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 P 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 P are the real roots of





Of course, the corresponding y coordinates for the normal points are given by



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 xnew = 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 gives a line so close to the normal line that it could easily be mistaken for it.


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