Looks can be deceiving!

Look at the graph in the previous post. I was right! They don't intersect at a common point, rather the limit of the consecutive intersections is \left(1, \frac{1}{6} \right) .

Close up of the intersections of y^n + x^n -6xy = 0 for the first ten integers.

This is just numeric, from Mathematica ... I'll try to prove it later...

By the way an answer to the question of fining a curve orthogonal to the follium of Descartes is x^3 +y^3 +6xy= the reflection in the line y=-x. Do you know of any other trivial answers?

Goodnight! Tomorrow: Back to the generating functions, Owen has been bugging me!

Orthogonal Curves.

Points that call out to you.

The Follium of Descarts and it's family based on varible exponents.

The Follium of Descarts and it's family based on varible exponents.

x^3+y^3 = 6xy, x^4+y^4 = 6xy, x^5+y^5 = 6xy ... x^n+y^n = 6xy

You get the idea. Naturally I want to know what those intersections are! I need to go for a run before it gets dark, but It seems that if we fix x=1 then as n \rightarrow \infty, y \rightarrow \frac{1}{6}

Maybe the spots that look like common points aren't after all. Maybe it's just a very tight sequence of intersections.