Snell's law, found by searching
At its defaults it draws snell's law, found by searching. Paths from a point in a medium of index 1 to a point in one of index 1.5, and the optical path length of each against where it crosses the boundary. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 55.80° and 33.46°, which satisfy n₁sin θ₁ = n₂sin θ₂ to 1.0e-8. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.
fermat-path is one function in lib/figures/optics.js —
rays, lenses, mirrors and what light does to a surface. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
Paths from a point in a medium of index 1 to a point in one of index 1.5, and the optical path length of each against where it crosses the boundary. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 55.80° and 33.46°, which satisfy n₁sin θ₁ = n₂sin θ₂ to 1.0e-8. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.
A lens with two flat faces
The options are the ones The channel with no walls passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Rays entering a rod whose refractive index falls parabolically from the axis outward, parallel to the axis and at -2.4, -1.2, 0, 1.2, 2.4 mm from it. The ray equation in such a medium is the harmonic oscillator's, so each path is a cosine of the same period whatever height it started at — which is exactly the condition for a focus, and the reason all of them cross the axis together at 12 mm. Nothing is curved anywhere: the faces are flat and the bending is done by the inside of the glass. Cut the rod at a quarter of the period and it images; cut it at half and it relays the beam parallel again, inverted. The period is a property of the profile alone, which is why a rod like this is specified by a length rather than by a curvature.
How steep a gradient has to be to matter
The options are the ones The channel with no walls passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The index gradient above a hot road against height, on a logarithmic scale, with two constants drawn across it for comparison. At the surface the road's gradient is 6.00e-4 per metre and it dies away over 5 cm. The lower line is the standard atmosphere's, 2.62e-8 per metre, computed from the barometric law and a 6.5 K/km lapse rate rather than quoted, and it points the other way — the index falls with height in the free atmosphere and rises with height above the road, which is why one mirage appears below the object and the other above it. The middle line is the Earth's own curvature, 1.57e-7 per metre. The road beats it by a factor of 3823; the atmosphere reaches 17% of it, which is the whole of why the optical horizon is further off than the geometric one.
Snell's law, found by searching
The options are the ones The path that does not change passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Paths from a point in a medium of index 1 to a point in one of index 1.5, and the optical path length of each against where it crosses the boundary. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 55.80° and 33.46°, which satisfy n₁sin θ₁ = n₂sin θ₂ to 1.0e-8. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.
Snell's law, found by searching
The options are the ones The path that does not change passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Paths from a point in a medium of index 1 to a point in one of index 2.42, and the optical path length of each against where it crosses the boundary. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 59.47° and 20.85°, which satisfy n₁sin θ₁ = n₂sin θ₂ to 5.5e-8. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.
Equal angles, found by searching
The options are the ones The path that does not change passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Paths from a source to a detector by way of a flat mirror, and the total length of each against where it meets the mirror. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 48.01° and 48.01°, which satisfy equality to 5.3e-8 radians. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.
What checks it
physicscheck asserts something about fermat-path that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
The channel with no walls
A pipe will not carry a note below its cutoff, and no length of pipe helps. Replace the walls with nothing but a region where the wave travels slightly slower, and the cutoff disappears — however weak the contrast and however thin the channel, at least one mode is bound. The difference is not a matter of degree; it is the difference between a boundary condition and a potential well.
OpticsThe path that does not change
Reflection, refraction and the angle of the rainbow are not three laws. They are one condition — that the optical path length is stationary — and the word stationary rather than shortest is the whole of what makes an elliptical mirror and a rainbow the same statement.
OpticsThe path that takes the longest time
Light is said to take the quickest route. Put a source and a detector in front of a mirror curved a little more than the ellipse through them, and the route it takes is slower than every route beside it — by construction, not by exception. The principle was never about least; it was about stationary, and the difference is where optics stops being geometry.
OpticsThe ray that bends without a surface
Snell's law is about a boundary, and light bends in air where there is no boundary anywhere. Let the index vary continuously and the law of angles becomes a differential equation — one that carries a conserved quantity, forbids the ray from reaching certain heights, and turns a hot road into a mirror a hundred metres long.
OpticsThe surface that images one point exactly
Fermat's principle says every ray from an object to its image must take the same time. Written as an equation, that is a curve Descartes found — a surface with no aberration at all, at any angle. It exists, it is easy to compute, and it images exactly one pair of points and nothing else.
WavesThe taper that matches every note
A quarter-wave layer cancels a reflection at one wavelength and only near it. Spread the same change of impedance over a distance instead, and the reflection vanishes for every wavelength shorter than about twice that distance — not by cancelling one echo against another, but by leaving no step anywhere for an echo to come from.