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Apollonius's Ellipse and Evolute Revisited - The Geometry

Frederick Hartmann and Robert Jantzen

For a given ellipse


and an arbitrary point O(x0,y0) in the plane, depicted in Figure 1, the equation of the normal line through this point from the "foot" P(x,y) of that normal line is easily found.

FIGURE 1:  The normal line segment from the given point O and the point P together with the tangent line segment from P to a point on the major axis, a configuration whose geometry led Apollonius to the hyperbola whose intersection with the ellipse determines its normal lines.

By implicit differentiation, the slope of the tangent line is dy/dx = -(b2 x)/(a2 y), whose negative reciprocal is the slope of the normal line

y0 - y = (a2 y)/(b2 x) (x0-x) .
Clearing fractions leads to the equation of the hyperbola of Apollonius
xy(a2-b2) -x0 a2 y +y0 b2 x = 0 .

Thus the "feet" of the normals are the intersections of the ellipse with a rectangular hyperbola which passes through both the origin and the point O(x0,y0) and whose axes are rotated by an angle p/4 with respect to the coordinate axes. The problem of counting the normals is reduced to finding the number of such intersections as O(x0,y0) moves around in the plane. Figure 2 illustrates the situation where O is sufficiently close to the center of the ellipse and in particular inside the evolute, which is shown in the figure and explained in the next section.

FIGURE 2:  For a point just slightly off center, the four feet of the normals to the ellipse shift slightly from the center point configuration, where the hyperbola of Apollonius degenerates to its vertical and horizontal asymptotes along the axes of the ellipse.

For simplicity we assume a > b throughout our discussion so that the major axis is the x-axis. Note that the focal distance of the foci from the origin along the x-axis is then c=(a2-b2)1/2.

Frederick Hartmann and Robert Jantzen, "Apollonius's Ellipse and Evolute Revisited - The Geometry," Loci (August 2010)