# Cardioid

Cardioid
Field: Geometry
Image Created By: Henrik Wann Jensen

Cardioid

When a light source illuminates the inner surface of a metal ring, the resulting shape is a cardioid formed by light rays.

# Basic Description

In geometry, the cardioid is defined by the path of a point on the circumference of a circle of radius $R$ that is rolling without slipping on another circle of radius $R$. Its name is derived from Greek work kardioedides for heart-shaped, where kardia means heart and eidos means shape, though it is actually shaped more like the outline of the cross section of an apple.

The cardioid was first studied by Ole Christensen Roemer in 1674 in an effort to try to find the best design for gear teeth. However, the curve was not given its name until an Italian mathematician, Johann Castillon, used it in a paper in 1741.

Since the cardioid is also a roulette, more specifically an

, and a special case of a of Pascal, it is believed that it could have originated from Etiene Pascal's studies.

# A More Mathematical Explanation

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## Generating a Cardioid Using Other Shapes

#### Envelope

A cardioid can be formed by a set of circles:

1. Draw a fixed base circle, C, and a point, P, on the circumference of the circle.
2. Draw the set of circles centered on C and passing through P. To find this set of circles, we can do the following:
a. Pick any point on C and mark it with a blue dot.
b. Draw a circle whose center is at C, and passes through P.
• Repeat a and b for every point on C and you will generate a set of circles, shown below.

There is only one curve that is tangent to every circle in the set, and it is a cardioid shown in pink, E.

The process of deriving a new curve from a given set of curves in this manner is called taking the of those curves. This image shows intermediate steps in the process of drawing the set of circles. As more circles are added, it becomes more clear that the envelope is a cardioid.

To draw circles and see how they form a cardioid, drag the red dot around the circle below:

#### Evolute

To generate a new cardioid from an existing cardioid, we can take the following steps:

1. We begin with a cardioid, C, and draw circles tangent to C.
2. Mark the center point of each circle tangent to C.

Once many tangents circles are marked, we can see that their center points will form a smaller, mirror image cardioid, E.

The process of drawing circles tangent to a curve and marking their midpoints is called taking the of a curve.

#### Caustic

A cardioid can also be constructed using a circle and a light source using the following steps:

1. Begin with a circle, C, made of material that reflects light.
2. Place it on a surface, like a table top.
3. Pick a point, P, on the circumference of the circle.
4. Fix a light source at P, so that light rays hit the inside of the circle.

Light rays will be reflected off the circle in many directions, and the envelope of these rays will be a cardioid.

The process of reflecting light off a curve so that light rays form a new shape is called generating the of a curve. A cardioid can be produced in this manner because a cardioid is the caustic of a circle when the light source is located on the circle itself.

#### Conchoid

To generate a cardioid using a circle, we can perform the following:

2. Mark a fixed point on the circle, P.
3. Draw line segments of length 2d that cross P and have a midpoint on the circle.

As more line segments are added to the figure, the resulting cardioid becomes apparent.

This process is called taking the of a circle. When the fixed point is located on the circle itself and the length is twice the diameter of the circle, the conchoid will be a cardioid with a diameter twice the original circle's diameter.

#### Pedal

A cardioid can be generated circle by performing the following:

1. Start with a circle, C, and fix a point, O on the circumference.
2. Choose another point, P, on the circumference.
3. Draw a line that is tangent to C at point P.
4. Mark a point, Q, on this tangent line such that PQ and OQ are perpendicular.

If we repeat steps 2-4 for every point on the parabola, a cardioid will result.

This process is called taking the of a circle with respect to a fixed point on the circle.

#### Inverse

A cardioid can be produced from a parabola using the following steps:

1. Begin with a parabola, C, with a focus at a point, O.
2. Draw a fixed circle with a center at O and a radius k.
3. Pick a point, Q, on the parabola and extend a line that crosses O and Q.
4. Mark a point, P, so that P is located on the line OQ and satisfies the equation $OP \times PQ = k^2$.

If we repeat this process for the other points on the parabola, the new curve that will result is a cardioid.

This procedure gives us the of a parabola with respect to its focus. The cusp of the resulting cardioid will lie at the center of the circle.

## Equations for a Cardioid

When $r$ is the radius of the moving circle, the Cardioid curve is given by:

### Parametric equation

$x = 2R {\cos} t (1 + {\cos t})$

$y = 2R {\sin} t (1 + {\cos t})$

To derive the parametric equations for a cardioid, we must parametrize the location of the point on the rolling circle that traces out a cardioid, $S$ in terms of the radius of the circles, $R$ and the angle of rotation, $a_1$. Using the image, we can see that the position of $S$ is given by the equations

$x = MN + NO + PQ$

$y = OP + QS$.

It remains to parametrize each component of these equations in terms of $R$ and $a_1$.

We know that the two circles, $c_1$ and $c_2$ have a radius of $R$. Let the center of $c_1$ be located at $(R, 0)$. Then, the center of $c_2$ is located a distance of $2R$ from the center of $c_1$. Since the position of $c_1$ is fixed, $MN$ will always be equal to $R$. To find the length of $NO$, we can use the right triangle $NOP$. The hypotenuse of $NOP$ will always be $2R$, and we have already let the angle of rotation be $a_1$. Then we can define $NO$ as $2R \cos (a_1)$ and the $OP$ is $2R \sin (a_1)$.

