Four perpendiculars to an ellipse

Stand somewhere inside an oval running track and ask in which directions one could walk to reach the track at right angles. There are always at least two: straight to the nearest point of the track, and straight to the farthest, since a shortest or longest line to a curve always meets it squarely. But from some places there are four. Which places, and why exactly four and not three or five?

Apollonius of Perga, who had already shown that one cone gives all four curves, answered this for the ellipse in the fifth book of his Conics, written around 200 BC — the most sophisticated of the eight books, devoted to what he called the minimum and maximum lines from a point to a conic, and what are now called its normals. The boundary between two and four normals is a curve with four cusps, and it has turned out to be one of the most interconnected objects in elementary geometry: the locus of centres of curvature, the envelope of the normals, the set where the distance function changes its shape. This essay draws all three and shows they are one curve.

The equation of a perpendicular #

Write the ellipse as the points $(a\cos t,\, b\sin t)$, here with $a = 2$ and $b = 1.2$ — the same parametrisation used to measure the length round an ellipse. The direction perpendicular to the curve at parameter $t$ is $(\cos t / a,\, \sin t / b)$, and a line in that direction through the curve passes through a given point $(u, v)$ exactly when

$$a u \sin t - b v \cos t - (a^2 - b^2)\sin t\cos t = 0 .$$

This is one equation for the unknown $t$, and the substitution $s = \tan(t/2)$ turns its sines and cosines into rational functions of $s$ and the whole into a polynomial of degree four. So there are at most four normals from any point. The figures find them by scanning $t$ round the ellipse in two thousand steps, locating every change of sign and refining each by bisection; the count is the number of roots.

From the point in the hero at $(0.55, 0.18)$ the equation has four roots: the four orange lines. From $(2.3, 0.9)$, outside the ellipse and well away from it, it has two. A quartic can have four real roots or two or none, and here at least two are always guaranteed, by the shortest and longest distances; the question is when the other two exist.

Normals and the foci #

The normal to an ellipse has a classical description that the equation hides. At each point the two lines to the foci make equal angles with the tangent — the reflection property that sends every ray from one focus to the other — so the normal bisects the angle between the two focal lines. And through every point of the ellipse passes a hyperbola with the same foci, crossing the ellipse at right angles, as the two confocal families showed; the normal to the ellipse is the tangent to that hyperbola. So the family of normals is the family of tangent lines to the confocal hyperbolas, each taken at the point where its hyperbola meets this ellipse.

These descriptions say where a single normal goes. They do not easily say how many pass through a given point, which is why Apollonius’s question needed a different kind of argument. The algebraic route is the cleanest: a polynomial equation whose roots are the feet, and a criterion for when its roots are real.

Where four becomes two #

Counting over the whole plane draws the boundary. The tinted region, where four normals exist, is bounded by a stretched star with four cusps, two on the long axis of the ellipse and two on the short axis. Every one of the 3,776 cells agrees with the test of which side of the star its centre lies on. The star is

$$(a x)^{2/3} + (b y)^{2/3} = (a^2 - b^2)^{2/3},$$

a stretched astroid, and here its cusps lie at $x = \pm 1.28$ on the long axis and at $y = \pm 2.13$ on the short one — outside the ellipse, which reaches only to $y = \pm 1.2$.

That last fact is the most surprising part of the picture. Near the flat sides of an ellipse, a point outside the curve can still see four perpendicular directions to it: two to the nearby flat side and two to the far side. Near the pointed ends, points well inside the ellipse see only two. The region of four normals is not the inside of the ellipse at all; it is long in the direction where the ellipse is short.

The discriminant draws the boundary #

The quartic in $s = \tan(t/2)$ has real coefficients depending on the point $(u, v)$, and as the point moves its roots move continuously. Two real roots can disappear only by first colliding and becoming a double root, after which they leave the real line as a pair of complex roots. So the boundary between two and four normals is the set of points where the quartic has a double root, which is where its discriminant vanishes — the same polynomial in the coefficients that what the coefficients already know used to read the number of real roots without finding them, and that a resultant computes from a polynomial and its derivative.

