the asteroid: special plane curves - college of the redwoods · 2008-12-16 · 1/17 the asteroid:...
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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