I draw a curve of the second order through 5 points. I can count how many times I get a hyperbola and how many times I get an ellipse. And what is the best way to calculate how many times you get hyperboles like 3 + 2, 4 + 1 and 5 + 0?
About this problem
https://sciendo.com/article/10.2478/rmm-2019-0003
V Points.sm (17 KiB) downloaded 27 time(s).

Hi Valery. One way is counting how many points are in each half-plane given by the vertical axis of the hyperbola. I can't upload files, I don't know why. This link is for a SMath live file, which do not work because it have plugins, but you can download it from there.

https://smath.com/en-US/cloud/worksheet/FNrErsL6

EDIT: I can upload files again.



V Points.pdf (100 KiB) downloaded 29 time(s).

V Points.sm (25 KiB) downloaded 30 time(s).

Best regards.
Alvaro.

Interesting problem!

I took a different approach from Alvaro to counting the points (mine is somewhat more long-winded).
I rotated the points and hyperbola so the separator is either vertical or horizontal.

However, I've been trying unsuccessfully to upload my SMath file for the last couple of days, so I guess the upload will have to wait!

Edit: Can now upload again!

V Points_2.sm (33 KiB) downloaded 26 time(s).

See please
https://community.ptc.com/sejnu66972/attachments/sejnu66972/PTCMathcad/178864/1/On%20the%20five%20points%20problem%20of%20V%20Ochkov.pdf
and
https://community.ptc.com/t5/Mathcad/Hyperbola-and-Ellipse-new-math-constant/m-p/495992