Worked out for this quartic — a computation quoted here, not repeated — the discriminant vanishes exactly on the astroid, and that is how the equation of the star is derived. The equation of the boundary was therefore in principle available to anyone with the algebra of the seventeenth century, though the computation is heavy by hand. It is the same principle as the sign that decides which curve a quadratic equation in two variables draws: a single polynomial in the coefficients separates the qualitatively different cases.

The normals sweep out the star #

Why does the count change on that particular curve? Consider all the normals of the ellipse at once, each drawn as a full line.

The lines crowd together along the star. Each one touches it, and the star is their envelope, the curve that every member of the family is tangent to. The number of normals through a point is the number of these lines passing through it, and that number can change only where the family folds over on itself, at the envelope: on one side of a fold a point lies on two more lines than on the other. Inside the star every point lies on four of the lines; outside, on two; and on the star itself two of the four have merged into a single line tangent to the star.

The same kind of envelope appeared in the caustics of an elliptical billiard, where the family was the trajectories of a ball and the envelope a confocal conic. A caustic in optics is the envelope of light rays, which is why sunlight reflected in a cup makes a bright cusped curve on the coffee: the rays pile up where the family folds, exactly as the normals do here.

The distance from a point, round the ellipse #

There is a third reading, which explains why the count goes up in pairs.

Walk round the ellipse and record the distance to a fixed point. Wherever this distance stops increasing or decreasing, the line from the point to the ellipse is perpendicular to it, because moving along the curve does not change the distance to first order. So the feet of the normals are the critical points of the distance. A smooth function round a closed loop has at least one minimum and one maximum, which accounts for the two normals that always exist. Inside the star, the distance has an extra dip and an extra rise — a second, shallower minimum and a second maximum — and four critical points in all.

Moving the point across the star merges a minimum with a maximum. The dip flattens until the minimum and maximum meet in a point of inflection and then both disappear, which is why the count changes by two and not by one. This is the simplest case of what catastrophe theory calls a fold, and the cusps of the star are the next simplest, cusp catastrophes, where three critical points merge into one. The same accounting of critical points — minima, maxima and the saddles between them — is what peaks, pits and passes did on surfaces.

Centres of curvature #

The star has a fourth description, the one that gives it its classical name. At every point of the ellipse there is a circle that fits it best, matching its direction and the rate at which it bends; the curve that bends as far as it has gone built whole curves from this rate. The centre of that circle lies on the normal, at distance equal to the radius of curvature.

As the point moves round the ellipse, the centre of curvature moves along the star. At the sharp ends of the long axis the best-fitting circle is small, of radius $b^2/a = 0.72$, and its centre is a cusp of the star on the long axis. At the flat ends of the short axis it is large, of radius $a^2/b \approx 3.33$, with its centre far beyond the ellipse on the other side, the cusp on the short axis. The locus of centres of curvature is called the evolute, and its coordinates are

$$\left(\frac{a^2 - b^2}{a}\cos^3 t,\; -\frac{a^2 - b^2}{b}\sin^3 t\right),$$

whose cubes of cosines and sines are what make it an astroid. The evolute and the envelope of the normals are the same curve for every smooth curve, not only the ellipse: the normal at a point and the normal at a neighbouring point meet, in the limit, at the centre of curvature, and the meeting points of neighbouring members of a family are exactly what an envelope is made of. The four cusps of the evolute correspond to the four vertices of the ellipse, the points of greatest and least curvature, and the four-vertex theorem says every closed convex curve has at least four such points — so every such curve’s evolute has at least four cusps.

Points on the star itself are the borderline cases. At an ordinary point of the star two of the four feet have merged, so there are three distinct normals, one of them counted twice — the normal that touches the star there. At a cusp three feet merge: the point is the centre of curvature at a vertex, and the normal at the vertex osculates the family to higher order. From the cusp on the long axis, for instance, the only feet are the two ends of the axis — the near vertex counted three times and the far one once — so the four roots of the equation give just one normal line, the axis itself. Fewer than two distinct feet, or three, occur only on the star, a set of zero area: almost every point of the plane has exactly two normals or exactly four.

