A caustic is the bright curve where a family of rays folds over itself. The caustics overview shows the pattern and the ray-density idea behind it, and the reflection caustics article works that idea through curved mirrors and the coffee-cup nephroid. This page steps back from any single mirror or water surface and builds the mathematics that all of those cases share.

The central object is the envelope of a one-parameter family of lines. Once the envelope is defined, the brightness, the smooth fold, and the sharp cusp all follow from the same few derivatives. The background needed is a derivative and the idea of a limit. Everything is built in two dimensions, where the pictures are clearest, and the three-dimensional version uses the same reasoning with one more parameter.

A Caustic Is the Envelope of a Line Family

A single ray is a line. To get a caustic, take a whole family of lines, one for each value of a parameter $t$. In optics the rays come from a source, a mirror, or a water surface, but the mathematics only needs the family itself. A convenient way to write a family is

$$y = t,x + f(t)$$

Here $t$ is the slope of the line, and $f(t)$ sets where the line crosses the vertical axis. Changing $f$ changes the whole family, and every smooth function $f$ produces a different set of lines and a different caustic. This slope-intercept form keeps the algebra to a minimum and appears throughout the rest of the page.

Now watch two neighbors. The line at $t$ and the line at $t + \Delta t$ are close together. Where they meet, two rays of the family pass through the same point, and that is where light concentrates. As $\Delta t$ shrinks toward zero, the meeting point slides toward a fixed location, and the collection of all these limits is a curve called the envelope of the family. Every line of the family touches the envelope at one point and is tangent to it there, the way a straight edge rests against a smooth curve.

The envelope is the caustic. The rays themselves do not draw the bright curve. The curve is the boundary that the rays graze as they fold past one another.

The Envelope Condition

Finding the envelope takes one subtraction. The line at $t$ and the line at $t + \Delta t$ both pass through their meeting point, so for that point $(x, y)$

$$y = t,x + f(t), \qquad y = (t + \Delta t),x + f(t + \Delta t)$$

Subtract the first equation from the second to get

$$0 = \Delta t,x + f(t + \Delta t) - f(t)$$

Divide by $\Delta t$ and let it go to zero. The difference quotient becomes the derivative, and the limit equation is

$$x = -f'(t)$$

Substituting this back into the line gives the height of the envelope point, $y = f(t) - t,f'(t)$. Together the two expressions trace the caustic as $t$ varies:

$$\bigl(x(t),, y(t)\bigr) = \bigl(-f'(t),\ f(t) - t,f'(t)\bigr)$$

The first equation is the one that matters. It picks out the parameter where the family stops sweeping sideways, so neighboring lines pile onto the same point instead of separating.

The same subtraction works for a family written implicitly as $F(x, y, t) = 0$, and it gives a rule that is worth memorizing. Every envelope point satisfies both

$$F(x, y, t) = 0, \qquad \frac{\partial F}{\partial t}(x, y, t) = 0$$

The first equation keeps the point on the line, and the second removes the parameter. For the slope-intercept family, $F = y - t x - f(t)$, so the second equation is $-x - f'(t) = 0$, exactly the result above. When two neighboring lines are parallel, their meeting point runs off to infinity, which is why a family of parallel lines has no envelope anywhere in the plane.

Two examples show how different the envelopes can look from almost the same recipe.

Take $f(t) = -t^2$, so the family is $y = t,x - t^2$. Then $f'(t) = -2t$, and the envelope is $(x, y) = (2t,, t^2)$, which is the parabola $y = x^2/4$. Every line of the family is tangent to that parabola, and the envelope is a smooth curve with no special point.

Take $f(t) = -t^3$, so the family is $y = t,x - t^3$. Then $f'(t) = -3t^2$, and the envelope is $(x, y) = (3t^2,, 2t^3)$. Eliminating $t$ from the two coordinates gives $4x^3 = 27y^2$, a semicubical parabola with a sharp point, or cusp, at the origin. The two families differ only in $f$, yet one envelope is smooth and the other has a corner.

