Slope of Tangent to an Ellipse

Date: 09/21/2001 at 12:48:23
From: Libby Phelps
Subject: Angle of an ellipse given a point on the ellipse

If I know a point on an ellipse, how do I find the angle of the 
ellipse at that point? I've tried using arctangent with the point on 
the ellipse and the center point, but I don't get the correct result.

Date: 09/21/2001 at 22:59:20
From: Doctor Peterson
Subject: Re: Angle of an ellipse given a point on the ellipse

Hi, Libby.

I suppose that by the "angle of the ellipse" you mean the slope of the 
tangent (or the angle of that tangent from the horizontal, which is 
the arctangent of the slope).

You can use calculus to determine the slope at a given point. 
Alternatively, you can use the fact that an ellipse is a "squashed" 
circle:

                    (x,ay/b)
          *********+
       ***    |   /*** +
      *      a|  /    *   +
     *        | /      *      +
    *         |/        *        +
    *---------*---------*-----------+--
    *         |    a    *
     *        |        *
      *       |       *
       ***    |    ***
          *********

                    (x,y)
         **********+
      ***    b|  /  ***  +
     *        | /      *      +
    *---------*---------*-----------+--
     *        |    a   *
      ***     |     ***
         ***********

The top picture is a circle, and the bottom one is an ellipse obtained 
by multiplying the y coordinate by b/a. The tangent will remain 
tangent, and will still have the same x-intercept, though the slope is 
b/a times the original slope. It is easy to find the slope of the 
tangent to the circle; so to find the slope of the ellipse at point 
(x,y) on the ellipse, consider the point (x,ay/b) on the circle, find 
the slope of the tangent there, and multiply that slope by b/a to find 
the slope of the tangent to the ellipse.

If you need help doing this, please write back and show me where you 
got.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/