Symbolic Rendering
One Sphere, Ten Ways of Knowing
An explainer comparing numerical rendering with
symbolic rendering,
in the spirit of Utah graphics meets Macsyma (today: Python + sympy).
in the spirit of Utah graphics meets Macsyma (today: Python + sympy).
Dr. L. Van Warren — Original Idea & Mathematical Development
Kimi K3 — Assistance with Mathematical Development
Kimi K3 — Assistance with Mathematical Development
❦
To Phong, who lit the first pixel honestly; to the Macsyma
builders, who taught machines algebra; to Cook, Porter & Carpenter, who priced
the integrals; and to the reader who checks the difference canvas —
this book is the ledger between two ways of knowing the same image.
this book is the ledger between two ways of knowing the same image.
How to read this book.
Each chapter is one self-contained page: §1–§3 derive the mathematics once,
§4 is a live browser simulator (numeric and symbolic side by side, the difference
underneath), §5 is a Colab laboratory of eight cells, §6–§9 are the ledger, the
pitfalls, the history, and the road. The discipline, stated once for all:
derive the anchor, run the regression, report the error in both currencies
(variance and bias), and never let a black difference canvas claim correctness —
it claims only that two formulations agree. Appendix A collates the contract;
Appendix B is the insurance policy that reruns itself. The Registry Check below
greets the bindings at the door.
§i · The chapter map
Part I — The Closed Form
·
Chapter 1 · index1.html
One Sphere, Two Computations
The image becomes an object: Piecewise carries the decision tree;
lambdify carries it to the frame.
anchors: silhouette theorem ρ = eR/√(e²−R²) · closed forms ν, r·V̂ ·
the 0·NaN pitfall, armed
·
Chapter 2 · index2.html
Half-Vectors, Three Channels, Two Lights
Blinn's half-vector; RGB as three expressions; illumination as a
symbolic Σ.
anchors: disable-by-substitution (c₂→0) · image affine in light color ·
cse tames the triple swell
·
Chapter 3 · index3.html
Shadows, or the Quantifier Eliminated
The blocker test is three signs, no square root; branch flips draw
the boundary curves.
anchors: the Shadow Lemma · the two-number ledger ·
chambers as semi-algebraic regions
·
Chapter 4 · index4.html
Reflection, or Recursion Becomes Composition
The mirror is the screen of Chapter 1; the bounce is a substituted
expression; the strict test replaces the ε-hack.
anchors: the mirror flip (change of variables) ·
the factoring self-hit root · composition census
Part II — The Function
·
Chapter 5 · index5.html
Texture, or Albedo Becomes a Function
The chart's seam and pole are theorems with closed-form images;
the bitmap is a piecewise polynomial; gamma is faced.
anchors: longitude theorem · pole corollary ·
E(a+b) ≠ E(a)+E(b) · the bitmap Nyquist knee
·
Chapter 6 · index6.html
Antialiasing, or the Pixel Is an Integral
Supersampling pays 1/√M; the floor checker's box integral is a
closed form; the first asymmetric chapter.
anchors: the checker box integral · the coverage doctrine ·
convergence, measured
Part III — The Integrals
·
Chapter 7 · index7.html
Area Lights, or Penumbra Is an Integral
The key light becomes a disk; the lit fraction integrates the
Shadow Lemma; variance meets bias.
anchors: on-axis μ̄ in closed form · the conic penumbra ·
ρ→0 regression
·
Chapter 8 · index8.html
Glossy Reflection, or the Lobe Integral
Roughness measured in radians; the mirror is the anchor; the
Lambert weights are exact.
anchors: exact stratum weights · the roughness ruler
σθ = 1/√(2n) · n→∞ regression
·
Chapter 9 · index9.html
Bump Mapping, or the Derivative Is the Texture
The rest stop: no new integral — the exact gradient versus the
finite-difference U-curve.
