Symbolic Rendering

Chapter 9 — Bump Mapping, or the Derivative Is the Texture

An explainer comparing numerical rendering with symbolic rendering,
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

Python-ready for Jupyter Notebooks in Google Colab.
Requires an internet connection for mathematical typesetting (MathJax).

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How to use this file. §1 what changes; §2 the scene; §3 the derivation — the perturbed normal, the exact tangent frame, the kink loci of a ridged height field. §4 is the live simulator with the ε slider (finite-difference step) under your hand. §5: Cells 1 and 3 replace Chapter 5's (bump composes orthogonally with Chapters 6–8's integrals; the rest stop builds on the point-light base). Cells 2, 4–8 are new. §6–§9 as ever.

☰ contents

What changes in Chapter 9
The scene (Chapter-9 assumptions)
The math, once, carefully
Live simulator
The Colab laboratory (Python / sympy)
The honest ledger — Chapter-9 deltas
Pitfalls gallery, continued
A brief history, continued
Roadmap

§1 · What changes in Chapter 9

Chapter 5's scene returns (two spheres, two point lights, sign-logic shadows, checker floor): the rest stop is built on the point-light base, and §6 notes how Chapters 6–8's integrals stack orthogonally. One change, and it is shading-side only:

Sphere A's normal is perturbed by a height field. $N' = \operatorname{normalize}(N + \beta\,g)$, where $g$ is the chart-space gradient of $h(u,v)$ mapped through the tangent frame. The geometry does not move — a fact with a theorem's weight this chapter.
The pipelines differ in exactly one subroutine. Numeric differentiates h by central differences with step ε (your slider: truncation error ∝ ε² on one end, roundoff ∝ 1/ε on the other — the classic U-curve, live). Symbolic differentiates h analytically: Chapter 5's Cell 8 gradient, composed through Chapter 4's machinery. The difference canvas now measures differentiation error — deterministic, tunable, and zeroable only on one side.
Bump moves no boundary. Silhouettes, seams, shadow curves: unchanged. Chamber flips are zero by construction — and the one view where ridges appear (the kink map) shows loci that Chapter 5 already proved are rays and circles.

§2 · The scene (Chapter-9 assumptions)

Chapter 5's scene; sphere A wears a height field $h(u,v)$ with amplitude β (default 0.35). Three height fields: sine (smooth everywhere — the control), ridges ($|\sin \pi k_u u \cdot \sin \pi k_v v|$ — creased along its zero lines: kinks with algebraic loci), and rings ($\sin 2\pi k_v v$ — latitude bands; isolates the v-direction and the pole). Sphere B, the floor, and both lights are unchanged. Bump is shading-side: silhouettes and shadows use the true geometry (the honest caveat — §7 lists what this simplification costs).

§3 · The math, once, carefully

The tangent frame, exact. With $N = (\sin\varphi\cos\theta, \sin\varphi\sin\theta, \cos\varphi)$ on sphere A, the chart directions are the orthonormal pair $$e_\theta = \frac{(-N_y,\; N_x,\; 0)}{\sin\varphi}, \qquad e_\varphi = \frac{(N_z N_x,\; N_z N_y,\; -\sin^2\varphi)}{\sin\varphi}, \qquad \sin\varphi = \sqrt{1 - N_z^2},$$ derived once (Cell 2), with a pole fallback confessed in §7. The perturbed normal: $$N' = \frac{M}{|M|}, \qquad M = N + \beta\,(h_u\, e_\theta + h_v\, e_\varphi).$$ The gradients. Sine: $h_u = \pi k_u \cos(\pi k_u u)\sin(\pi k_v v)$, $h_v = \pi k_v \sin(\pi k_u u)\cos(\pi k_v v)$. Ridges: $h = |s_1 s_2|$ gives $h_u = \pi k_u \cos(\pi k_u u)\, s_2\, \sigma$, $h_v = \pi k_v s_1 \cos(\pi k_v v)\, \sigma$ with $\sigma = \operatorname{sign}(s_1 s_2)$ — undefined on the zero lines $u = m/k_u$ (rays, by Chapter 5's longitude theorem) and $v = m/k_v$ (circles, by its corollary): the kink loci are algebraic, as advertised. Rings: $h_u = 0$, $h_v = 2\pi k_v \cos(2\pi k_v v)$.

