(This is actually a re-post, I accidentally addressed Brad only, sorry

for that).

Hi Brad and others,

Hi Dominik,

Some of these graphs are looking cool, but I'm not sure why your

worried about integers (m,n). If you include (m,n) then every point

will actually intersect infinitely many curves. The slopes of these

curves at the point of intersection are different per curve, but

reduce to one particular line over a finite field. So (m,n) could

really be considered extraneous, and parameter k quite sufficient.

you have to include at least m\in{-1,0}, n\in{-1,0} - you can see this

in[1] (I didn't want to draw my own, so I am reusing yours here).

Another thing to think about is intersection geometries combining

elliptic curves with a non-linear addition curve. For example, Edwards

curves have a hyperbolic addition rule. In this case the quadratic

hyperbolas are more similar to cubic elliptic curves than to lines.

They don't wrap the torus nicely, again require a big set to account

for all intersections.

Drawings of linear intersections over finite fields are now becoming

standard fare. We have seen continuous interpretation of hyperbolic

intersection in a number of references. I have yet to see hyperbolic

intersection over finite field, so perhaps this would be an

interesting direction to go in terms of depiction?

Now you have unveiled my secret long-term plan. Yes, I want to slowly go

through all the basics using simple Weierstrass form, then continue with

Montgomery form and finally show the hyperbolic intersections of twisted

Edwards over finite field (there are few things to be solved in that

case, but I am looking forward those challenges :) ).

Speaking of slowly going through all the basics, I produced another

article[2] and accompanying video[3] - this time about point at

infinity. Truth is, I am not satisfied with this one and I am thinking

about creating another one showing the actual intersections at point at

infinity on the projective plane.

Cheers,

Dominik

[1]

Loading Image...[2] https://trustica.cz/en/2018/03/29/elliptic-curves-point-at-infinity/

[3]