All animations · In the chapter · Open full MP4
Animation 13
A function value is translated by the function's slope
symmetry-translation-tangent-zoom.mp4
These are sparse samples decoded from the current MP4, not newly rendered illustrations. They can support checks of the sampled states and labels, but cannot establish continuous motion, timing, transitions, or the absence of problems between samples. Use the full MP4 when judging those properties.
1280 × 720 · 24 fps · 15 s · 360 frames · 12 samples
MP4 SHA-256 a4db667ace47c5f9a12f280dbb6ac8ef4002e3511aedf2c5824cc67cc5fa9b40
Frame manifest · Context markdown · Download complete review packet
Chapter context
Section: Generator of Translations. Excerpts are verbatim; line numbers refer to the included chapter markdown.
Caption
A function value is translated by the function's slope
Image description
Tangent approximation under local zoom contact sheet
Before the animation
Chapter lines 574–574
We have a function representation. Now we want to find an infinitesimal generator that acts on functions to produce a translation. We can do so by considering what happens to a function's value under "tiny" displacements. In that case, the new value $f(x-a)$ is close to the old value $f(x)$, and correction is given, in the infinitesimal limit, by the slope at $x$. This is the same idea that any curve becomes flat when zoomed in sufficiently:
After the animation
Chapter lines 582–582
That is the visual version of the first-order Taylor expansion that relates a function's derivatives to its value under translation:
Chapter lines 584–592
Generation source
Run the main script without arguments. It draws the translation-and-tangent-zoom frames with Pillow and encodes them with FFmpeg. It has no local Python imports or input movie.
Mapping evidence and limits
render() sets the exact symmetry-translation-tangent-zoom output stem. Constants specify 1280 x 720, 24 fps, 360 frames. No matching validation JSON is present; attribution is by source/output name and pipeline, not a recorded movie hash.
Source SHA-256 values identify the exact downloadable bytes in this packet. The source mapping and recorded checks explain the likely generation pipeline; they do not prove that these exact source bytes produced the movie. A GitHub link pinned to a commit is provided only when the delivered source bytes exactly match that path at the build's Git HEAD.
- scripts/generate_symmetry_translation_tangent_zoom_animation.py · GitHub at 974f9de35254
SHA-256 3ed12cd6638ecf0dbef49b37d1613cacc0f4dbf94abea8feb907f9efc83c6679
Decoded contact sheet
Extraction method: Twelve evenly spaced decoded frame indices, including first and last. Native-resolution JPEGs from the encoded MP4; timestamps read from FFmpeg showinfo. No source rerendering. Frame indices are zero-based. Sparse samples do not establish continuous motion or capture every transition.. Frame indices are zero-based.
Full-size sampled frames