The parabola, three or one #

For a parabola the normal equation is a cubic, so the count is three or one. For $y = x^2/2$, the normal at $(s, s^2/2)$ passes through $(u, v)$ when $s^3 + 2(1 - v)s - 2u = 0$, and its discriminant vanishes on the evolute,

$$27x^2 = 8(y - 1)^3,$$

a curve with a single cusp at height 1 on the axis, the centre of curvature at the vertex. Above the evolute three normals; below it, one. A point on the axis below height 1 has only the normal straight down the axis; above it, two more appear, symmetric about the axis.

The evolute here is Neile’s semicubical parabola, and it has its own place in history: William Neile found its length exactly in 1657, the first curved arc whose length was expressed by algebra alone, at a time when many believed no algebraic curve could be rectified. That its arc length is algebraic is no accident. The length of an evolute between two points equals the difference of the radii of curvature at the corresponding points of the original curve, since the original is unwound from its evolute like a string from a spool. The same unwinding gives every evolute a length that can be read from the curve it came from.

Four questions with one astroid for an answer #

Four different questions have the same answer. Where do two normals become four? Where does the family of normals fold? Where do the centres of curvature lie? Where does the distance from a point change its number of minima? All four give the same four-cusped star. A geometer asking about perpendicular lines, an analyst asking about critical points, an optician asking about caustics and a differential geometer asking about curvature are describing one curve.

And Apollonius found it first, without coordinates, calculus or any notion of curvature. His fifth book determines, by synthetic arguments about areas and similar figures, exactly which points have how many normals, and the boundary he describes is the evolute — nineteen centuries before Christiaan Huygens defined evolutes in his work on pendulum clocks in 1673. How much of the modern picture Apollonius saw has been debated by historians since the book was recovered from Arabic translation in the seventeenth century.

Double roots on the evolute itself #

The counts come from scanning the parameter in finite steps, and a pair of roots closer together than one step would be missed. For points exactly on the evolute the equation has a double root, and the scan cannot reliably see it; that is why the agreement over the grid is between cell centres and the strict inequality, which no centre happened to sit on. The figures also use a single ellipse, with semi-axes 2 and 1.2. For an ellipse with $a < \sqrt{2}\,b$ the evolute lies entirely inside the ellipse, and the surprise of the region reaching outside disappears; the shape of the picture depends on the eccentricity.

Feet found by scanning, Apollonius quoted #

Computed here: the feet of the normals by sign changes on 2,000 points of the parameter and bisection; the count at points inside and outside the astroid, checked against the formula; the count at 3,776 grid cells, all agreeing with the astroid test; the envelope of 72 normals; the distance round the ellipse and its critical points; the circles of curvature and the position of the vertex’s centre at $a - b^2/a$; and the normals to the parabola from the cubic, three and one.

Quoted: Apollonius’s fifth book, Neile’s rectification, Huygens’s definition of evolutes, the four-vertex theorem and the vocabulary of catastrophe theory.

Still open: how many normals in higher dimensions #

From a point in space, how many normals can be drawn to an ellipsoid? The analogue of the quartic is a polynomial of degree six, and six normals are possible from points inside the region bounded by the ellipsoid’s focal surface. In general, for a smooth algebraic variety the maximum number of normals from a generic point is a fixed number, its Euclidean distance degree, introduced by Jan Draisma, Emil Horobeț, Giorgio Ottaviani, Bernd Sturmfels and Rekha Thomas in 2016 and now studied in optimisation and statistics, where it measures how hard it is to find the nearest point of a model to data. For many varieties of practical interest — sets of low-rank matrices, varieties from statistics — the Euclidean distance degree is known only by computer for small cases, and how many of the critical points can be real, the question Apollonius answered for the ellipse, is open for most of them.

The star of the normals #

From a point inside the four-cusped evolute of an ellipse four normals meet the curve; from outside, two. The evolute is at once the locus of centres of curvature, the envelope of the normals and the place where the distance from a point gains or loses a minimum and a maximum together. Counting normals over the plane finds the boundary exactly on the astroid, reaching outside the ellipse across its flat sides, and for the parabola the same construction gives a cubic and Neile’s semicubical parabola.