Everywhere a throw can reach

A projectile launched at a fixed speed can be aimed anywhere, and the set of places it can then reach is not the whole plane.

That much is obvious. What is not obvious is that the region has a sharp edge with an equation, that the edge is itself a parabola, that its focus is the muzzle, and that a point strictly inside it can be hit in exactly two ways rather than one.

The boundary has a name — the parabola of safety, Torricelli’s, from 1644 — and it is one of the earliest examples of a construction that turns up everywhere once it is recognised. What follows is the construction, the arithmetic that produces it, and the reason the same object appears in the sky whenever it rains.

The boundary is not a trajectory #

Every trajectory in the family starts at the origin and ends on the ground. The boundary starts at the maximum height directly overhead, comes down on both sides, and reaches the ground at the maximum range. It is not any of the launches and no launch follows it.

Work in units where the launch speed and gravity are both one, so the greatest range is $1$ and the greatest height is $\tfrac12$. A launch at angle $\theta$ traces the parabola every projectile traces,

$$y = x\tan\theta - \frac{x^2}{2\cos^2\theta},$$

and the question is what the highest of these is above a given abscissa. Differentiating with respect to the angle at fixed $x$ gives the tangency condition $\tan\theta = 1/x$ — steep launches govern the near ground, shallow ones the far — and substituting it back gives the boundary,

$$y = \frac{1}{2} - \frac{x^2}{2}.$$

The generator finds that maximum numerically, by searching over launch angles at each abscissa, and compares it with the formula. They agree to the last digit the arithmetic has, which is what makes the curve a measurement of the family rather than a claim about it.

Two features of the boundary are worth naming. Its vertex is at $v^2/2g$, which is exactly how high a vertical throw goes — the one launch whose apex lies on the boundary because it touches it at the top. And its focus is the launch point itself: every point of the parabola is as far from the gun as from the horizontal line at height $v^2/g$, which is the height a projectile would reach if the whole of its kinetic energy went into climbing at twice the actual speed.

That focus is not a coincidence and it has a physical reading. The directrix at $v^2/g$ is the level at which a projectile of this speed would have zero kinetic energy if gravity had somehow been allowed to act on it for the whole journey twice over; more usefully, $v^2/g$ is the total energy per unit mass expressed as a height, counting the kinetic energy already present. A point on the boundary is exactly as far from the muzzle as it is below that level, so reaching the edge of the reachable set is a statement about spending the whole energy budget on the trip — and the focus–directrix definition of a parabola is the geometric form of that statement. Every conic in mechanics arrives this way, as a locus defined by an energy that has run out somewhere.

Two ways in, one way to the edge, and no way beyond #

The nicest way to see what the boundary is doing is to ask, for a given target, how many launches reach it.

Substituting $\tan\theta = t$ into the trajectory and rearranging gives a quadratic in $t$:

$$\frac{x^2}{2}t^2 - xt + \left(y + \frac{x^2}{2}\right) = 0.$$

A quadratic has two roots, one root, or none, and the discriminant decides which. Working it out, the condition for real roots is exactly $y \le \tfrac12 - \tfrac{x^2}{2}$, which is the boundary again — arrived at from a completely different direction, without differentiating anything or thinking about envelopes.

So the boundary is where a discriminant vanishes. Inside it there are two launch angles; on it the two coincide; outside it they are complex, which is the algebra’s way of saying no aiming works.

The two solutions inside are the flat trajectory and the lobbed one, and everything an artillerist argues about lives in the difference between them. The flat one arrives sooner and harder and with a shallower impact angle; the lobbed one takes longer, clears whatever is in the way, and comes down more nearly vertically. Mortars exist because the second root exists, and so does the distinction between direct and indirect fire, which is a distinction between the roots and not between the weapons.

The two roots differ by more than their shape. For a target at half the maximum range on level ground, the flat root is at $15°$ and the lobbed one at $75°$; their flight times are in the ratio of their vertical launch components, so the lobbed shell is in the air $3.7$ times as long, and it arrives at $75°$ from the horizontal where the flat one arrives at $15°$. Against a wall the flat shot is useless and the lobbed one is the only option; against a moving target the flat shot is the only option, because a four-fold longer flight is a four-fold larger error in where the target will be. The apex heights differ by more still — $\tan^2$ of the launch angle, a factor of nearly fourteen — which is why one of the pair can clear a ridge and the other cannot.

For a target on the ground the two roots are complementary angles, which is the pairing the first rung on this ladder is about. For a target above the ground they are not complementary, and the symmetry that makes the ground case memorable is a special case rather than the rule.

Why a boundary made this way is bright #

Here is the connection that makes the safety parabola more than a curiosity about cannons.

