A cardioid (from the Greek καρδία "heart") is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion.[1]

The name was coined by de Castillon in 1741[2] but had been the subject of study decades beforehand.[3] Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk.

A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid (any 2d plane containing the 3d straight line of the microphone body). In three dimensions, the cardioid is shaped like an apple centred around the microphone which is the "stalk" of the apple.

Cardiod animation
cardioid generated by a rolling circle on a circle with the same radius


Generation of a cardioid and the coordinate system used

Let be the common radius of the two generating circles with midpoints , the rolling angle and the origin the starting point (see picture). One gets the

and the representation in


Introducing the substitutions and one gets after removing the square root the implicit representation in

Proof for the parametric representation

The proof can be done easily using complex numbers and their common description as complex plane. The rolling movement of the black circle on the blue one can be split into two rotations. In the complex plane a rotation around point (origin) by an angle can be performed by the multiplication of a point (complex number) by . Hence the

rotation around point is,
rotation around point is: .

A point of the cardioid is generated by rotating the origin around point and subsequent rotating around by the same angle :


Herefrom one gets the parametric representation above:

(The following formulae were used. See trigonometric functions.)

Metric properties

For the cardioid as defined above the following formulas hold:

  • area ,
  • arc length and
  • radius of curvature

The proofs of these statement use in both cases the polar representation of the cardioid. For suitable formulas see polar coordinate system (arc length) and polar coordinate system (area)

proof of the area formula
proof of the arc length formula
proof for the radius of curvature

The radius of curvature of a curve in polar coordinates with equation is (s. curvature)

For the cardioid one gets


Chords of a cardioid

Chords through the cusp

  • C1: chords through the cusp of the cardioid have the same length .
  • C2: The midpoints of the chords through the cusp lie on the perimeter of the fixed generator circle (see picture) .
proof for C1

The points are on a chord through the cusp (=origin). Hence

proof for C2

For the proof the representation in the complex plane (see above) is used. For the points


the midpoint of the chord is

which lies on the perimeter of the circle with midpoint and radius (see picture).

Cardioid as inverse curve of a parabola

cardioid generated by the inversion of a parabola across the unit circle (dashed)
  • A cardioid is the inverse curve of a parabola with its focus at the center of inversion (see graph)

For the example shown in the graph the generator circles have radius . Hence the cardioid has the polar representation

and its inverse curve


which is a parabola (s. parabola in polar coordinates) with the equation in cartesian coordinates.

Remark: Not every inverse curve of a parabola is a cardioid. For example, if a parabola is inverted across a circle whose center lies at the vertex of the parabola, then the result is a cissoid of Diocles.

Cardioid as envelope of a pencil of circles

cardioid as envelope of a pencil of circles

In the previous section if one inverts additionally the tangents of the parabola one gets a pencil of circles through the center of inversion (origin). A detailed consideration shows: The midpoints of the circles lie on the perimeter of the fixed generator circle. (The generator circle is the inverse curve of the parabolas's directrix.)

This property gives rise to the following simple method to draw a cardioid:

1) Choose a circle and a point on its perimeter,
2) draw circles containing with centers on , and
3) draw the envelope of these circles.
proof with envelope condition

The envelope of the pencil of implicitly given curves

with parameter consists of such points which are solutions of the non-linear system

  • (envelope condition).

( means the partial derivative for parameter .

Let be the circle with midpoint and radius . Then has parametric representation . The pencil of circles with centers on containing point can be represented implicitly by


which is equivalent to

The second envelope condition is


One easily checks that the points of the cardioid with the parametric representation

fulfill the non-linear system above. The parameter is identical to the angle parameter of the cardioid.

