is not the Maclaurin trisectrix, which is a cubic curve arising in a different construction.[1](https://mathworld.wolfram.com/Limacon.html) • [10](https://ijgeometry.com/wp-content/uploads/2017/10/86-92.pdf)
## The limaçon trisectrix and angle trisection
For the case \( b = 2a \), the curve solves a classical construction problem: Pascal's limaçon allows the trisection of any angle.[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lima%C3%A7on) • [10](https://ijgeometry.com/wp-content/uploads/2017/10/86-92.pdf) The construction works as follows. Take the base circle with horizontal diameter \( AB \) and let \( AR \) be a line making the angle to be trisected with \( AB \). The ray from the pole \( O \) through the point where \( AR \) meets the trisectrix cuts off an angle \( P_1AO \) equal to one third of \( RAO \).[8](https://www.maplesoft.com/applications/Preview.aspx?id=4872) The proof rests on isosceles triangles: by the construction, \( P_1Q = OQ = AO \), so triangles \( OQP_1 \), \( OAQ \), and \( OAR \) chain equal angles together and divide the original angle in thirds.[8](https://www.maplesoft.com/applications/Preview.aspx?id=4872) Because the trisecting point is found on a fixed curve rather than by compass and straightedge alone, the curve was studied as far back as [Archimedes](https://www.edgepedia.org/archimedes) as well as by Étienne Pascal in 1630.[7](https://mathworld.wolfram.com/LimaconTrisectrix.html)
## History and etymology
The word "limaçon" comes from the Latin *limax*, meaning "snail".[1](https://mathworld.wolfram.com/Limacon.html) According to MathWorld, the curve was first investigated by [Albrecht Dürer](https://www.edgepedia.org/albrecht-durer), who gave a method for drawing it in *Underweysung der Messung* (1525) under the name Arachne; it was rediscovered by Étienne Pascal (1588–1651), a magistrate and amateur mathematician, father of [Blaise Pascal](https://www.edgepedia.org/blaise-pascal), and named by Gilles Personne Roberval in 1650.[1](https://mathworld.wolfram.com/Limacon.html) • [4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml) The 1911 Britannica instead states that the curve was invented by Blaise Pascal and further investigated and named by Gilles Personne de Roberval.[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lima%C3%A7on) Modern specialist references credit Étienne, not Blaise.[2](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on) • [4](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml)
## Applications: limaçon microcavities
The limaçon has a modern role in optics. A limaçon-shaped optical microcavity has a boundary \( r(\varphi) = R(1 + \varepsilon\cos\varphi) \), where \( R \) is the mean radius and \( \varepsilon \) the deformation parameter; the asymmetric deformation breaks the rotational symmetry of a circular disk while preserving high-quality whispering-gallery-type modes, and light escapes in a directed beam rather than radiating uniformly.[11](https://www.sciencedirect.com/science/article/abs/pii/S0030399224017572) • [12](https://iopscience.iop.org/article/10.1088/1367-2630/11/12/125018/pdf)
The foundational 2009 demonstration reported a chaotic limaçon microlaser with quality factor 23,000 for a mode volume below 0.2 µm³, spontaneous emission coupling efficiency of about 6%, and unidirectional output with divergence angle below 40°.[5](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.80.041807) Limaçon-shaped quantum cascade lasers emitting near \( \lambda \approx 10 \) µm achieved beam divergence of about 33° for deformations \( 0.37 < \varepsilon < 0.43 \), while retaining high-\( Q \) whispering-gallery-like modes for \( \varepsilon < 0.50 \).[12](https://iopscience.iop.org/article/10.1088/1367-2630/11/12/125018/pdf)
