Assuming the two fixed points have coordinates (0,0) and (0,−2), the teardrop's equation is: x⁴ + 2·x²·y² + y⁴ + 10·x² − 6·y² + 8·y − 3 = 0. The sage commands used to perform the computation are here:https://gist.github.com/Gro-Tsen/35c2e3ed4ac8a3b04229f489f5b195df …
Defn: Defined on coordinates by sending (s : t) to (54*s^4 - 36*s^2*t^2 - 16*s*t^3 - 2*t^4 : -63*s^4 - 84*s^3*t - 26*s^2*t^2 - 20*s*t^3 + t^4 : 45*s^4 - 12*s^3*t + 22*s^2*t^2 + 4*s*t^3 + 5*t^4)
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A final picture: parametric_plot( (2*(27 - 18*t^2 - 8*t^3 - t^4)/(45 - 12*t + 22*t^2 + 4*t^3 + 5*t^4), -(63 + 84*t + 26*t^2 + 20*t^3 - t^4)/(45 - 12*t + 22*t^2 + 4*t^3 + 5*t^4) ), (t,-1e5,1e5), plot_points=1e6)pic.twitter.com/2Wt2w12BBE
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Just for fun, here's a "conformal neighborhood" of the curve. The small gap in the upper left is where |t|>100. I'm impressed that you guys could come up with both implicit and parametric formulae for the curve. I didn't know the latter was even possible.pic.twitter.com/r0pH4tRMSL
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