Cardioid as envelope of a pencil of lines

Cardioid as envelope of a pencil of lines

A similar and simple method to draw a cardioid uses a pencil of lines. It is due to L. Cremona:

  1. Draw a circle, divide its perimeter into equal spaced parts with points (s. picture) and number them consecutively.
  2. Draw the chords: . (i.e.: The second point is moved by double velocity.)
  3. The envelope of these chords is a cardioid.
Cremona's generation of a cardioid

The following consideration uses trigonometric formulae for . In order to keep the calculations simple, the proof is given for the cardioid with polar representation (see section Cardioids in different positions).

equation of the tangent

of the cardioid with polar representation :

From the parametric representation

one gets the normal vector . The equation of the tangent is:

With help of trigonometric formulae and subsequent division by , the equation of the tangent can be rewritten as:

equation of the chord

of the circle with midpoint and radius : For the equation of the secant line passing the two points one gets:

With help of trigonometric formulae and the subsequent division by the equation of the secant line can be rewritten by:

Despite the two angles have different meanings (s. picture) one gets for the same line. Hence any secant line of the circle, defined above, is a tangent of the cardioid, too:

  • The cardioid is the envelope of the chords of a circle.

The proof can be performed with help of the envelope conditions (see previous section) of an implicit pencil of curves:

is the pencil of secant lines of a circle (s. above) and

For fixed parameter t both the equations represent lines. Their intersection point is


which is a point of the cardioid with polar equation

Cardioid as caustic: light source , light ray , reflected ray
Cardioid as caustic of a circle with light source (right) on the perimeter

Cardioid as caustic of a circle

The considerations made in the previous section give a proof for the fact, that the caustic of a circle with light source on the perimeter of the circle is a cardioid.

  • If in the plane there is a light source on the perimeter of a circle which is reflecting any ray, then the reflected rays within the circle are tangents of a cardioid.

As in the previous section the circle may have midpoint and radius . Its parametric representation is

The tangent at circle point has normal vector . Hence the reflected ray has the normal vector (see graph) and contains point . The reflected ray is part of the line with equation (see previous section)

which is tangent of the cardioid with polar equation

from the previous section.

Remark: For such considerations usually multiple reflections at the circle are neglected.

Cardioid as pedal curve of a circle

Point of cardioid is foot of dropped perpendicular on tangent of circle

The Cremona generation of a cardioid should not be confused with the following generation:

Let be a circle and a point on the perimeter of this circle. The following is true:

  • The foots of perpendiculars from point on the tangents of circle are points of a cardioid.

Hence a cardioid is a special pedal curve of a circle.


In a cartesian coordinate system circle may have midpoint and radius . The tangent at circle point has the equation

The foot of the perpendicular from point on the tangent is point with the still unknown distance to the origin . Inserting the point into the equation of the tangent yields

which is the polar equation of a cardioid.

Remark: If point is not on the perimeter of the circle , one gets a limaçon of Pascal.

The evolute of a cardioid

evolute of a cardioid
magenta: one point P, its centre of curvature M and its osculating circle

The evolute of a curve is the locus of centers of curvature. In detail: For a curve with radius of curvature the evolute has the representation

with the suitably oriented unit normal.

For a cardioid one gets:

  • The evolute of a cardioid is another cardioid one third as large (s. picture).

For the cardioid with parametric representation

the unit normal is

and the radius of curvature

Hence the parametric equations of the evolute are

These equations describe a cardioid a third as large, rotated 180 degrees and shifted along the x-axis by .

(Trigonometric formulae were used: )

Orthogonal trajectories

orthogonal cardioids

An orthogonal trajectory of a pencil of curves is a curve which intersects any curve of the pencil orthogonally. For cardioids the following is true:

  • The orthogonal trajectories of the pencil of cardioids with equations
are the cardioids with equations

(The second pencil can be considered as reflections at the y-axis of the first one. See diagram.)

For a curve given in polar coordinates by a function the following connection to cartesian coordinates hold:

and for the derivatives

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point :

For the cardioids with the equations and respectively one gets:


(The slope of any curve is dependent from only, and not from the parameters  !)

That means: Any curve of the first pencil intersectcs any curve of the second pencil orthogonally.

