14 Guidance as a geometric measurement

Guidance rendering is a measurement layer with a camera model, clipping, sampling and units. It must preserve the spatial information actually simulated. A downstream model may change appearance; Rheon should export reproducible geometric evidence.

Camera coordinates

For camera coordinates $(X,Y,Z)$ with positive Z forward,

$$u=f_xX/Z+c_x,\qquad v=f_yY/Z+c_y.$$

Axial depth $Z$ differs from radial range $\sqrt{X^2+Y^2+Z^2}$ off axis. Name the representation, handedness, image origin, clipping and pixel-center convention. Mirroring or shifts may be camera-contract defects rather than fluid errors.

Linearized projection gives

$$\delta u\approx(f_x/Z)\delta X-(f_xX/Z^2)\delta Z.$$

The same world-space error moves more pixels near the camera, and depth errors shift off-axis silhouettes. Evaluate actual camera distributions, not just a world-space norm.

Surface and volume outputs

Liquid can export silhouette, depth and normals in a declared frame. Smoke may integrate extinction. Piecewise constant extinction along a ray gives

$$T=\exp\left(-\sum_i\sigma_i\Delta s_i\right), \qquad \alpha=1-T.$$

Increasing samples without adjusting segment length changes opacity incorrectly. Concentration-to-extinction is a rendering model, not conservation. A diffuse volume has no unique surface; first-hit depth needs a named threshold or accumulated-opacity rule.

Graphics depth buffers may be nonlinear in distance. Exporting raw values as metre depth is incorrect without conversion. Validate planes at known distances and off-axis spheres. Include validity masks rather than ambiguous zero depth.

Metrics and integration

Silhouette IoU is $|S\cap T|/|S\cup T|$, with an explicit both-empty case. It can hide a displaced thin filament in a large body. Add boundary-distance percentiles and maximum displacement. Compare depth on mutually valid pixels and report missing surfaces separately. Normals use angular error.

Rest scenes under a fixed camera expose flicker from grid crossings, mesh ambiguity and thresholds. Filtering can reduce flicker while shifting edges, which must remain in the error budget.

Export camera calibration, world scale, dimensions, channel semantics, masks, physical time, state ID and renderer version. Resizing requires updated intrinsics or a stated resampling transform. A depth image without its mapping is ambiguous even when attractive.

The first renderer should be simple and deterministic. Its acceptance is geometric agreement with known primitives and state, not photorealism. Integrated diffusion experiments need separately pinned models and seeds.

Read the field before interpreting the picture

The sticky example beside this chapter is the existing 24³ carrier-flow lesson: it displays a computed tracer and instantaneous velocity from that bounded grid model. It is not the native material/SPH preview, a CAD flow solution, or a pressure-load measurement. The additions below explain how to ask better questions of each model; they add no new live material or CAD solver.

Start with a field’s meaning. A passive tracer marks where transported material has been represented; opacity is a display rule applied to that marker. Neither is automatically mass density. A velocity arrow represents a vector, while its length may be fixed or proportional to speed. An instantaneous streamline is tangent to one frozen velocity field; it does not record a parcel’s past motion. The engineering-visualization guide develops sampling, sections, pressure, loads and export from these distinctions.

Rheon has two separate data paths. The native material preview exposes particle position, velocity, mass, temperature and material ID. Its per-particle density and pressure are private, and the reported maximum pressure is evaluated before the position and wall-temperature update; accepted speed describes a later stage. A pressure maximum cannot become a synchronized pressure slice merely by coloring accepted particle positions. The carrier path instead has staggered face velocities and a pressure correction. Its correction is not particle pressure, and its aligned face flux is not SPH mass crossing. See the material contract and source-anchored inventory.

For every probe, ask: what quantity, where, in which units, at what physical time and evaluation stage, and from which data? Mark it native, derived, prescribed or unavailable. Show the sampling rule, validity and pressure reference when applicable. An unavailable value must remain distinguishable from zero. A color range belongs to the measurement: lock it across compared runs, and show numerical values alongside color.

Predict a measurement before adding a view

Consider a unit box with constant density. Prescribe three fields: rest, uniform velocity $(1,0,0)\,\mathrm{m/s}$, and shear $(y\,\mathrm{s}^{-1},0,0)$ with $y$ measured in metres. A density-only image is identical in all three cases. Through the unit face normal to positive x,

$$Q=\int_A\mathbf u\cdot\mathbf n\,dA$$

is respectively 0, 1 and 0.5 m³/s. Reversing the normal reverses the signs. These are prescribed analytic fields, not three native simulations satisfying the same box-wall conditions. This small example teaches why a slice, arrows and an oriented section answer questions that opacity cannot.

The seven lesson specifications are a proposed progression, not seven implemented browser controls. The analytic companion checks selected mathematical reference calculations; it does not call Rheon or validate its physical models. Detailed controls, assumptions and failure probes are in the worked lessons.

Keep geometry and engineering claims separate

A CAD design surface, the boundary used by the solver and a display mesh may all differ. A smooth surface colored from nearby fluid samples is not yet a conservative force integral. Pressure loads require a declared pressure reference, consistently oriented numerical boundary and supported stress model. Uniform pressure cancels on a closed surface with consistent normals; it generally does not cancel on an open patch. Wall shear additionally requires a qualified viscous wall treatment and wall gradient. Box-contact impulses are a different diagnostic, not a wall-shear map. These distinctions follow the separate pressure, viscous and moment conventions used in established analysis tools.

The rotating emitter is a prescribed source. Its pressure-head speed, midpoint orientation and fixed-volume parcel credits do not provide a nozzle collision solid or simulated nozzle walls. The source definitions and accepted publication are anchored in emitter geometry, profile, particle admission and transactional publication.

The first useful native extension is an unquantized, stage-labelled particle inspector and export, followed by reconstruction with visible support and invalid regions. Add histories only after retaining quantitative state, and loads only after establishing their geometry and stress contract. An analytic check verifies a calculation; a relevant refinement study estimates numerical error; matched experiments address physical validation. The source bibliography and implementation order preserve the evidence and prerequisites behind that sequence.

Draft checkpoint: inspect a retained obstacle projection

Unmerged PR50, inspected at 3f49a158a0f1bec93f47edf6cc42ad86242e4aa5, adds a native obstacle projection inspector. Its 12×8×6 grid has 0.125 m spacing and a fixed aligned obstacle. One accepted pressure projection over 0.05 s supplies a retained result; this is not an advancing wake or a viscous wall-load computation. Its scope and validation record records the fixture, gauge, rendering choices and remaining integration work.

A practical inspection sequence is to probe a fluid cell, rotate the camera, and revisit the same cell: presentation changes must not change its value. Then move the slice. Scalar colors retain a full-state valid-cell range, so a slice change cannot silently rescale a small feature into a large one. Pressure means gauge-fixed correction in Pa; speed is reconstructed from adjacent face velocities, not a native face flux. Solid cells report unavailable, not zero. Velocity arrows encode velocity times 0.1 s, with physical aspect ratio and camera foreshortening; they are neither normalized directions nor trajectories.

The lab’s snapshot and camera are separate from live appearance history. Scrubbing an SPH appearance frame cannot retime the retained projection or recover quantitative particle fields. The draft uses CPU-drawn slices and wire overlays, and requires full-fluid or solid cells; it supplies no arbitrary CAD import, cut-cell reconstruction, calibrated pressure/viscous load, VTK exchange or GPU-performance result. Its reported local PR47 combination remains a tested draft integration, not a merge or shared accepted API. The existing browser chapter still presents its separate bounded carrier-flow lesson.