One obvious generalisation is the locus of a point with constant sum of distances from three given points. With just two points, cancellation yielded a nice quadratic polynomial; but with three, this luck runs out and you are lumbered with an octic. For example, if the three given points are (0,0), (1,0), (0,1), and their distances sum to d, the equation of the curve becomes 0 = 2 3 4 4 3 6 7 5 2 2 4 - 64 y x - 64 y x - 40 d y - 8 d - 24 y - 72 x y + 48 x y 4 2 2 2 4 4 3 2 4 8 + 48 x y + 64 x y - 32 d x + 16 d y x - 64 y x - 64 y x + 9 y 2 6 5 2 3 4 2 3 2 2 + 64 d y x - 24 y x - 72 y x - 72 y x + 96 d y + 32 d x y 5 2 2 2 5 2 3 2 3 2 2 + 56 x d - 64 d x + 80 y x + 96 d x - 96 d x y - 64 d y 2 2 2 4 6 2 3 3 4 4 3 + 32 d y x - 72 d x + 8 d x - 96 d y x - 72 x y - 40 d x 6 3 2 2 4 2 2 6 4 2 4 2 + 8 d y + 112 x d y + 64 d x - 28 d x + 64 d y - 40 d y x 2 4 4 2 4 2 3 2 2 2 2 2 + 56 x d y - 40 d x y + 56 y x d + 112 y x d - 144 d x y 4 4 6 8 6 4 4 6 2 8 4 4 + 30 d x + 16 y + 9 x + 16 x + 30 d y + 36 x y + d + 54 x y 2 6 4 2 2 4 2 5 2 2 4 2 2 4 2 + 36 x y + 60 d x y - 72 y d + 56 y d - 84 d x y - 84 d y x 4 4 2 6 6 2 6 2 7 5 - 32 d y + 16 d - 28 d y - 12 d x - 12 d y - 24 x + 80 y x 3 3 6 + 160 y x - 24 x y . (with proportional spacing turned off!) For 3 general points there are 2355 terms ... Don't tell me --- that was, like, so much more than you needed to know! Fred Lunnon On 11/11/12, Henry Baker <hbaker1@pipeline.com> wrote:
What would be the appropriate generalization of an ellipse to more than 2 foci ?
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