A boundary of a family of curves, touched by each member once, is called an envelope. Envelopes have a characteristic property: near the envelope, two members of the family are close together, and on it they coincide. Anything that counts family members per unit area therefore diverges there.

For the projectile that count is a probability. Fire shells at the same speed and random angles and ask where they land. The landing distance is $R(\theta) = \sin 2\theta$ in the units above, and the density of landing points goes as $|\mathrm{d}R/\mathrm{d}\theta|^{-1}$, which blows up where the range is stationary — at 45°, the maximum. A wide spread of angles about 45° produces a narrow pile of impacts at maximum range, and the pile has an integrable singularity at its edge, falling as the inverse square root of the distance from it.

The exponent is worth extracting because it is the same one everywhere. Near a stationary point the range is quadratic in the angle, $R \approx R_{\max} - c(\theta - \theta^*)^2$, so the angles that land within a distance $\delta$ of the maximum form an interval of width $\sqrt{\delta/c}$ — and the density of impacts per unit distance is therefore proportional to $\delta^{-1/2}$ — the same square-root pile-up a stationary phase produces in a wave. A square-root divergence is what a smooth maximum always produces when its argument is sampled uniformly, and it is integrable, so the total number of shells landing near the edge is finite even though the density is not.

Rays of light do exactly this. A family of rays with a boundary produces a surface where neighbouring rays cross, and because the number of rays per unit area diverges there, so does the brightness. Such a surface is a caustic, and the argument that produces it is the argument above with light substituted for shells.

A caustic is what makes such a boundary bright, and the safety parabola is one in exactly the sense a rainbow is. Rays entering a raindrop at a range of impact parameters leave over a range of angles, and near the stationary point of that relation many entering rays leave in nearly the same direction — so the light piles up there and nowhere else. The parabola does the same thing with trajectories rather than rays: near the envelope, a spread of launch angles arrives at nearly the same place.

The rainbow is that caustic, and its angle is where the deviation is stationary. The bright rim of light thrown on the bottom of a cup by its curved side is another.

The same construction is what a spherical mirror’s aberration is. Rays reflected from a sphere do not cross at a point; they cross on a cusped surface which is the envelope of the reflected family, and the cusp’s shape is the aberration. In each of these three cases the interesting object is not any member of a family of curves but the curve the family cannot cross, and it is found the same way in all of them.

What the divergence really signals is that the description has stopped being adequate rather than that anything is infinite. Rays are a short-wavelength approximation, and where the ray density blows up the approximation fails; the wave treatment replaces the infinity with a finite peak of a definite width, decorated with fringes on the bright side. For light that repair is the Airy function and the fringes are the supernumerary bows below a rainbow. For projectiles there is no wave to appeal to, and the corresponding repair is statistical: shells are not launched at perfectly equal speeds, and the spread in speed smears the singular pile-up into a peak of finite height whose width measures the gun rather than the geometry.

The mathematics does not distinguish between the cases. What the projectile family and the ray family share is a one-parameter set of curves with a boundary, and every conclusion above — the tangency, the vanishing discriminant, the divergent density, the two-solutions-becoming-one — holds for both. That is why the same square-root singularity turns up in the intensity beside a rainbow, in the pile-up of impacts at maximum range, and in the fringe spacing of the supernumerary bows where the wave treatment repairs the divergence the ray treatment produces.

Where it stops, and it stops early #

Air. The whole construction assumes the only force is gravity. Add drag and the trajectory is no longer a parabola, no longer symmetric, and no longer given in closed form; the reachable region still exists and still has a boundary, but the boundary must be found numerically and is not a conic.

The size of the correction is set by the drag force at launch measured in weights, and the second rung on this ladder works it out: for a golf ball that parameter is about two, and the optimum angle drops well below 45°. A safety parabola drawn for a real projectile is a considerable overestimate of what it can reach.

The ground. The construction assumes a flat Earth and a uniform gravitational field, which is exactly the approximation that fails when the range becomes comparable with the planet.

It stops early, and the first thing to go is the parabola itself. A projectile’s true path is a conic with a focus at the Earth’s centre, so it does not close and is not a parabola; the classroom parabola is what a small piece of one ellipse looks like when the ground beneath it is treated as flat and gravity as uniform. Over a cricket pitch the difference is unmeasurable, and over a few hundred kilometres it is the whole answer.

The true path of a thrown stone is an ellipse with one focus at the centre of the Earth, and the parabola is its small-arc limit. How small the correction is can be put as a number. The parabolic and elliptic paths for a launch of range $R$ on a planet of radius $a$ differ in apex height by a fraction of order $R/a$; for a cricket ball thrown eighty metres on the Earth that is about $10^{-5}$, which on an apex of twenty metres is a fifth of a millimetre — comfortably below the effect of the seam. For an intercontinental ballistic missile it is the whole problem, and the reachable region becomes a spherical cap whose boundary is set by a rather different piece of arithmetic. In that regime the maximum-range angle is no longer 45° either: it falls as the range approaches half the planet’s circumference and the trajectory begins to spend its time in a genuinely curved field.

