Why does the bright curve have such a sharp, cusped shape instead of being a blurry spot?

The cusped shape is a geometric singularity where many reflected rays accumulate along a single curve, called an envelope. This high density of rays creates a region of intense brightness. It's not blurry because the model uses perfect reflection and infinitely thin rays; in reality, light's wave nature would slightly blur the sharpest cusps due to diffraction.

Is this only about coffee cups, or does it apply elsewhere?

The phenomenon is universal for curved reflective or refractive surfaces. You see similar caustic patterns at the bottom of a swimming pool, from a wine glass, or when light passes through irregular glass. In astronomy, gravitational lensing can create giant caustics of distorted starlight. The coffee cup is a simple, accessible example of this broader principle.

What does moving the 'table line' up and down demonstrate?

Changing the table height shows how the caustic pattern evolves in space. The classic bright curve (the nephroid) only forms at a specific distance. Moving the line reveals that the reflected rays cross and re-distribute, demonstrating that a caustic is a specific surface in 3D space, not just a 2D pattern.

The simulator uses 'parallel rays.' How does sunlight, which comes from a finite-sized sun, affect the pattern?

Sunlight is not perfectly parallel due to the sun's angular size (about 0.5 degrees). This angular spread acts to blur or 'smear' the theoretically perfect caustic, making it less sharp. The simulator's parallel-ray assumption is an idealization that clearly reveals the underlying geometric structure.

Coffee-Cup Caustics

Parallel beam, circular cup segment, reflected-ray density on a table line.

About this model

Coffee-Cup Caustics visualizes the formation of a bright, focused curve of light—a caustic—created when parallel rays, like sunlight, reflect off the inside of a circular cup. The core physics is governed by the law of reflection, where the angle of incidence equals the angle of reflection relative to the surface normal. For a circular arc, the reflected rays do not converge to a single point but envelope a curve known as the nephroid, a specific type of caustic. The simulator maps this by calculating the reflection of many incoming rays. For a ray hitting the cup at a point defined by angle θ from the vertical, the direction of the reflected ray depends on the local normal. The resulting pattern on a horizontal 'table' line below the cup is a concentration of rays, with the highest density defining the caustic curve. This model simplifies reality by assuming perfectly specular reflection, a perfectly circular cross-section, and perfectly parallel incoming light. It ignores diffraction, scattering, and the effects of coffee liquid. By interacting with it, students learn how simple reflection from a curved surface generates complex, singular intensity patterns, explore the geometric construction of caustics, and see a direct application of envelope theory and ray optics. It connects the abstract mathematics of derivatives (the caustic is the envelope of the family of reflected rays) to a tangible, everyday phenomenon.

Who it's for: Undergraduate physics or engineering students studying geometric optics, calculus-based physics, or mathematical methods, as well as advanced high school students in AP Physics or calculus courses.

Key terms

Caustic
Law of Reflection
Geometric Optics
Envelope (Mathematics)
Specular Reflection
Nephroid
Ray Density
Singularity (Optics)

How it works

A classroom-famous optics artifact: the same envelope idea as an astroid, here built from many reflected rays instead of full wave theory.

Frequently asked questions

Why does the bright curve have such a sharp, cusped shape instead of being a blurry spot?: The cusped shape is a geometric singularity where many reflected rays accumulate along a single curve, called an envelope. This high density of rays creates a region of intense brightness. It's not blurry because the model uses perfect reflection and infinitely thin rays; in reality, light's wave nature would slightly blur the sharpest cusps due to diffraction.
Is this only about coffee cups, or does it apply elsewhere?: The phenomenon is universal for curved reflective or refractive surfaces. You see similar caustic patterns at the bottom of a swimming pool, from a wine glass, or when light passes through irregular glass. In astronomy, gravitational lensing can create giant caustics of distorted starlight. The coffee cup is a simple, accessible example of this broader principle.
What does moving the 'table line' up and down demonstrate?: Changing the table height shows how the caustic pattern evolves in space. The classic bright curve (the nephroid) only forms at a specific distance. Moving the line reveals that the reflected rays cross and re-distribute, demonstrating that a caustic is a specific surface in 3D space, not just a 2D pattern.
The simulator uses 'parallel rays.' How does sunlight, which comes from a finite-sized sun, affect the pattern?: Sunlight is not perfectly parallel due to the sun's angular size (about 0.5 degrees). This angular spread acts to blur or 'smear' the theoretically perfect caustic, making it less sharp. The simulator's parallel-ray assumption is an idealization that clearly reveals the underlying geometric structure.

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Coffee-Cup Caustics

About this model

How it works

Frequently asked questions

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Coffee-Cup Caustics

About this model

How it works

Frequently asked questions