[math-fun] Picture of Steve Gray's oval
Fred Lunnon sent me a picture of an oval based on Steve Gray's suggestion.
What about the locus of points such that d(A)+kd(B) is constant,
I pulled out a bunch of quotes from the list to put it in context: http://www.tiac.net/~sw/2007/02/Steve_Gray_oval/index.html --Steve
My oval equation works in 3 dimensions with no change, giving an oval of rotation. With the right choice of k [as in d(A) +c*d(B) = 2b (string length), and a good choice for 2a (A,B separation)] it can probably be made to look just like a chicken egg. Think of the locus point X as having a small, frictionless ring that the string slides through. What would the locus of d(A) + c*d(B) + e*d(C) = constant look like in 3-D? An oval of nonrotation? A lumpy egg from a sick chicken? This can be implemented mechanically with a string that starts at A, goes through a ring at point X (the locus point), loops back to a ring at B, goes straight to a ring C, then back to X where the end is tied. It has to slide frictionlessly through the small rings at B, C, and X. If this is not clear, let me know and I will try to confuse it further. If it's not correct, it's someone else's fault. Steve Gray Steve Witham wrote:
Fred Lunnon sent me a picture of an oval based on Steve Gray's suggestion.
What about the locus of points such that d(A)+kd(B) is constant,
I pulled out a bunch of quotes from the list to put it in context:
http://www.tiac.net/~sw/2007/02/Steve_Gray_oval/index.html
--Steve
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The polynomial equation is degree 8 and rather large: too large, in fact, for either Maple's implicitplot3d() to cope with even approximately, or to be cut-and-pasted into Richard Morris's Algebraic Surface Solver. The equation with square roots plots OK under Maple, but an exported graphics file does not permit interactive viewing, and is not terribly informative by itself. In the special case when all the weights are equal, and the "focal" points lie on an equilateral triangle, the surface resembles a rounded, 3-cornered cushion: at the critical length where the string is tightly stretched at the foci, they develop pointed cusps. Apart from loss of symmetry, I can't see anything surprising will develop when the parameters are further varied! WFL On 2/23/07, Steve Gray <stevebg@adelphia.net> wrote:
My oval equation works in 3 dimensions with no change, giving an oval of rotation. With the right choice of k [as in d(A) +c*d(B) = 2b (string length), and a good choice for 2a (A,B separation)] it can probably be made to look just like a chicken egg. Think of the locus point X as having a small, frictionless ring that the string slides through. What would the locus of d(A) + c*d(B) + e*d(C) = constant look like in 3-D? An oval of nonrotation? A lumpy egg from a sick chicken? This can be implemented mechanically with a string that starts at A, goes through a ring at point X (the locus point), loops back to a ring at B, goes straight to a ring C, then back to X where the end is tied. It has to slide frictionlessly through the small rings at B, C, and X. If this is not clear, let me know and I will try to confuse it further. If it's not correct, it's someone else's fault.
Steve Gray
Steve Witham wrote:
Fred Lunnon sent me a picture of an oval based on Steve Gray's suggestion.
What about the locus of points such that d(A)+kd(B) is constant,
I pulled out a bunch of quotes from the list to put it in context:
http://www.tiac.net/~sw/2007/02/Steve_Gray_oval/index.html
--Steve
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participants (3)
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Fred lunnon -
Steve Gray -
Steve Witham