The Cardioid

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The Cardioid
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DESCRIPTION

• The word cardioid comes from the Greek root
  cardi meaning heart. The Cardioid curve is a
  special case of the epicycloid and the limacon of
  Pascal. It can also be defined as the curve
  traced by a point of a circle that rolls around
  the circumference of a fixed circle of equal
  radius without slipping.

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History

• Studied by Ole Christensen Roemer in 1674, it was
  discovered during an effort to try to find the
  best design for gear teeth. The curve was given
  its name by de Castillon in the philosophical
  ransaction of the Royal Society of 1741. The arc
  length was later discovered in 1708 by La Hire.
• However, since this cardioid is also a special
  case of the limacon of Pacscal, it is believed by
  some to have been originated from Etiene Pascals
  studies. (1588-1640)

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• Start with two circles with centers C¹ and C²
  with radius R.
• Circle C² is tangent to circle C¹ at point T and
  center C² is 2R from C¹.

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• Let angle TC¹D be the angle made by the line
  going through the centers of the two circles and
  the x-axis. Let it be known as q1 in our
  explanation.
• So far, we know that the coordinates of C² are as
  follows

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• To find the coordinates of the point A we can do
  what we did previously, but we have two unknown
  angles AC²E(q3) and AC²B(q²).
• Draw a line parallel to the x-axis through C² and
  point D (L1).

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• Now draw two adjacent isosceles-right triangles
  with their height being the distance from point T
  to the x-axis at point H, their bases are each R
  (along the segment C¹ to C²), and their
  hypotenuse be the distance from each circles
  center to point H.

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• Using the SAS theorem (side-angle-side) we know
  that since these two triangles share the right
  angle, their base and their height, that their
  other angles must be equal.
• In turn, using another geometric proof, we now
  can see that q³ is equal to the sum of q¹ and q²,
  AND that q¹ and q² are equal.

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• By making a second triangle you can see that from
  C² to point A is

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• Finally, let t be equal to the angle q¹.
• Therefore, the parametric equations (and
  coordinates of the point A that travels around
  the circle) for the Cardioid are

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Cardioid Using Parametric Equations

• tlinspace(0,2pi)
• R1
• x2Rcos(t)Rcos(2t)
• y2Rsin(t)Rsin(2t)
• plot(x,y)
• grid on
• axis square
• title('The Cardioid Using Parametric Equations')

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Cardioid Using Polar Coordinates

• thetalinspace(0,2pi)
• r2(1cos(theta))
• polar(theta,r)
• title('The Cardioid Using Polar Coordinates')

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LOCUS!
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Sources

• Xah Special Place Curveshttp//www.best.com/xah
  /SpecialPlaneCurves_dir/Cardioid_dir/cardioid.html
  
• Peter Gent Cardioidhttp//online.redwoods.cc.ca.
  us/instruct/darnold/CalcProj/Sp98/PeterG/Cardioid.
  html
• Sean Larson Introducing the Cardioidhttp//onlin
  e.redwoods.cc.ca.us/instruct/darnold/CalcProj/Sp98
  /seanL/cardioidf.html
• Limaconhttp//mathworld.wolfram.com/Limacon.html
• Cardioidhttp//en.wikipedia.org/wiki/Cardioid