Cardioid

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Cardioid

A Cardioid is a pattern defined by the path of a point of the circumference of a circle that rotates around another circle.


Contents

Basic Description


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


A More Mathematical Explanation

Generating a Cardioid Using Other Shapes

Using different methods we can generate the cardioid im [...]

Generating a Cardioid Using Other Shapes

Using different methods we can generate the cardioid image.

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. Pick any point on C and mark it with a blue dot.
  3. Draw a circle whose center is at C, and passes through P.

Repeat 2 and 3 for many points on C, and you will generate a set of circles, shown in the image 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 envelope 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.

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 evolute 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 diffuse 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.

Image:Cardioidcaustic1.jpg

The process of reflecting light off a curve so that light rays form a new shape is called generating the caustic 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:

  1. Start with a circle, C, with diameter, d.
  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.

Image:Stepcardioid.jpg

This process is called taking the conchoid 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. [1]

Pedal

A cardioid can be generated using a 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 circle, a cardioid will result.

Image:pedal 1.jpg

This process is called taking the pedal 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.

Image:inverse4.jpg

This procedure gives us the inverse 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 vertically opposite angles, which are always congruent. The two angles labeled  a_1 are also congruent because they are alternate exterior angles, 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 side-side-angle rule 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 double angle formulas 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})
Then we substitute the parametric equations 2R \cos t (1+cos t ) for  x
 2R \sin t (1 + cos t) for y
After Substituting we have this [(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 deduced that this equation will generate the same cardioid as the parametric equations we derived previously.

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

One popular type of microphone is the cardioid microphone. 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. This image should be interpreted as follows:

  • Each point on the black cardioid corresponds to a combination of an angle at which the sound enters the microphone and a volume of sound being input.
  • Each point along this cardioid will produce the same sound level output of the microphone's speakers.

You might notice that a sound being picked up from the top of the microphone (at 0°) can be input at almost 0 decibels and still produce the same volume as if a sound that was 6 decibels louder was picked up from the side of the microphone (at 90°). If you are not familiar with the decibel scale used for sound, this may not seem particularly noteworthy, However, this is the same as saying 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.


The main advantage of cardioid microphones over other types of microphones is their extremely "tight" cardioid pattern. Cardioid microphones are designed to only pick up things that close to it, allowing for it to be used with minimal sound interference. This ability makes cardioid and hyper-cardioid microphones especially useful for live performances and loud settings. In these situations, a cardioid microphone can deliver clearer sound even when there are a variety of loud sounds nearby.

Cardioid microphones are the industry standard choice for live settings, but do not have the same quality as other types of microphones. Typically in a studio setting more a sensitive microphone would be used.

Cardioid's in the Mandelbrot Set

The Mandelbrot set is a fractal, which shows the same patterns at any level of 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. The largest bulb of the Mandelbrot set is in the shape of a cardioid, shown in black in this image.

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.


Image Gallery

References

  1. http://mypages.iit.edu/~maslanka/conchoids2009.pdf


Teaching Materials

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Future Directions for this Page

This page was created by Liza, worked on by Becky, and still needs some changes:

  • Work on the more mathematical section:
  • Specifically, the caustic section is probably the most confusing. We're still unsure as to what exactly makes the cardioid get formed by light rays. The images in this section (there are two right next to each other-- the black one and the one with the metal ring on the table) need explanations that are more clear. The issue is that the one of the metal ring is computer generated and we don't know where the light source was positioned.
  • Equation of a cardioid section
  • Equations should be numbered
  • Prove/derive the polar equations for a cardioid
  • Why it's interesting
  • Cardioid microphone section is still confusing
  • The wikipedia page on cardioids says that the image of any circle through the origin under the map z -> z^2 is a cardioid. Someone could expand on this idea and write a section on this.



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