4 cardioids in polar representation and their position in the coordinate system

In different positions

Choosing other positions of the cardioid within the coordinate system results in different equations. The picture shows the 4 most common positions of a cardioid and their polar equations.

In complex analysis

Mandel zoom 00 mandelbrot set
Boundary of the central bulb of the Mandelbrot set is a cardioid.

In complex analysis, the image of any circle through the origin under the map is a cardioid. One application of this result is that the boundary of the central bulb of the Mandelbrot set is a cardioid given by the equation

The Mandelbrot set contains an infinite number of slightly distorted copies of itself and the central bulb of any of these smaller copies is an approximate cardioid.

The caustic appearing on the surface of this cup of coffee is a cardioid.


Certain caustics can take the shape of cardioids. The catacaustic of a circle with respect to a point on the circumference is a cardioid. Also, the catacaustic of a cone with respect to rays parallel to a generating line is a surface whose cross section is a cardioid. This can be seen, as in the photograph to the right, in a conical cup partially filled with liquid when a light is shining from a distance and at an angle equal to the angle of the cone.[4] The shape of the curve at the bottom of a cylindrical cup is half of a nephroid, which looks quite similar.

Cardioid construction
Generating a cardioid as pedal curve of a circle

See also


  1. ^ Weisstein, Eric W. "Parabola Inverse Curve". MathWorld.
  2. ^ Lockwood
  3. ^ Yates
  4. ^ "Surface Caustique" at Encyclopédie des Formes Mathématiques Remarquables

Further reading

  • Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 24–25. ISBN 0-14-011813-6.

External links

AKG (company)

AKG Acoustics (originally Akustische und Kino-Geräte Gesellschaft m.b.H., English: Acoustic and Cinema Equipment) is an acoustics engineering and manufacturing company. It was founded in 1947 by Dr. Rudolf Görike and Ernest Plass in Vienna, Austria. It is a subsidiary of Harman International Industries, a division of Samsung Electronics.The products currently marketed under the AKG brand mostly consist of microphones, headphones, wireless audio systems and related accessories for professional and consumer markets.

Audio feedback

Audio feedback (also known as acoustic feedback, simply as feedback, or the Larsen effect) is a special kind of positive loop gain which occurs when a sound loop exists between an audio input (for example, a microphone or guitar pickup) and an audio output (for example, a power amplified loudspeaker). In this example, a signal received by the microphone is amplified and passed out of the loudspeaker. The sound from the loudspeaker can then be received by the microphone again, amplified further, and then passed out through the loudspeaker again. The frequency of the resulting sound is determined by resonance frequencies in the microphone, amplifier, and loudspeaker, the acoustics of the room, the directional pick-up and emission patterns of the microphone and loudspeaker, and the distance between them. For small PA systems the sound is readily recognized as a loud squeal or screech. The principles of audio feedback were first discovered by Danish scientist Søren Absalon Larsen, hence the name "Larsen Effect".

Feedback is almost always considered undesirable when it occurs with a singer's or public speaker's microphone at an event using a sound reinforcement system or PA system. Audio engineers typically use directional microphones with cardioid pickup patterns and various electronic devices, such as equalizers and, since the 1990s, automatic feedback detection devices, to prevent these unwanted squeals or screeching sounds, which detract from the audience's enjoyment of the event and may damage equipment. On the other hand, since the 1960s, electric guitar players in rock music bands using loud guitar amplifiers, speaker cabinets and distortion effects have intentionally created guitar feedback to create different sounds including long sustained tones that cannot be produced using standard playing techniques. The sound of guitar feedback is considered to be a desirable musical effect in heavy metal music, hardcore punk and grunge. Jimi Hendrix was an innovator in the intentional use of guitar feedback, alongside effects units such as the Univibe and wah-wah pedal in his guitar solos to create unique sound effects and musical sounds.

