(This is actually a re-post, I accidentally addressed Brad only, sorry for that).
Hi Brad and others, On 03/24/2018 11:45 PM, Brad Klee wrote: > 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] https://ptpb.pw/AfTT.png [2] https://trustica.cz/en/2018/03/29/elliptic-curves-point-at-infinity/ [3] https://youtu.be/WnBEZ0qNdV0 _______________________________________________ Curves mailing list Curves@moderncrypto.org https://moderncrypto.org/mailman/listinfo/curves