# 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.

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

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.