C1000 (microphone)

The C1000 is a cardioid/hypercardioid condenser microphone produced by AKG Acoustics.

It uses either phantom power or an internal 9-volt battery for when phantom power is inconvenient or unavailable. This flexibility is also useful when using consumer recording gear that frequently lacks phantom power. The pattern is switchable with the addition or removal of an optional capsule attachment.

Coin rotation paradox

The coin rotation paradox is the counter-intuitive observation that, when one coin is rolled around the rim of another coin of equal size, the moving coin completes two full rotations after going all the way around the stationary coin.

Georg Neumann

Georg Neumann GmbH (Neumann), founded in 1928 and based in Berlin, Germany, is a prominent manufacturer of professional recording microphones. Their best-known products are condenser microphones for broadcast, live and music production purposes. For several decades Neumann was also a leading manufacturer of cutting lathes for phonograph disks, and even ventured into the field of mixing desks for a while.

Jupiters Darling

Jupiters Darling is the thirteenth studio album by the American rock band Heart, released in 2004 through the label Sovereign Artists. Sovereign Artist's Marketing Director, Paul Angles, simultaneously released their album via file sharing networks, which were included in an amicus brief to the US Supreme Court. Two promo singles were released simultaneously with the album: "The Oldest Story in the World" which peaked at No. 22 on the Mainstream Rock Chart and "The Perfect Goodbye". Heart performed the song "The Perfect Goodbye" with country star Wynonna Judd on CMT's Crossroads in the summer of 2004.This album continued the move back to Heart's hard rock and folk rock roots, although only peaking at number 94 on the U.S. Billboard 200 Since then the Sovereign Artists record company closed and stopped doing business. It has been reported (via band sources on that Sovereign still owes Heart thousands of dollars.

This album has no RIAA certification.

The album cover bears an image of the Mandelbrot set, rotated so the main cardioid is oriented the same way a heart would normally be, with the cusp at top. The album cover art is seen briefly in the 2005 film Elizabethtown, for which Nancy Wilson did soundtrack music, and at the time was married to the film's director/screenwriter Cameron Crowe.


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 (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.

Mandelbrot set

The Mandelbrot set is the set of complex numbers for which the function does not diverge when iterated from , i.e., for which the sequence , , etc., remains bounded in absolute value.

Its definition and name are due to Adrien Douady, in tribute to the mathematician Benoit Mandelbrot. The set is connected to a Julia set, and related Julia sets produce similarly complex fractal shapes.

Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point , whether the sequence goes to infinity (in practice — whether it leaves some predetermined bounded neighborhood of 0 after a predetermined number of iterations). Treating the real and imaginary parts of as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence crosses an arbitrarily chosen threshold, with a special color (usually black) used for the values of for which the sequence has not crossed the threshold after the predetermined number of iterations (this is necessary to clearly distinguish the Mandelbrot set image from the image of its complement). If is held constant and the initial value of —denoted by —is variable instead, one obtains the corresponding Julia set for each point in the parameter space of the simple function.

Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization and mathematical beauty.


A microphone, colloquially named mic or mike (), is a device – a transducer – that converts sound into an electrical signal. Microphones are used in many applications such as telephones, hearing aids, public address systems for concert halls and public events, motion picture production, live and recorded audio engineering, sound recording, two-way radios, megaphones, radio and television broadcasting, and in computers for recording voice, speech recognition, VoIP, and for non-acoustic purposes such as ultrasonic sensors or knock sensors.

Several types of microphone are in use, which employ different methods to convert the air pressure variations of a sound wave to an electrical signal. The most common are the dynamic microphone, which uses a coil of wire suspended in a magnetic field; the condenser microphone, which uses the vibrating diaphragm as a capacitor plate; and the piezoelectric microphone, which uses a crystal of piezoelectric material. Microphones typically need to be connected to a preamplifier before the signal can be recorded or reproduced.

ORTF stereo technique

The ORTF stereo microphone system, also known as Side-Other-Side, is a microphone technique used to record stereo sound.

