The Hyperbola

[John Harris]

I've looked at the wikipedia entry for hyperbola but I can't think how to rotate one or shift its origin.

If I have three coordinates, how do I express the hyperbola that passes through all of them?

Let's say

x,y=-1,0
x,y=0,1
x,y=infinity,10

What form of notation should I be looking for?

 

[John Harris]

Well yes, Denis. What I doubtless had drummed into me as a teenager is how to rotate by 45 degrees and shift an axis one bit to the right, but it' evaporated. It's why I came here for a clue. What would you suggest?

 

[John Harris]

It's a practical question, I'm writing a simulation page to show, among other things, the effects of alternative progressive tax systems and an assortment of poverty traps. I'm stuck without this hyperbolic approach to a maximum figure, and the variable initial gradient at which the tax is applied. I do wish I had a comprehensive text book to crib from but it's frustrating to know there are people who could get me back on track but they choose not to for reasons of principle.

 

[HallsofIvy]

So you want a hyperbola that passes through the points (1, 0) and (0, 1) and such that y approaches 10 as x goes to infinity. Any hyperbola with vertical and horizontal asymptotes is of the form y= B/(x- a)+ C The fact that "y approaches 10 as x goes to infinity means that C= 10. So we have y= B/x- a)+ 10. y= 0 when x= 1 means B/(1- a)+ 10= 0 and y= 1 when x= 0 means B/(0- a)+ 10= 1. That last equation gives -B/a= -9 or B= 9a. Putting that into the first equation, 9a/(1- a)+ 10= 0. 9a/(1- a)= -10. 9a= 10(1- a)= 10- 10a. Adding 10a to both sides, 19a= 10 so a= 10/19. And since B= 9a, B= 90/19. The equation is y= (90/19)/(x- 10/19)+ 10.

 

[John Harris]

Thank you, that seems to cover exactly what I need - I'll slip those bits into javascript and see what plots I can generate. I think you have a typo at "through the points (1, 0) and (0, 1)" which ought to read "through the points (-1, 0) and (0, 1)". I'll be a few hours working on my code. I'm very grateful.