You have probably seen caustics without stopping to name them: shimmering light networks on a swimming-pool floor, a bright curved arc inside a coffee cup, a sharp patch of focused sunlight beneath curved glass. Those bright lines and patches appear when reflected or refracted light gets concentrated into a smaller region than the surface it came from. Where rays spread out, a surface looks dimmer; where they bunch together, light piles up and can draw strikingly bright, sharp-edged patterns.

These patterns are geometry made visible by light transport: the shape of a mirror or water surface is written out in light. The optics behind everyday examples like these follow from simple ray tracing and a few geometric rules.

We will work through three cases, each with its own interactive visualization: reflection from a curved mirror with a movable light source, the circular reflection that draws the classic coffee-cup curve, and refraction through a wavy water surface onto a pool floor.

How Rays Create Brightness

The simplest mental model is to emit many rays and track where they land. The full set of rays we are tracing is called a ray family. If neighboring rays end up close together, more light lands in a smaller area, and that densely packed zone is the caustic.

In geometric optics, the caustic is the curve that bounds the bright region. That curve is the envelope of the ray family, the line the rays skim as they bunch together. You do not need advanced differential geometry to use this idea in practice; in simulations a simple approximation works well: trace many rays and look for where adjacent rays intersect or cluster.

Two mechanisms produce most everyday caustics:

• Reflection: a ray bounces off the surface, leaving at the same angle it arrived, measured against the surface.
• Refraction: direction changes by Snell’s law when crossing media with different refractive indices, and the refraction article works through the full mechanics.

Each rule acts at a single point on the surface, yet the brightness pattern emerges from all those points working together. Small normal changes across a curved surface can drastically reshape where rays converge.

Reflection Caustics from a Curved Mirror

A flat mirror changes nothing about how the rays spread: diverging rays keep diverging and evenly spaced rays stay evenly spaced, so light never piles up. Curvature breaks that uniformity. When neighboring points of a mirror tilt in slightly different directions, their reflected rays get steered onto crossing paths, and where many crossings bunch together, a bright line appears. This visualization lets you watch that happen. Drag the orange source dot to move the light, and adjust the mirror curvature slider; every change to either input reshapes where the reflected rays converge.

Each blue line is one ray traveling from the source to its landing point on the mirror, and the red line leaving that same point is that ray after reflection. The action is in the red family: pick two neighboring red lines and follow them, and where they cross, a purple dot appears. A crossing pair means two adjacent parts of the ray bundle have squeezed together, so the dots outline where light concentrates. The curve they trace is the caustic your eye would perceive as a bright line.

With the source centered above the mirror, the caustic forms a symmetric, pointed arc. Drag the source sideways and the arc slides the opposite way, leaning and stretching further on one side until its symmetry disappears. Raise the curvature and neighboring normals tilt apart faster, bending the reflected rays more sharply, so the arc spreads across a wider region and drifts up toward the mirror. Lower the source and the incoming rays arrive at a wider spread of angles, which spreads the reflected family and pushes parts of the caustic upward.

Watch what the curvature slider actually changes: not the mirror’s character, but how quickly neighboring normals tilt. At the gentlest setting the reflected rays barely turn, and their crossings fall far below the picture in a loose bundle; raise the curvature and the concentration climbs back toward the mirror and spreads into a wider arc. The parabola drawn here is one convenient shape, nothing more.

Any smooth surface whose normals turn gradually from point to point steers neighboring reflected rays onto crossing paths, and that alone is enough to form a caustic. A parabola, a circle, or any irregularly curved mirror will do; the shape sets where and how sharply light concentrates, not whether a caustic forms at all.

Reflection Caustics Inside a Coffee Cup

The coffee cup is a curved mirror you hold in your hand. The bright arc that runs around the inside of the cup is light reflecting off its curved wall. Every small stretch of that wall acts like a tiny flat mirror, and because the wall curves, those mirrors face slightly different directions. Rays from a source inside the cup bounce off the wall and gather along a single bright curve, the same envelope idea from the earlier section. In the visualization below, a point source sits inside a circular cup and emits rays in every direction; purple dots mark where neighboring rays cross after their first bounce, tracing that bright curve.

Leave the source near the center and the reflected rays cross into a tidy, roughly symmetric pattern. Drag the orange dot toward the wall and the bright curve slides away from it, drifting toward the opposite side of the cup and stretching into a wider arc. Move the source up or down and the depth changes the angle at which rays meet the wall, reshaping the entire reflected family.