It was devised around 1960 at the Office de Radiodiffusion Télévision Française

(ORTF) at Radio France.

ORTF combines both the volume difference provided as sound arrives on- and off-axis at two cardioid microphones spread to a 110° angle, as well as the timing difference as sound arrives at the two microphones spaced 17 cm apart. The microphones should be as similar as possible, preferably a frequency-matched pair of an identical type and model.

The result is a realistic stereo field that has reasonable compatibility with mono playback. Since the cardioid polar pattern rejects off-axis sound, less of the ambient room characteristics is picked up. This means that the mics can be placed farther away from the sound sources, resulting in a blend that may be more appealing. Further, the availability of purpose-built microphone mounts makes ORTF easy to achieve.

As with all microphone arrangements, the spacing and angle can be manually adjusted slightly by ear for the best sound, which may vary depending on room acoustics, source characteristics, and other factors. But this arrangement is defined as it is because it was the result of considerable research and experimentation, and its results are predictable and repeatable.

These interchannel differences are not the same as interaural differences, as produced by artificial head recordings. Even the spacing of 17 cm is not strictly based on interaural ear spacing. The recording angle for this microphone system is ±48° = 96°.

Proximity effect (audio)

The proximity effect in audio is an increase in bass or low frequency response when a sound source is close to a directional or cardioid microphone.

Shure SM57

The Shure SM57 is a low-impedance cardioid dynamic microphone made by Shure Incorporated and commonly used in live sound reinforcement and studio recording. It is one of the best-selling microphones in the world. It is used extensively in amplified music and has been used for speeches by every U.S. president since its introduction in 1965. In 2004, honoring its four decades of "solid, dependable performance", it was inducted into the first-ever TEC Awards TECnology Hall of Fame.

Shure SM58

The Shure SM58 is a professional cardioid dynamic microphone, commonly used in live vocal applications. Produced since 1966 by Shure Incorporated, it has built a strong reputation among musicians for its durability and sound, and half a century later it is still considered the industry standard for live vocal performance microphones. The SM58 and its sibling, the SM57, are the best-selling microphones in the world. The SM stands for Studio Microphone.Like all directional microphones, the SM58 is subject to proximity effect, a low frequency boost when used close to the source. The cardioid response reduces pickup from the side and rear, helping to avoid feedback onstage. There are wired (with and without on/off switch) and wireless versions. The wired version provides balanced audio through a male XLR connector. The SM58 uses an internal shock mount to reduce handling noise.

A distinctive feature of the SM58 is its pneumatic suspension system for the microphone capsule. The capsule, a readily replaceable component, is surrounded by a soft rubber balloon, rather than springs or solid rubber. This gives notably good isolation from handling noise, one reason for its being a popular microphone for stage vocalists. Microphones with this feature are intended primarily for hand-held use, rather than on a stand or for instrument miking.

The SM58 is unswitched, while the otherwise identical SM58S has a sliding on-off switch on the body. Other suffixes refer to any accessories supplied with the microphone: when a cable is provided, the model is actually SM58-CN, while the SM58-LC has no provided cable; the SM58-X2u kit consists of the SM58-LC and an inline X2u XLR-to-USB signal adaptor (capable of providing phantom power for condenser microphones, and offering an in-built headphone jack for monitoring).

Sinusoidal spiral

In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates

where a is a nonzero constant and n is a rational number other than 0. With a rotation about the origin, this can also be written

The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including:

The curves were first studied by Colin Maclaurin.

Soundfield microphone

The Soundfield microphone is an audio microphone composed of four closely spaced subcardioid or cardioid (unidirectional) microphone capsules arranged in a tetrahedron. It was invented by Michael Gerzon and Peter Craven, and is a part of, but not exclusive to, Ambisonics, a surround sound technology. It can function as a mono, stereo or surround sound microphone, optionally including height information.

