Date: Nov 11, 1997 3:05 PM
Author: John Conway
Subject: Re: construction of triangle of given perimeter, given point and angle



On Tue, 11 Nov 1997, Eileen M. Klimick Schoaff wrote:

> 
> A parabola, on the other hand, is determined by any 4 of its points. 
>  John Conway>
> 
> Am I missing something here?  Don't 3 points determine a parabola if the axis
> of symmetry is either vertical or horizontal?

   Yes they do.   But not if it isn't.   You did miss something!

  But if we consider any axes,
> then there are an infinite number of parabolas passing through three points. 

    Yes, this is true.  But I spoke of 4 points, not 3.

> The generic equation is ax^2 + bxy +cy^2 + dx + ey + f = 0.

    This is the general conic, which is usually an ellipse or hyperbola
   rather than a hyperbola.

  If Jon Roberts is
> considering parabolas of the form y = ax^2 + bx + c, then knowing 3 points
> gives you three equations with three unknowns which can easily be solved --
> unless there is no solution.
> 
> In the April 1997 issue of the Mathematics Teacher, a colleague of mine, Dr.
> Ellie Johnson, wrote an article "A Look at Parabolas with a Graphing
> Calculator".  In this article she using the calculator to generate many
> solutions to the generic equation.  Of course this just shows that given three
> points and restricting yourself to a parabola of the form y = ax^2 + bx + c,
> you can derive the equation.  That does not, of course, construct it.
> 
> Does the fourth point determine whether the axis of symmetry is vertical,
> horizontal, or rotated?

   Yes, roughly speaking.

  In ax^2 + bxy +cy^2 + dx + ey + f = 0, it looks like
> you need more than 4 points to determine a, b, c, d, e, f.

    You do indeed need 5 points to determine the general conic.

> Then again, I am only a math education person and do not have a PhD in math so
> I am probably far in the dark.
> 
> Eileen Schoaff
> Buffalo State College
> 
    I think as a math educator you really SHOULD have known of the
existence of conics other than parabolae!
 
             John Conway