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The Asteroid: Special Plane Curves

Benjamin O’Hanen and Matthew Wisan

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Ole Roemer

Figure 1: The man who found the Asteroid.

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Double Generation

Figure 2: Double Generation of the Asteroid.

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Parameterizing the Asteroid Part 1.1

Figure 3: Initial Setup.

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Parameterizing the Asteroid 1.2

Figure 4: Large Circle Parametrization

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x1 =3a cos(θ) (1)

y1 =3a sin(θ) (2)

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Parameterizing the Asteroid Part 2

Figure 5: Smaller Circle Parametrization.

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x2 =a cos(α) (3)

y2 =a sin(α) (4)

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Parameterizing the Asteroid Part 2.2Parameterized in terms of α, and θ.

x =3a cos(θ) + a cos(α) (5)

y =3a sin(θ) + a sin(α) (6)

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Relating α to θ

Figure 6: Relating the Angles α to θ.

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In terms of theta (θ) only,

x =3a cos(θ) + a cos(−3θ) (7)

y =3a sin(θ) + a sin(−3θ). (8)

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Simplifying Parametric Equations, for xFor the X-value equation we have,

x =3a cos(θ) + a cos(−3θ)

=3a cos(θ) + cos(2θ + θ)

=3a cos(θ) + a(cos(2θ) cos(θ)− sin(2θ) sin(θ))

=3a cos(θ) + a(cos3(θ)− sin2(θ) cos(θ)− 2 sin2(θ) cos(θ)

=a cos(θ)(3 + cos2(θ)− 3 sin2(θ))

=a cos3(θ) + a cos(θ)(3− 3 sin2(θ))

=a cos3(θ) + a cos(θ)(3(cos2(θ) + sin2(θ))− 3 sin2(θ))

=a cos3(θ) + a cos(θ)(3 cos2(θ))

=a cos3(θ) + 3a cos3(θ)

=4a cos3(θ)

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Simplifying Parametric Equations, for yAnd for the Y-value equation

y =3a sin(θ) + a sin(−3θ)

=3a sin(θ)− sin(2θ + θ)

=3a sin(θ)− a(sin(2θ) cos(θ) + cos(2θ) sin(θ))

=3a sin(θ)− a(2 sin(θ) cos2(θ) + cos2(θ) sin(θ)− sin3(θ))

=a sin(θ)(3− 3 cos2(θ) + 3 sin2(θ))

=a sin3(θ) + a sin(θ)(3(cos2(θ) + sin2(θ))− 3 cos2(θ))

=a sin3(θ) + a sin(θ)(3 sin2(θ))

=a sin3(θ) + 3a sin3(θ)

=4a sin3(θ)

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Asteroid Graphed

Figure 7: The Asteroid.

x =4a cos3(θ)

y =4a sin3(θ)

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Cartesian Equations

x2/3 + y2/3 =(4a)23 cos2(θ) + (4a)

23 sin2(θ)

=(4a)23(cos2(θ) + sin2(θ))

=(4a)23

Now replacing 4a with R we get

x2/3 + y2/3 = R23 (9)

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Bibliography

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References[1] Arnold, David. 1997, Special Plane Curves, Assignment,

http://online.redwoods.cc.ca.us/instuct/darnold/MULTCALC/CURVES/urves.htm

[2] Lockwood, E.H. 1967, A Book of Curves, Cambridge Univer-sity Press, New York

[3] Lawrence, J.Dennis 1972, A Catlog of Special Plane Curves,Dover Publications, Inc., New York

[4] Westfall, Richard S. 2006, The Galileo Project,http://galileo.rice.edu/lib/catalog.html

[5] Roemer Picture From a Unpronoucable Danish Website,http://www.danskekonger.dk/biografi/andre/pict/roemer.jpg