Space cardioid

The space cardioid is a 3-dimensional curve derived from the cardioid. It has a parametric representation using trigonometric functions.


A subwoofer (or sub) is a loudspeaker designed to reproduce low-pitched audio frequencies known as bass and sub-bass, lower in frequency than those which can be (optimally) generated by a woofer. The typical frequency range for a subwoofer is about 20–200 Hz for consumer products, below 100 Hz for professional live sound, and below 80 Hz in THX-approved systems. Subwoofers are never used alone, as they are intended to augment the low frequency range of loudspeakers that cover the higher frequency bands. While the term "subwoofer" technically only refers to the speaker driver, in common parlance, the term often refers to a subwoofer driver mounted in a speaker enclosure (cabinet), often with a built-in amplifier.

Subwoofers are made up of one or more woofers mounted in a loudspeaker enclosure—often made of wood—capable of withstanding air pressure while resisting deformation. Subwoofer enclosures come in a variety of designs, including bass reflex (with a port or vent), using a subwoofer and one or more passive radiator speakers in the enclosure, acoustic suspension (sealed enclosure), infinite baffle, horn-loaded, and bandpass designs, representing unique trade-offs with respect to efficiency, low frequency range, cabinet size and cost. Passive subwoofers have a subwoofer driver and enclosure and they are powered by an external amplifier. Active subwoofers include a built-in amplifier.The first subwoofers were developed in the 1960s to add bass response to home stereo systems. Subwoofers came into greater popular consciousness in the 1970s with the introduction of Sensurround in movies such as Earthquake, which produced loud low-frequency sounds through large subwoofers. With the advent of the compact cassette and the compact disc in the 1980s, the easy reproduction of deep and loud bass was no longer limited by the ability of a phonograph record stylus to track a groove, and producers could add more low frequency content to recordings. As well, during the 1990s, DVDs were increasingly recorded with "surround sound" processes that included a low-frequency effects (LFE) channel, which could be heard using the subwoofer in home theater systems. During the 1990s, subwoofers also became increasingly popular in home stereo systems, custom car audio installations, and in PA systems. By the 2000s, subwoofers became almost universal in sound reinforcement systems in nightclubs and concert venues.

Surround sound

Surround sound is a technique for enriching the fidelity and depth of sound reproduction by using multiple audio channels from speakers that surround the listener (surround channels). Its first application was in movie theaters. Prior to surround sound, theater sound systems commonly had three "screen channels" of sound, from loudspeakers located in front of the audience at the left, center, and right. Surround sound adds one or more channels from loudspeakers behind the listener, able to create the sensation of sound coming from any horizontal direction 360° around the listener. Surround sound formats vary in reproduction and recording methods along with the number and positioning of additional channels. The most common surround sound specification, the ITU's 5.1 standard, calls for 6 speakers: Center (C) in front of the listener, Left (L) and Right (R) at angles of 60° on either side of the center, and Left Surround (LS) and Right Surround (RS) at angles of 100–120°, plus a subwoofer whose position is not critical.

Surround sound typically has a listener location or sweet spot where the audio effects work best, and presents a fixed or forward perspective of the sound field to the listener at this location. The technique enhances the perception of sound spatialization by exploiting sound localization; a listener's ability to identify the location or origin of a detected sound in direction and distance. This is achieved by using multiple discrete audio channels routed to an array of loudspeakers.


Maxime Rodolphe Nouchy, known as Maxim Nucci and Yodelice (born 23 February 1979 in Créteil), is a French singer-songwriter who performs in English. He has released five albums as of 2014: "Maxim Nucci" (2006), Tree of Life (2009), Cardioid (2010), "Square Eyes" (2013) and "Like a Million Dreams" (2014). The songs belong to folk, rock and pop music. He is also known for his acting performance in Guillaume Canet's film Little White Lies (French title: Les Petits Mouchoirs) with Marion Cotillard in 2010. The song "Talk to me" was featured.


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