anchors: bump moves no boundary (flips = 0 by theorem) ·
∂N′/∂β in closed form · β→0 regression
·
Chapter 10 · index10.html
Motion & Depth, or the Camera Gains a Clock and a Pupil
The aperture is Chapter 7's disk; the shutter is an interval;
the product integral is five-dimensional.
anchors: switching-time quadratic · the bokeh law ·
swept silhouette of degree ≤ 4 · two regressions
Back Matter — The Contract
·
Appendix A · indexA.html
The Contract
The claims and the non-claims; 27 named theorems; the regression
chain; the health battery, live.
anchors: 16 live checks · the collated ledger ·
errata in open court
·
Appendix B · indexB.html
The Machine-Checked Archive
Every verification from every chapter, one notebook, runnable
end-to-end, with the three tolerance classes named.
anchors: 44 checks · fixed seeds · confessed tolerances ·
the tally
§ii · The ledger of the whole
In the manner of Appendix A: what each chapter added, what it paid, and its signature measurement — the number the chapter exists to print.
| Ch. | Added | Paid | Signature measurement |
|---|---|---|---|
| 1 | the image as expression | derivation seconds; expression swell | max |num − sym| ≈ 1e-15; ρ = 1.032796 |
| 2 | RGB Σ; substitution as switch | triple swell, cse-clawed | substitution ≡ omission at 1e-12; ∂²I/∂c² = 0 |
| 3 | shadows without square roots | the totality tax (~2×) | lemma: 200,000/200,000; flips ~0.5% on the curves |
| 4 | reflection by composition | composition swell (~3×) | mirror flip max |Δt| ≈ 1e-15; acne roots factor exactly |
| 5 | texture, bitmap, gamma order | the transcendental chart | max |Δ(u,v)| ≈ 1e-16; knee at k = NB/2 |
| 6 | the pixel as integral | band machinery on ~2% | box integral vs 64² quadrature ≈ 1e-16; error ~1/√M, plotted |
| 7 | the source integral; two currencies | K² both sides | μ̄ exact to 1e-8; RMS(variance) vs RMS(bias), priced |
| 8 | the lobe integral; exact weights | K² × source loop | weights vs quadrature ≈ 1e-10; >90% first-ring mass at n=4096 |
| 9 | the exact gradient | nothing — the rest stop | 1 eval vs 4; U-curve bottom ≈ ε=1e-6; ∂N′/∂β to 1e-9 |
| 10 | the pupil and the shutter | 5-D quadrature | coverage vs scan ≈ 1e-3; bokeh law exact to 1e-9 |
| A | the contract, collated | 16 checks, every load | battery verdict, timestamped, on the page |
| B | the archive | 44 checks, fixed seeds | 44/44 — or the failing row, named |
§iii · The archive at a glance
Appendix B's three courts of evidence, with their case counts — stated here so the cover carries the rules of the whole:
| Court | Tolerance | Hears | Checks |
|---|---|---|---|
| Exact | ==, or a nonzero constant contradicted |
identities in exact arithmetic; determinism; gamma non-commutativity | 6 |
| Machine | ≤ 10⁻¹² … 10⁻⁹ | equalities true in ℝ, evaluated in float64 — closed forms, derivatives, the bokeh law | 17 |
| Statistical | scaled to discretization, printed with the number | quadrature cross-examinations; the 200,000-configuration census; scan verifications | 21 |
§iv · The registry check
Twelve files make one book. This page greets its siblings at the door: a live reachability probe of every chapter and appendix, run on load, re-runnable at a click. (Served over HTTP each file reports present; opened from the local filesystem, fetch is sandboxed and the probe reports unverified — the pages are unharmed either way, and the card grid above links them regardless.)
standing by…
| file | title | status | note |
|---|
Cite any file by its name (index1.html … indexB.html) when requesting revisions; every page's section ids are stable within it.
❦
The volume is twelve files; the contract is one sentence.
Derive the anchor · run the regression · report the error in both currencies ·
let the page show its work.Set by the Press · the difference canvas is the conscience ·
the checks, being re-runnable, continue without us.