Wegert

Interactive phase portraits of complex rational functions.

Add zeros ○ and poles × on the complex plane. Drag them, pan, and pinch to zoom. The portrait and the displayed formula update from the same factor state.

Android test build: v0.1.50 APK · all releases

The APK is a prerelease signed with the repository's public test key.

Read the picture

For a complex function f(z):

• hue records arg f(z)
• lightness repeats with log10 |f(z)|
• zeros are rings
• poles are crosses

The reusable colour core is code/wegert_color.glsl. Function construction stays upstream of that boundary.

This project follows the phase-portrait approach developed by Elias Wegert. See Visual Complex Functions: An Introduction with Phase Portraits.

More Wegert: Complex Beauties calendars, 2011–2024 · complete 2024 calendar PDF · research profile · visual.wegert.com.

The 2024 calendar credits Elias Wegert, Gunter Semmler, Pamela Gorkin, and Ulrich Daepp.

Controls

• tap ○ or ×, then tap the portrait to add a zero or pole
• drag a marker to move that factor
• drag empty space to pan
• pinch to zoom
• clear removes all zeros and poles
• three-finger tap restores g(z) = (z - 1)(z - 2)(z - 5)
• placing a zero on a pole, or a pole on a zero, cancels one factor

Reference render

The preserved reference animation keeps the roots of

g(z) = (z - 1)(z - 2)(z - 5)

fixed while the codomain is multiplied by exp(i theta).

MP4 · static PNG · R renderer

More rendered examples are in rendered_images/.

Code

Readable source lives in code/.

• wegert_color.glsl — phase/log-modulus to colour
• wegert_function.c — rational-function state
• wegert_scene.c — zeros, poles, and view state
• wegert_portrait_renderer_gles.c — GLES portrait renderer
• wegert_render_frames.c — offscreen RGB24 frame stream
• _/build/ — Android packaging, tests, CI, and release machinery

The root-level C/GLSL names remain entry-point symlinks into code/.

Detailed rendering, gesture, test, emulator, and release notes are in docs/technical-notes.md.

Build

Requirements: Android SDK 36, NDK r29 (29.0.14206865), CMake 3.22.1, JDK 17, Gradle 9.5.1, and Android Gradle Plugin 9.3.1.

cd _/build
gradle :app:assembleDebug

The debug APK is written to:

_/build/app/build/outputs/apk/debug/app-debug.apk

It contains arm64-v8a and armeabi-v7a for devices, plus x86_64 for CI emulation.

Scope

Wegert's reusable boundary is the phase-portrait presentation: colour mapping, coordinates, markers, and ordinary zero/pole interaction.

Analytic Continuation owns whole-plane evolution of f = R exp(q).

Lacunary owns bounded/local continuation, charts, branches, sheets, and monodromy experiments.