Some demonstrations for algebraic surfaces

Better documentation will be written later.

There are currently two scripts here:

Projections of surfaces, and cusps of the branch curve

This script demonstrates how generic projections of smooth surfaces from the projective space P³ give cusp points on a branch curve (Arnold et al¹ call this singularity a "Whitney pleat", paragraph 1.6, page 14 )

Pinch points of algebraic surfaces

This script visualizes the pinch point of an algebraic surface (which is locally formally isomorphic to the Whitney umbrella surface)(Wikipedia: Pinch point, Wikipedia:Whitney umbrella; Griffiths and Harris², Chapter 4, paragraph 6, "Ordinary Singularities of Surfaces"; Arnold et al¹-- Part I, paragraph 1, page 24.)

References

The scripts are written in the surf programming language

Sample output on YouTube

• Cusp formation, semi-transparent version
• Cusp formation, non-transparent version
• Whitney umbrella / pinch point of a surface, animated

Some previews

Formation of a cusp, one shot:

Formation of a cusp, animation:

(works in some browsers)

Pinch point, one shot:

Pinch point, animation:

(works in some browsers)

Usage

For the formation of a cusp, (semi-transparent variant):

      $ cd formation-of-a-cusp-on-a-branch-curve/transparent
      $ surf-alggeo-nox transparent.surf 
      $ cd frames
      $ ./animate-with-ffmpeg-to-mp4.sh
      $ mv -v animated.mp4  whitney-umbrella.mp4

For the pinch point / Whitney umbrella:

      $ surf-alggeo-nox whitney-umbrella.surf
      $ cd frames
      $ ./animate-with-ffmpeg-to-mp4.sh
      $ mv -v animated.mp4  whitney-umbrella.mp4

Requirements

• surf

• ffmpeg (for creating animations)

**Installing surf and ffmpeg **

In the Ubuntu derivatives,

   # apt install -y  surf-alggeo-nox  
   # apt install -y  ffmpeg

Footnotes

1. Vladimir Arnold et al,
   Singularities of Differ entiable Maps, Volume 1,
   Classification of Critical Points, Caustics and Wave Fronts, 2012,
   point of view of Singularity theory. ↩ ↩²

2. Phillip Griffiths, Joseph Harris, "Principles of Algebraic Geometry",
   Chapter 4, paragraph 6, "Ordinary Singularities of Surfaces",
   page 617 (2011 ed.) --
   point of view of algebraic geometry; ↩