Normals From A Point To An Ellipse

Propositions 11 of Book 1 of Euclid's Elements describes how to 
draw a line perpendicular to a given line through a given point ON
the line, and Proposition 12 describes how to do the same thing
for a given point NOT on the given line.  These two constructions
are about equally trivial because they both have unique solutions.

If we replace the "given line" with an ellipse, the analog of
Proposition 11 is still easy, because for any given point on the
ellipse there is a unique line through that point perpendicular
to the ellipse.  We merely need to bisect the rays from the given
point to the two foci of the ellipse.  However, the analog of
Proposition 12 is substantially less trivial, because in general
there can be FOUR distinct lines that are each perpendicular to 
the given ellipse and that pass through the given point.  This 
multiplicity of "normals" is obvious if the given point is inside 
the ellipse, as shown below

   

However, even for external points, it is possible (though not
necessary) for there to be four distinct normals to an ellipse.
For example, there are four "normals" from the point (0.5,1.8) to 
an ellipse centered at the origin, with major axis of length a=2 
along the x axis, and minor axis of length b=0.7, as illustrated 
in the figure below:

   

Consequently, the construction of a normal to an ellipse through
a given point not on the ellipse involves the solution of a
general quartic.

Consider an ellipse centered at the origin in standard orientation,
with major and minor dimensions a,b.  Given a point P1 at the
coordinates (X,Y) not on the ellipse, the values of x of the points 
on the ellipse where the ellipse is perpendicular to the line through
P1 are the real roots of

      f(x)  =  c4 x^4  + c3 x^3 + c2 x^2 + c1 x + c0 = 0

where
         c4 =   (a^2 - b^2)^2
         c3 = - 2a^2 X (a^2 - b^2)
         c2 =   a^2 [a^2 X^2 + b^2 Y^2 - (a^2 - b^2)^2]
         c1 =   2a^4 X (a^2 - b^2)
         c0 = - a^6 X^2

Of course, the corresponding y coordinates for the normal points
are given by y = +- b sqrt[1 - (x/a)^2] with appropriate choice of
sign, depending on whether it is a convex or concave point of
normalcy.  The quartic can have either two real roots, or four
(not necessarily distinct).  For practical purposes we can use
Newton's method to quickly find a real root using the iteration
x_new = x - f(x)/f'(x), but care must be taken to ensure that we
are iterating toward the appropriate root.

Incidentally, there's a simple way of determining a very close 
estimate of the location of the "near side" normal point.  For any 
external point P, let the lines from the two foci F1-P and F2-P 
strike the ellipse at the points A and B respectively, and then
let C denote the intersection of the lines F1-B and F2-A.  The line
C-P is very nearly normal to the ellipse.  This construction can be 
performed with just quadratics, and is give a line so close to the
normal line that it could easily be mistaken for it.