Elementary Solid Geometry and Mensuration

Safe domain and elementary geometry

A classical problem of mechanics involves a projectile fired from a given point with a given velocity whose direction is varied. This results in a family of trajectories whose envelope defines the border of a “safe” domain. In the simple cases of a constant force, harmonic potential, and Kepler or Coulomb motion, the trajectories are conic curves whose envelope in a plane is another conic secti...

Conformal Geometry and Elementary Particles.

Elementary Surprises in Projective Geometry

The classical theorems in projective geometry involve constructions based on points and straight lines. A general feature of these theorems is that a surprising coincidence awaits the reader who makes the construction. A classical example of this is Pappus’s theorem. One starts with 6 points, 3 of which are contained on one line and 3 of which are contained on another. Drawing the additional li...

Elementary Differential Geometry: Curves and Surfaces

First of all, I would like to thank my colleague Lisbeth Fajstrup for many discussion about these notes and for many of the drawings in this text. The VIDIGEO-project has provided interactive and dynamical software for illustrations of curves and associated objects which are used in the first chapters of these notes; the interactive versions are accessible at www.math.aau.dk/ ˜ raussen/VIDIGEO/...

Elementary Complexity and Geometry of Interaction

We introduce a geometry of interaction model given by an algebra of clauses equipped with resolution (following [Gir95]) into which proofs of Elementary Linear Logic can be interpreted. In order to extend geometry of interaction computation (the so called execution formula) to a wider class of programs in the algebra than just those coming from proofs, we define a variant of execution (called w...