Cardioids & Limaçons

USING GEOMETER’S SKETCHPAD TO SUPPORT MATHEMATICAL THINKING

Shelly Berman p. 1 of 5 Jo Ann Fricker Cardioid Envelope.doc

Cardioids & Limaçons Construction of a Limaçon as the boundary of an envelope:

♦ Imagine a stationary circle C1, with a point P riding around its

circumference.

♦ Now imagine that point P is the center of a dynamic circle whose radius is defined by P and a second, stationary point Q.

♦ The envelope generated by the circles as P moves around C1 is a limaçon1.

♦ If the point Q is on circle C1, the boundary of the envelope forms a cardioid2.

♦ If the point Q is outside of circle C1, the boundary of the envelope forms a convex limaçon.

♦ If the point Q is inside of circle C1, the boundary of the envelope forms a dimpled limaçon.

1 The limaçon of Pascal was first investigated by Dürer, who gave a method for drawing it in Underweysung der Messung (1525). It was rediscovered by Étienne Pascal, father of Blaise Pascal, and named by Gilles-Personne Roberval in 1650 (MacTutor Archive). 2 The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741.

C1

Cardioid

P

Q

C1

Convex Limacon

Q

PC1

Dimpled Limacon

Q

P



Alternate Construction of a Cardioid as a locus: • Imagine two tangent circles:

a stationary circle C1, with another circle C2 rolling around the circumference.

• Now imagine a point P on

the rolling circle. The path described by P as C2 rolls around C1 is an epicycloid.

♦ If the radius of C1 is the

same as the radius of C2 then the epicycloids is a cardioid.

C2

C1

Roll for CARDIOID

P

Cardioid



Adapted from Exploring Precalculus with The Geometer’s Sketchpad 4.0 http://www.keypress.com/catalog/products/software/Prod_GSPModExpPrecalc.html

f θ( ) = a+a⋅cos θ( )

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Cardioids

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

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90°30°-180°

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2

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450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

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315°300°

285°270°255°240°

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Hide PolarHide Cartesian

Animate

f θ( ) = a-a⋅cos θ( )


Family of Cardioids


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90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

n = 1.00f θ( ) = a+a⋅cos θ-n⋅45( )


Family of Cardioids


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90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

f θ( ) = -a-a⋅cos θ( )


Family of Cardioids


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

f θ( ) = -a+a⋅cos θ( )


Family of Cardioids


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

f θ( ) = -a-a⋅cos θ-n⋅45( )n = 1.00


Family of Cardioids


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate




f θ( ) = -a-a⋅sin θ( )


Family of Cardioids


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animatef θ( ) = a-a⋅sin θ( )


Family of Cardioids


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

f θ( ) = -a-a⋅sin θ-n⋅45( )n = 1.00


Family of Cardioids


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animatef θ( ) = a+a⋅sin θ-n⋅45( )n = 1.00


Family of Cardioids


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

f θ( ) = a+a⋅sin θ( )


Family of Cardioids


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate f θ( ) = -a+a⋅sin θ( )


Family of Cardioids


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate




f θ( ) = a+b⋅sin θ( )b = 3.00


Family of Limacons


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 1.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

f θ( ) = a+b⋅sin θ( )b = 4.00


Family of Limacons


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 1.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

f θ( ) = a+b⋅sin θ( )b = 2.00


Family of Limacons


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

f θ( ) = a+b⋅sin θ( )b = 2.00


Family of Limacons


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = 3.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

f θ( ) = a+b⋅sin θ( )b = 2.00


Family of Limacons


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = -3.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animatef θ( ) = a+b⋅sin θ( )b = 3.00


Family of Limacons


--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

a = -2.0045°

-90° 540°450°360°270°180°60°0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°


Animate

Documents

Cardioids & Limaçons