The visualization places the source inside the cup so you can drag it, but the classical version of this pattern starts from the opposite extreme: rays arriving from far away, effectively parallel like sunlight. Bounce those parallel rays off a perfect circle and they gather into a kidney-shaped curve called a nephroid. Real cups are not perfect circles, the surface ripples, and sunlight is not perfectly parallel, so the curve you actually see is usually a distorted version of that shape. Either way the cause is the same: the wall curves smoothly, so it bends reflected rays from every direction onto one structured bright path.

Refraction Caustics on a Pool Floor

The light networks on a swimming-pool floor are the refraction side of the same story. Sunlight arrives as a nearly parallel family of rays and enters the water, where each ray bends according to Snell’s law, the rule explained in depth in the refraction article . If the surface were flat and still, every ray would bend by the same amount and the floor would light up evenly. A wavy surface changes that: each wave slope tilts the boundary a little differently, so neighboring rays bend by slightly different angles. Those small differences accumulate, and where many refracted rays land close together the floor looks bright, while where they spread out it looks dark.

The visualization below models that arrangement. Blue rays come in from the sun, bend at the wavy surface, and land on a floor strip whose brightness shows how many rays arrived at each spot.

Raise the wave amplitude and the surface tilts more steeply, bending rays by a wider spread of angles and deepening the contrast between bright and dark bands. Raise the wave frequency and more wave crests fit across the pool, so the bands become finer and more closely spaced. Rotate the sun angle and the whole pattern slides sideways, moving where the concentration peaks land. Raise the water’s refractive index and the stronger bending relocates where rays land, which can pull the concentration regions tighter.

Even this simplified 2D model reproduces what makes a real pool floor shimmer: a moving, wavy surface converts smooth sunlight into complex, ever-shifting light textures.

The Common Math Behind All Three Visualizations

Each visualization in this article runs the same pipeline, and that pipeline is ray tracing in miniature: the core loop renderers use to build pictures. Follow one ray and you see it in full: it starts at the light, meets the surface, picks up a new direction, and travels on to its landing spot. This section breaks that loop into concrete steps and shows the math behind the redirect rule, the step where reflection and refraction part ways.

1. Start with rays from the light: every ray begins at the light source, or arrives parallel from the sun in the pool. The scenes choose their rays differently: the mirror and the pool pick evenly spaced points along the surface the rays will meet, while the cup emits evenly spaced directions from the source.
2. Find the normal: work out which direction the surface faces at the point where each ray meets it. The mirror’s normal follows from the slope of its curve, the circle’s normal always points along its radius, and the water’s normal follows the local wave slope.
3. Redirect the ray: replace the incoming direction with the reflected or refracted one, using the rules below.
4. Trace and count: send each redirected ray to its landing spot and count how densely the rays arrive there. That density field is what your eye perceives as caustic brightness.

For reflection, the redirect rule is the one that governed the mirror and the cup: the ray bounces off at the same angle it arrived, measured on either side of the normal, like a ball bouncing off the floor. The formula below writes that rule in vector form, taking the incoming direction $\mathbf{i}$ and the surface normal $\mathbf{n}$ and returning the outgoing direction $\mathbf{r}$:

$$\mathbf{r} = \mathbf{i} - 2(\mathbf{i} \cdot \mathbf{n})\mathbf{n}$$

Reflecting the ray across the normal keeps the angle with the normal unchanged, which is the equal-angle rule stated in words.

For refraction, the redirect rule is Snell’s law, where $\theta_1$ and $\theta_2$ are the angles the ray makes with the normal on either side of the surface, and $n_1$ and $n_2$ are the two refractive indices:

$$n_1 \sin\theta_1 = n_2 \sin\theta_2$$

Read it as a balance between the two sides of the surface: the ray bends more sharply when it arrives at a steeper angle or when the jump in refractive index is larger, which is why the pool pattern shifts when you raise the water’s refractive index.

Everything you have seen reduces to this pipeline: send rays from the light onto the surface, turn its normals into new ray directions, trace the rays, and watch the family concentrate. The computation is identical in all three cases; only the surface itself changes, and its shape decides the normals the pipeline works with.

Recap

Caustics are the bright lines and patches that appear when reflection or refraction concentrates light into a smaller region than the surface it came from. The same geometric mechanism sits behind all of the everyday examples: a curved mirror whose normals keep turning, a circular cup wall that sorts reflected rays onto one bright curve, and a rippling water surface that bends sunlight into moving bands.

If you keep one model in mind, make it this: trace many rays and watch how the density of their landings evolves. Where neighboring rays fold together and cluster, a caustic emerges.

That model is simple, yet it links the patterns you see in daily life to the same ray-tracing thinking used in optics tools and renderers.