From Ray Families to Caustics

The envelope becomes a caustic as soon as the lines are rays. A mirror or a water surface turns an incoming family into an outgoing one, and as long as the surface is smooth, the outgoing rays form exactly the kind of one-parameter family studied above. The parameter can be the position along the surface, the angle of the incoming ray, or any other label. The envelope condition does not care which.

The explorer draws a family of reflected rays from a point source. The purple dots mark where neighboring reflected rays cross, which is the finite-step version of the limit that defines the envelope. With few rays the dots are scattered. As the ray count rises, the dots crowd onto a single curve, and that curve is the caustic. Drag the source and the envelope moves, because the family of reflected directions changes with the source position.

Reflection and refraction differ only in how each outgoing line is computed. A reflection caustic is a catacaustic and a refraction caustic is a diacaustic, but the envelope mathematics is identical. The reflection caustics article writes the envelope condition for a reflected family in determinant form, starting from the vector reflection law. That determinant and the derivative above are two spellings of the same singularity, one chosen to match the ray geometry and one chosen to match the line family. The light refraction article develops the corresponding direction rule, Snell’s law, for the refraction case.

Why a Caustic Is Bright

The envelope tells you where light concentrates, but it does not yet say how bright the caustic is. A short calculation with a screen fills that gap.

Put a straight screen across the rays and let the ray at parameter $t$ meet the screen at position $X(t)$. A small interval of the parameter, from $t$ to $t + dt$, carries a fixed slice of the ray fan. On the screen that slice lands in an interval of length $|X'(t)|,dt$, where the prime is a derivative with respect to $t$. The light per unit screen length is therefore proportional to

$$\rho(t) = \frac{1}{|X'(t)|}$$

The density is small where the rays spread apart and large where they crowd together. Where $X'(t) = 0$ the density blows up, and that is the caustic. In two dimensions the single derivative becomes the Jacobian determinant of the map from ray parameters to screen position, and the caustic is the set where that determinant vanishes.

The screen formula agrees with the envelope from the previous section. For the parabola family $y = t,x - t^2$, take a horizontal screen at height $y_0$. The ray at $t$ meets it at

$$X(t) = \frac{y_0}{t} + t, \qquad X'(t) = 1 - \frac{y_0}{t^2}$$

The derivative vanishes when $t^2 = y_0$. At that parameter the contact point of the ray with the envelope is $(2t,, t^2) = (2t,, y_0)$, which lies on the screen. The brightest point on the screen is exactly where the envelope crosses it, and the smooth rise in density on the approach is what makes a caustic a bright line rather than a uniform wash.

The pool explorer estimates this density directly. It counts how many refracted rays land in each narrow strip of the floor and paints brightness in proportion to the count. The bright bands are the places where the count is highest, which is the discrete form of $1/|X'(t)|$. Raising the wave amplitude or frequency tilts the surface more sharply and deepens the contrast between bright and dark bands, because the density ratio, not the total number of rays, sets the brightness.

Fold and Cusp: The Two Shapes of a Caustic

A more general view packages a ray family with a single function. Let $S(x, y, t)$ be the optical path length, or any distance-like quantity, from a source labeled by $t$ to the point $(x, y)$. A ray reaches $(x, y)$ when the path is stationary in the parameter, which is the condition

$$\frac{\partial S}{\partial t} = 0$$

This is Fermat’s principle written for a whole family at once. The caustic is where the stationary path becomes degenerate, so the second derivative vanishes as well:

$$\frac{\partial S}{\partial t} = 0, \qquad \frac{\partial^2 S}{\partial t^2} = 0$$

These two equations are the general envelope condition in disguise, and they make the local shapes easy to separate.

