calculus-2014-05-21

DRAFT c○ 2014 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse
6. Creating String Art
Here is a list of some beautiful algebraic curves in the plane:
• Rose Curve : (x 2 + y 2 ) 3 = 4x 2 y 2
• Hyperbola: x 2 /a 2 − y 2 /b 2 = 1, choose a and b
• Nephroid: (x 2 + y 2 − 4a 2 ) 3 = 108a 4 y 2 , choose a
• Lemniscate x 4 = x 2 − y 2
• Folium of Descartes x 3 + y 3 − 3axy = 0, choose a
• Serpentine Curve x 2 y + a 2 y − abx = 0, choose a and b
• Trisectrix of Maclaurin 2x(x 2 + y 2 ) = a(3x 2 − y 2 ), choose a
• Ambersand Curve (y 2 − x 2 )(x − 1)(2x − 3) = 4(x 2 + y 2 − 2x) 2
• Bean Curve x 4 + x 2 y 2 + y 4 = x(x 2 + y 2 )
• Bicuspid Curve (x 2 − a 2 )(x − a) 2 + (y 2 − a 2 ) 2 = 0, choose a
• Three-leaved Clover x 4 + 2x 2 y 2 + y 4 − x 3 + 3xy 2 = 0
• Deltoid Curve (x 2 + y 2 ) 2 + 18a 2 (x 2 + y 2 ) − 27a 4 = 8a(x 3 − 3xy 2 ), choose a
• Devil’s Curve y 2 (y 2 a 2 ) = x 2 (x 2 b 2 ), choose a and b
• Hippopede (x 2 + y 2 ) 2 = cx 2 + dy 2 , choose c and d
• Limacon (x 2 + y 2 − ax) 2 = b 2 (x 2 + y 2 ), choose a and b
• Astroid (x 2 + y 2 − 1) 3 + 27x 2 y 2 = 0
• Butterfly Curve x 6 + y 6 = x 2
Of course, this is a just a small list to give you some ideas. There are an unlimited number of
others. Two particularly useful libraries of curves are the National Curve Bank available at http:
//curvebank.calstatela.edu/index/index.htm and the Famous Curve Index available at http:
//www-history.mcs.st-and.ac.uk/Curves/Curves.html.
66. Independent Investigation: Find the graph of a function or an algebraic curve
that you really like and use GeoGebra to make your own piece of string art. You don’t
need to just take two lines to “attach” your strings. You can use a box or circle or
anything you want. Be creative! Did you use equal spacing on your line segments or
not?
67. Independent Investigation: Take your above curve and materials, like wood,
nails and string, or paper, thread and a needle to actually make your piece of string art.
Be creative!
6.1. Open Question. Is it always possible to find a frame of line segments for an algebraic
curve so that the tangent lines intercept the line segments with equal spacing? Or at least in a nice
pattern?
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