Local families
A terminal beam, cone or sheet can keep most of a traced prefix. Validate the moved segment and its new connection; do not retrace an unchanged, already-valid prefix.
Path sampling / The rotational atlas
A family with a way through.
Two real scattering events, one light, one camera. Three rotational charts share the work; a distance–distance plane gives the family another way to make the connection. These are restricted physical films, separate from the earlier fixed-angle algebra test below.
Loading the recorded physical experiment…
Path sampling / Four ways to approach a hard connection
Make a rare path easier to find.
Give the medium its own subpath, preserve a nonlinear specular solution, meet along an intersection of sheets, or release coordinates farther apart. Each changes the geometry of the search. The studies expose a useful construction and a failure case; they are proposals and diagnostics, not four finished general transport estimators.
Specialized sampling / Let the camera define the problem
A narrow task can deserve its own estimator.
Tilt-shift is a deliberately specialized direction: use a tilted focus plane and finite aperture to organize the connections that matter to that camera. A time gate imposes a different restriction—arrival time. These browser studies visualize those constraints; complete camera-weighted transport estimators remain future work.
Visibility / The cost after the connection
Remember the openings.
Solving the connection does not tell us whether light can get through. A primitive can generate many candidate paths, only for walls to reject them. The expensive case is not simply “more bounces”: it is a family that moves several segments and must check them again.
Interactive schematic
A hit on the primitive can still be a miss for light.
Changed segments4potential visibility checks
Clear connection
V₀ V₁ V₂ V₃ V₄ = 1
One blocked leg makes the whole path contribute zero. A renderer can stop testing at the first rejection.
Retained geometryMoved / new legBlocked legMedium-owned segment
A dependency illustration, not a timed renderer. Unchanged segments can be reused only when their visibility is already valid and the scene is unchanged. Switching blockers requires revalidation even for an unmoved prefix.
Separated coordinates
Generalized plane and rotational families can move intermediate vertices. Crossed sheets and independent medium bridges add their own joins. Count the affected segments, not just the number of bounces.
Structured media
Binary visibility is only one factor. Moved segments through a heterogeneous medium may also need new transmittance estimates. The shadow atlas below addresses the binary, opaque part.
A luminous sheet becomes a map of where light can pass. Broad, stable openings make nearby coordinates agree; fine silhouettes and moved geometry make a cache less reliable. Compare the exact field, its cached prediction and a signed correction below.
Loading the shadow study…
This experiment caches one conditional visibility leg. Extending it to a whole path means controlling the estimator of the whole visibility product; multiplying correlated corrected leg estimates does not automatically preserve the expectation.
Continue inside RadianceLab
From a useful construction to a native experiment.
The native research window contains progressive rotational and three-scatter sheet renderers, plus a visibility-cache instrument. Choose a favorable or difficult scene, compare the methods, and save the linear film with its settings. These are deliberately bounded experiments inside the application; the recorded studies above retain their original provenance.
Open the recording guide ↗ · What the native methods compute ↗
Cylinder · distance before spinCone · distance at spinHyperboloid · distance after spinPlane · both distances
What changes?
The path stays.
The solved coordinates move.
Two flight lengths and two azimuths describe the same short light path. Each method samples two coordinates and solves the other two together with the camera distance. The colored sheets are those solved families. The plane releases both distances. Try Shared blind direction, then switch the plane off: three decorative surfaces alone do not guarantee a well-conditioned estimator.
The cone uses the final azimuth and final distance. The cylinder moves the preceding distance. The hyperboloid moves the preceding azimuth. They use one consistent rotating frame.
One constraint, complementary charts
X = t₁ b(φ₁) + t₂ c(φ₁, φ₂)
X = o + s d
The determinant J = |(Xᵤ × Xᵥ) · d| measures how transversely a sheet crosses the ray. Near tangency, J approaches zero.
For these bounded uniform proposals, balance weights are proportional to the residual sampling density times J. All valid isolated roots contribute.
A more intricate surface is not automatically a better estimator. The useful question is whether another chart covers its expensive or unstable connections.
A numerical experiment, not just a drawing
Do the estimates agree?
The independent reference integrates a different pair of coordinates. The three estimators and their MIS combination target the same bounded, smooth, attenuated path integral. These measurements test the connection algebra; they are not a benchmark of the complete RadianceLab renderer.
Loading the saved experiment…
A small conditional transport film
One exposure for every method. No denoising.
A fourth chart has a specific job. A distance–distance photon plane can cover a configuration where all three rotational determinants vanish. Adding it everywhere did not improve the first equal-query test. The useful next experiment is to select it where the trio becomes poorly conditioned. Native capture record.
Second experiment / the shadow atlas
Predict cheaply. Check selectively.
A small cache stores visibility on the primitive’s own coordinates. Cache-only rendering changes the answer. A randomized exact residual can restore its expectation, at the cost of additional variance—including negative individual samples.
estimate = prediction + B / ε × (exact − prediction), B ∼ Bernoulli(ε)
Loading the saved visibility experiment…
One fixed swept surface and one visibility leg against two boxes. This is a controlled cache experiment; it does not yet accelerate full scene transport or cache heterogeneous transmittance.
The next useful experiments
Follow the hard connection.
Use the family where it helps.
Target difficult camera connections, select the plane when the rotational charts become poorly conditioned, and measure error at equal time before integrating the suite more broadly.
Measure the life of a useful cache.
Start with broad, stable openings. Count cache construction, exact corrections and invalidation after geometry changes alongside the visibility queries saved.
Carry one constraint further.
The six interactive studies stay deliberately bounded. Choose a useful scene, supply its missing transport factors and sampling measure, then compare against an independent reference.
Scope, assumptions and the published foundation
The coordinate atlas uses Canvas 2D projection of exact parameterized surfaces. Decorative glow is a display effect. The probe uses analytical quadric intersections; browser geometry is independently checked against derivatives and known paths. The recorded numerical experiments are separate Python implementations in RadianceLab’s research directory. Each of the six browser studies states its own computed quantities and remaining limits.
The live coordinate diagram and older conditional benchmark hold the two polar angles fixed and restrict both flight lengths to a finite interval. That benchmark checks a conditional integral without specular interfaces, occluders or a full heterogeneous transport ledger. The physical two-scatter films above come from a separate experiment with their own stated transport factors and limits. A general light-transport estimator still needs complete sampling densities, scattering and camera factors, transmittance, visibility, and path-class coverage.
The cone/cylinder/hyperboloid family follows the 2019 Photon Surfaces framework. The new experiment studies this small combination and its conditioning in the current project. No claim of a newly invented primitive or general rendering speed-up is made.