The simplest generating function is $S = t^3/3 + x,t$. The stationary condition is $t^2 + x = 0$, and the degeneracy condition is $2t = 0$, so $t = 0$ and $x = 0$. The caustic is the vertical line $x = 0$. Just to its left, where $x < 0$, the equation $t^2 = -x$ has two solutions, so two rays pass through each point. Just to its right there are none. A point crossing a fold loses two rays at once, which is why the brightness piles up along a smooth, one-sided bright edge.

The next generating function is $S = t^4/4 + x,t^2/2 + y,t$. The stationary condition is $t^3 + x,t + y = 0$, and the degeneracy condition is $3t^2 + x = 0$. Solving the second for $x$ gives $x = -3t^2$, and substituting into the first gives $y = 2t^3$. Eliminating $t$ produces

$$4x^3 + 27y^2 = 0$$

which is the same cusp curve met in the previous example, reflected across the vertical axis. Inside the wedge between its two branches the cubic $t^3 + x,t + y$ has three real roots, so three rays pass through a point. Outside the wedge it has one. A cusp is where two fold edges meet and the ray count changes by two at a single point.

Shape	Generating function	Caustic	Rays through a nearby point
Fold	$t^3/3 + x,t$	$x = 0$	Two on one side, none on the other
Cusp	$t^4/4 + x,t^2/2 + y,t$	$4x^3 + 27y^2 = 0$	Three inside the wedge, one outside

The nephroid that a coffee cup draws ends in exactly these cusp points at its two tips. The reflection caustics article derives the nephroid and locates its cusps, and the cusp above is the local shape at each of them.

These two shapes are not just examples. Whitney’s theorem says that the only stable singularities of a smooth map from a line to the plane are the fold and the cusp. A small change to the optical system can move or bend a fold or a cusp, but it cannot turn it into a third kind of point. Any more degenerate caustic, where higher derivatives also vanish, is unstable: a slight nudge splits it into folds and cusps. That is why the sharp points on real caustics always look like cusps, and why a real cup that is not a perfect circle still produces the same local shapes.

The generating-function picture also points past the ray model. A fold produces an Airy pattern and a cusp a Pearcey pattern once the wave nature of light is included. The Airy disk article covers the diffraction side of that story.

Where the Ray Model Runs Out

The density formula divides by zero at the caustic, and that division is a warning rather than a result. Geometric optics has no finite brightness to offer at a fold, because the rays really do converge to a single point. Light has a wavelength, so the true caustic is a bright band of finite width with fringes near a fold and a more complicated pattern near a cusp. The ray envelope still marks where the band sits, which is why the geometric picture stays useful even where its brightness prediction fails.

A few families have no caustic at all. Parallel rays share one direction, so their lines never cross and the envelope sits at infinity. A flat mirror turns every ray by the same amount and inherits that same uniformity. The other degenerate case is a pencil, a family whose lines all pass through one common point. Then the envelope collapses to that point. A caustic needs the family to turn, and it needs the turning to vary from ray to ray.

When the reflected rays diverge instead of converge, the envelope lies behind the mirror and is called virtual. No rays cross in front of the surface, but the same equations produce the curve, and the fold still governs where the reflected light appears to come from. The reflection article collects these edge cases in more detail.

Summary

Caustics are the envelopes of ray families. Write a family of lines as $y = t,x + f(t)$, and the envelope is

$$\bigl(x, y\bigr) = \bigl(-f'(t),\ f(t) - t,f'(t)\bigr)$$

or, in general, the solution set of $F = 0$ and $\partial F/\partial t = 0$. The envelope is bright because neighboring rays land close together, with density proportional to $1/|X'(t)|$ on a screen, and the caustic is where that density diverges. Locally a caustic is either a fold or a cusp, the two stable shapes that a one-parameter family can produce, with the cusp governed by the curve $4x^3 + 27y^2 = 0$. Reflection and refraction differ only in how the turning at the surface is computed, so a single envelope calculation covers both.