And the speed is not always fixed. Everything here follows from one number being held constant, which is the right idealisation for a gun and the wrong one for a rocket, where the reachable set is decided by a fuel budget rather than by a muzzle velocity.

The second thing to go is the premise that the launch is instantaneous. A rocket’s reachable set is bounded by a total change in speed rather than by a speed, because it goes on pushing after it has left — and almost nothing about the projectile envelope survives that substitution. The safety parabola is a statement about objects given all their momentum at once, which covers thrown stones and shells and nothing that carries its own engine.

The idea generalises, and the generalisation has a name #

An envelope is what happens when a family of solutions to a differential equation has a singular member — one that is not obtained by choosing a constant in the general solution, but is instead tangent to all of them. Clairaut studied exactly this in the 1730s, in an equation now named after him, whose general solution is a family of straight lines and whose singular solution is their envelope.

That is a formal reason to expect envelopes wherever a parameter is being swept, and they duly appear: in the shape of a lens’s focal region, in the boundary of a set reachable by a controlled system, in the ridge lines where a fold of a mapping projects to the plane. The last of those is the general theory — Whitney’s, from 1955, and Thom’s afterwards — which classifies the ways a family can fold and shows that the fold and the cusp are the only two that survive small changes. A caustic is a fold, generically, and a cusped caustic like the mirror’s above is where two folds meet.

The projectile case is the easiest one in which every part of that machinery is visible at once and none of it is needed: the family is explicit, the envelope is a conic, the discriminant is a quadratic’s, and the whole of it can be checked by hand.

The picture and what it hides #

The envelope is not a barrier. Nothing is different about the air just beyond it. What the boundary separates is not two kinds of place but two kinds of aiming problem — two solutions from none — and the same point in space moves from one side to the other if the muzzle velocity changes by a per cent. The reachable region grows as the square of the speed in both directions, so a one per cent faster gun reaches two per cent further and two per cent higher, and the whole boundary scales rather than deforming: every safety parabola is the same parabola under a change of units, which is the sense in which there is only one figure here at all.

And it is a boundary for one launcher. Two guns of different speeds have different envelopes, and a target unreachable by one is comfortably inside the other’s. The safety parabola is a statement about a weapon rather than about a battlefield, which is presumably why the name survived from a period when the distinction was being made for the first time in print.

What the picture hides is that it is one-dimensional underneath. A vertical throw is a particle in a potential that rises linearly with height, and the vertex of the safety parabola is one of that problem’s turning points — the height at which a given energy runs out. Everything the envelope says about the horizontal is bookkeeping laid over that single fact, which is why the maximum range and the maximum height are the same number in disguise.

The one thing the figures genuinely cannot show is the third dimension. Everything here is drawn in the vertical plane containing the launch direction, and the real reachable set is that region rotated about the vertical axis — a paraboloid, not a parabola. Rotating it changes none of the arithmetic and does change one thing worth noting: the volume near the boundary is larger than the two-dimensional picture suggests, so the pile-up of impacts near maximum range is spread around a ring rather than concentrated at a point, and the density singularity is correspondingly softened.

What sweeping a parameter always produces #

It is worth stating the moral separately from the projectile, because the projectile is the easy case and the moral is general.

Whenever a physical family is generated by turning one knob — a launch angle, an impact parameter, a ray height, a control input — three things happen together at the edge of what the family covers. Two members of the family merge, so a count of solutions drops by two. A derivative with respect to the knob vanishes, so whatever quantity the family maps onto is stationary. And a density computed per unit of that quantity diverges, as an inverse square root, because a quadratic maximum is being sampled.

Those are the same event described three ways, and recognising any one of them predicts the other two. An artillerist who notices that two firing solutions have merged knows without further calculation that the range is stationary and that shells will pile up; an optician who notices a bright line in a beam knows that two rays have merged there and that a small change of geometry will split it into two. The habit worth acquiring is the translation, not any of the three individual facts.

The ladder from here #

Later rungs on this anchor: the envelope with drag, computed rather than drawn, and how much smaller it is than Torricelli’s; the reachable set of a projectile launched from a moving platform, where the family gains a second parameter and the boundary becomes a surface; the Magnus force on a spinning ball, which bends the trajectory out of the vertical plane and makes the reachable set three-dimensional in earnest; and the orbital limit, where the conic closes and the notion of range stops meaning anything.

The neighbouring ladders are the angle that throws furthest, which is the boundary’s meeting with the ground; the angle that drag moves, which is the first thing to spoil it; and the fold caustic, where the same singularity is met in light and repaired by treating the light as a wave.