To find the lengths of $PQ$ and $QS$, we can take similar steps. The triangle $PQS$ has an angle equal to $a_3$, which is equal to the sum of $a_1$ and $a_2$. To see why, notice that the two angles labeled $a_2$ are , which are always congruent. The two angles labeled $a_1$ are also congruent because they are , which are always congruent. Using this, we know that $PQ$ is equal to $R \cos (a_1 + a_2)$ and $QS$ is given by $R \sin (a_1 + a_2)$. Then, we can show that triangles $NTU$ and $PTU$ are congruent. We know that both triangles have one leg equal to $R$, and they share the leg $TU$. Since both triangles also have right angles between these sides, we know from the that they must be congruent. Therefore, $a_1$ is, in fact, equal to $a_2$, and we can let both $a_1$ and $a_2$ be equal to $t$.

Using substitution, we have

$x = R + (2R) \cos (t) + R \cos (2t)$

$y = (2R) \sin (t) + R \sin (2t)$

Then, we can factorize $R$.

$x = R (1 + 2 \cos (t) + \cos (2t))$

$y = R (2 \sin(t) + \sin (2t))$

Using the to further simplify the equations

$x = R (1 + 2 \cos (t) + 2 {\cos}^2 (t) - 1)$

$y = R (2 \sin(t) + 2 \sin (t) \cos(t) )$

Finally, we can simplify to the parametric form

$x = 2R \cos(t) (1+cos(t))$

$y = 2R \sin(t)(1 + cos(t))$

### Cartesian equation

$({x^2} + {y^2} - 2Rx)^2 = 4{R^2}({x^2} + {y^2})$.

We can show that the cartesian equation generates a cardioid by showing that it is equivalent to the parametric equations we derived above.

We can begin with the cartesian equation

$({x^2} + {y^2} - 2Rx)^2 = 4{R^2}({x^2} + {y^2})$.

and substitute the parametric equations, $2R \cos t (1+cos t )$ for $x$ and $2R \sin t (1 + cos t)$ for $y$.

After substituting, we have

$[(2R \cos t (1+cos t ))^2 + (2R \sin t (1 + cos t))^2 - 2R(2R \cos t (1+cos t ))]^2 = 4R^2[(2R \cos t (1+cos t ))^2 + (2R \sin t (1 + cos t))^2]$.

If we can simplify this equation to show that the two sides are, in fact, equal, we will have shown that this equation will generate the came cardioid as the parametric equations we derived in the previous section.

After expanding the left side of the equation,

$[(4R^2 \cos^2 t)(1 + \cos t)^2 + (4R^2 \sin^2 t)(1+ \cos t)^2 - 2R(2R \cos t (1 + \cos t))]^2$

we can begin to simplify. By factoring out $4R^2(1 + \cos t)^2$, we have

$[4R^2 (1 + \cos t)^2( \cos^2 t + \sin^2 t) - 2R(2R \cos t (1 + \cos t))]^2$

Using the pythagorean identity, this simplifies to

$[4R^2 (1 + \cos t)^2(1) - 2R(2R \cos t (1 + \cos t))]^2$.

Then, after expanding and combining like terms, we are left with

$16R^4 (1+cos t)^2$.

Now it remains for us to show that the right side of the equation is equal to the left side.

We have $4R^2[(2R \cos t (1+cos t ))^2 + (2R \sin t (1 + cos t))^2].$

After distributing the exponents,

$4R^2[4R^2 \cos^2 t (1 + \cos t)^2 + 4R^2 \sin^2 t (1 + \cos t)^2]$.

We can factorize $4R^2 ( 1 + \cos t)^2$, which leaves us with

$16R^4(1 + \cos t) [\cos^2 t + sin^2 t]$.

Again, using the pythagorean identity, we can see that this is equal to $16R^4 (1+cos t)^2$.

Since we have shown that the cartesian equation $({x^2} + {y^2} - 2Rx)^2 = 4{R^2}({x^2} + {y^2})$ is equal to the parametric equations we derived above, we know it will generate a cardioid.

### Polar equation

$P = 2R (1 - {\cos} {\theta})$

# Why It's Interesting

#### Cardioid Microphone

The cardioid microphone is popular type of microphone often used for live performances. It is particularly useful in these situations because it is most sensitive to sound coming from the front, so it picks up almost exclusively the desired sound, while minimizing ambient noise.

The cardioid microphone is not shaped like a cardioid. In fact, it is so named because the sound pick-up pattern is roughly heart shaped.

The image on the left shows a cardioid microphone's polar pattern, which indicates how sensitive it is to sounds arriving at different angles. These polar patterns represent the locus of points that produce the same signal level output in the microphone if a given sound pressure level is generated from that point. For example, a person talking at normal volume into the microphone at 0° will about 6 dB louder than someone talking at the same volume at 90° (into the side of the microphone). It may not appear that the sound intake is much less on the sides, but if two persons were speaking equidistant from the microphone, one directly at 0° and the other 90°, the person at 90° would sound as if he were twice as far from the microphone as the person at the front.

Cardioid microphones are able to reject sound that arrives from some directions allowing them to deliver clear sound even when there are a variety of undesired sounds nearby, like rustling papers during a speech or screaming fans during a concert.

#### Cardioid's in the Mandelbrot Set

The largest bulb of the Mandelbrot set is a cardioid, shown in black in this image. The Mandelbrot set is a fractal, so it exhibits the same pattern when viewed at any magnification. As a result, the set actually contains an infinite number of copies of the largest bulb, and the central bulb of any of these smaller copies is an approximate cardioid.

#### Caustics

A cardioid is the caustic of a circle when a light source is on the circumference of the circle. We can see this in a conical cup partially filled with coffee. When a light is shining from a distance and at an angle equal to the angle of the cone, a cardioid will be visible on the surface of the liquid.

A metal ring can also be used to create a cardioid, as in the main image on this page. When light is reflected onto the inner side of the cylinder before being focused onto the table, a cardioid caustic will appear on the table.