Difference between revisions of "Cardioid"

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===Parametric equation ===
 
===Parametric equation ===
  
<math>x = 2r {\cos}  t (1 + {\cos t})</math>
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<math>x = 2R {\cos}  t (1 + {\cos t})</math>
  
<math>y = 2r {\sin}  t (1 + {\cos t})</math>
+
<math>y = 2R {\sin}  t (1 + {\cos t})</math>
  
 
{{HideShowThis|ShowMessage=Click here to see a derivation of these equations.|HideMessage=Hide|HiddenText=  
 
{{HideShowThis|ShowMessage=Click here to see a derivation of these equations.|HideMessage=Hide|HiddenText=  

Revision as of 16:07, 21 July 2010

Inprogress.png
Cardioid
Metalring.jpg
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

Cardioid 1.gif
Apple cardioid3.jpg

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 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

Envelope

A c [...]

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.

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

Step by step envelope2.jpg

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.

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

Applet from http://web.me.com/paulscott.info/DC/cardioid/cardioid-gen.html.

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.

Cardioidevolute4.jpg

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.

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.

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.

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

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.

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.

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})

Paraderive5.jpg
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-angle-side 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 2(t))

Using the double angle formulas to further simplify the equations

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

 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

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


Why It's Interesting

Cardioid Microphone

Frontmic1.jpg

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.

Cardioid mic2.jpg

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.


Mandelbrot set.gif

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

Caustic coffee.jpg


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.


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