Theorem (Bump moves no boundary). The chamber masks are functions of the geometry $(x, y, \theta)$ alone; $N'$ enters only the shading factors. Hence the chamber partition — and therefore the flip census — is independent of β. ∎ The difference canvas is thus freed to measure one thing only: the gradient.

The finite-difference U-curve. Central differences err by $\frac{\varepsilon^2}{6}h'''$ (truncation) plus $O(\varepsilon_{\text{mach}}/\varepsilon)$ (roundoff). With $h \sim \sin(\pi k_u u)$, $h''' \sim (\pi k_u)^3 \approx 2{,}000$ at $k_u = 8$: truncation dominates until $\varepsilon \sim 10^{-6}$, roundoff drowns everything below $\sim 10^{-9}$ — and at a ridge, the central difference straddles the crease and returns a number that is not a derivative at any order. Cell 2 and Cell 7 plot the U; the simulator hands you the slider.

Three gifts of Chapter 9: (1) an exact derivative doing production work — the symbolic object's first win on both axes (accuracy and cost: one analytic evaluation against four extra function evals; Cell 4 prices it); (2) $\partial N'/\partial\beta$ in closed form — $(I - N'\!N'^{\!\top})\,g/|M|$ — verified to 10⁻¹² (Cell 5), the inverse-rendering thread from Chapter 1's Cell 6 pulled one chapter further; (3) the β→0 regression: the image must return Chapter 5, and does, to machine precision (Cell 6). Counterpoint: bump is a shading fiction. The self-shadowing of the bumps, their occlusion, and their silhouette are all absent — the surface pretends to be rough only to the lights. §7 prices the fiction, and Cell 8 names the honest version (displacement) as the place where boundaries start depending on h — and Chapter 3's machinery must move with them.

§4 · Live simulator

Sphere A's normals ripple; the geometry does not. Numeric differentiates by ε-steps you control; symbolic carries ∂h exactly. The ledger reports max/RMS on sphere A (where the difference lives) — and flips, which this chapter pins at zero by theorem. The bump view paints |∇h|; the kink view paints the ridge loci, where finite differences go to be wrong.

light 1 light 2 color material A material B ambient model sphere B floor

resolution height field display view

initializing…

NUMERIC — gradient by central differences, step ε (your slider)

SYMBOLIC — the exact analytic gradient, composed

|difference| ×10² — pure differentiation error; watch it ride the ε U-curve

numeric: – symbolic: – max |ΔI| on A: – RMS |ΔI| on A: – flips: 0 (theorem)

hover — pixel → world, surface, ∇h from both pipelines, kink flag, and both RGB triples.

Try this: (1) Height field sine, then sweep FD step ε from 1e-1 down: the difference canvas darkens along the truncation slope, bottoms out near 1e-6, and brightens again past 1e-8 — the roundoff cliff, live. The symbolic side never moves. (2) ridges + kink view: the white creases are exactly where the two pipelines part — central differences straddle the crease; the exact gradient knows it is undefined there. (3) rings + magnify the pole: the pinch returns (Chapter 5's theorem), now in the gradient. (4) bump β to 0: both panels return Chapter 5 — the regression, live on a slider.

§5 · The Colab laboratory (Python / sympy)

Cumulative on the Chapter-5 base (the integral chapters stack orthogonally — §6): Cells 1 and 3 replace Chapter 5's; Cells 2, 4–8 are new. Three run paths per cell — 📋 Copy code, ⬇ notebook (File → Upload notebook in Colab), or paste into colab.new.