Recent devices extend the platform. In 2024, injection limaçon microlasers with InGaAs quantum well-dots lased at temperatures up to 90 °C for the first time, with quality factor estimated at least \( 10^5 \) and minimum linewidth about 200 MHz; in the far field one dominant directional lobe appears, whereas quadrupole-shaped cavities emit two opposite lobes.[11](https://www.sciencedirect.com/science/article/abs/pii/S0030399224017572) A thermo-optically tuned limaçon quantum-dot microlaser under continuous-wave electrical pumping achieved single-mode lasing with side-mode suppression ratio above 25 dB and peak output power of about 0.4 mW, 1.5 times that of a comparable circular microdisk, with mode-hop-free wavelength tuning over 4.6 nm at a coefficient of about 0.3 nm mA⁻¹; the asymmetric geometry lifts the degeneracy of high-\( Q \) whispering-gallery modes, enabling the single-mode operation.[13](https://iopscience.iop.org/article/10.1088/1361-6463/ae8915) In sensing, 3D micro-printed polymer limaçon whispering-gallery-mode laser sensors showed a lasing threshold of 3.87 µJ/mm² and a linewidth of about 30 pm, detecting human IgG with a detection limit of around 70 ag/mL.[14](https://opg.optica.org/ol/abstract.cfm?uri=ol-50-11-3481) [Experiments](https://www.edgepedia.org/experiments) on limaçon semiconductor microcavities have also revealed spatiotemporal dynamics, with edge-emission fluctuations dominated by alternate oscillation between two output beams with periods as short as a few nanoseconds.[15](https://opg.optica.org/ol/abstract.cfm?uri=ol-48-3-574) Transformation-optics analysis of limaçon cavities defined by the conformal map \( \zeta = \beta(\eta + \varepsilon\eta^2) \) shows that the bidirectionality factors of high-\( Q \) modes are not directly proportional to their quality factors, with the best coexistence of strong bidirectionality and high \( Q \) found for anisotropic whispering-gallery modes.[16](https://www.nature.com/articles/s41598-019-44768-1)
## References
1. [Limaçon, Wolfram MathWorld](https://mathworld.wolfram.com/Limacon.html)
2. [Pascal limaçon, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Pascal_lima%C3%A7on)
3. [Limaçon, 1911 Encyclopædia Britannica (Wikisource)](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lima%C3%A7on)
4. [Limaçon of Pascal, Mathcurve](https://www.mathcurve.com/courbes2d.gb/limacon/limacon.shtml)
5. [Chaotic microcavity laser with high quality factor and unidirectional output, Physical Review A (2009)](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.80.041807)
6. [Generalization of the pedal concept in bidimensional spaces. Application to the limaçon of Pascal, Dyna 88(216) (2021)](http://www.scielo.org.co/pdf/dyna/v88n216/2346-2183-dyna-88-216-196.pdf)
7. [Limaçon Trisectrix, Wolfram MathWorld](https://mathworld.wolfram.com/LimaconTrisectrix.html)
8. [The Limaçon and Angle Trisection, Maplesoft Application Center](https://www.maplesoft.com/applications/Preview.aspx?id=4872)
9. [Why do certain limaçons have a dimple, Teaching Mathematics and its Applications (2006)](https://faculty.kfupm.edu.sa/math/hassanaz/teaching%20articles/limacon.pdf)
10. [A short proof of the trisection property of Pascal's limaçon, International Journal of Geometry 6(2) (2017)](https://ijgeometry.com/wp-content/uploads/2017/10/86-92.pdf)
11. [Far-field patterns and lasing threshold of limaçon- and quadrupole-shaped microlasers with InGaAs quantum well-dots (2024)](https://www.sciencedirect.com/science/article/abs/pii/S0030399224017572)
12. [Deformed microcavity quantum cascade lasers with directional emission, New Journal of Physics (2009)](https://iopscience.iop.org/article/10.1088/1367-2630/11/12/125018/pdf)
13. [Thermo-optically tuned single-mode limaçon microlaser on quantum-dots, Journal of Physics D](https://iopscience.iop.org/article/10.1088/1361-6463/ae8915)
14. [3D micro-printed polymer limaçon-shaped whispering-gallery-mode microlaser sensors for label-free biodetection, Optics Letters (2025)](https://opg.optica.org/ol/abstract.cfm?uri=ol-50-11-3481)
15. [Spatiotemporal lasing dynamics in a Limaçon-shaped microcavity, Optics Letters (2024)](https://opg.optica.org/ol/abstract.cfm?uri=ol-48-3-574)
16. [Optimization of conformal whispering gallery modes in limaçon-shaped transformation cavities, Scientific Reports (2019)](https://www.nature.com/articles/s41598-019-44768-1)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Analytic and coordinate geometry*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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