Interactive Raylib visualizer for building intuition about tangent vectors, covectors, and 2-forms on a curved surface.

The program renders a moving point on a manifold, the tangent plane at that point, and several linked views that explain how:

• tangent vectors live in T_pM
• a covector acts as a linear measurement on the tangent plane
• a 2-form returns signed oriented area

There are three interactive scenes:

1. Tangent vectors: shows a point moving along the manifold, the attached tangent plane, and tangent vectors whose evaluated scalar output is shown by color.
2. Cotangent space: shows a covector as a family of equal-value lines on the tangent plane, plus probe vectors and a flat inset view.
3. 2-forms: shows two tangent vectors spanning a parallelogram whose signed area is measured by a 2-form.

• A C compiler such as cc or clang
• raylib
• pkg-config

The current Makefile obtains both compiler and linker flags from pkg-config, so raylib needs to be installed in a way that pkg-config --cflags raylib and pkg-config --libs raylib succeed.

make
make run

To remove the built binary:

make clean

• 1, 2, 3: switch between the three scenes
• q: quit
• space: pause or resume the animation
• r: reset the app state
• left mouse drag: orbit the camera
• mouse wheel: zoom
• click scene tabs in the top bar: switch scenes
• click AUTO, W1, W2, W3 in the top bar: change the covector family used in the cotangent scene
• drag the Manifold color slider: adjust the surface tint
• drag the Curve position slider: scrub the point along the path
• hover vector tips or highlighted geometry: inspect the currently measured value

• The window is resizable and uses MSAA when available.
• The executable built by make is named differential_1_forms.
• The window title in the program is currently Differential 1-Form Visualizer, even though the app also includes a 2-form scene.