  A tutorial introduction to projective geometric algebra (PGA), a modern,
coordinate-free framework for doing euclidean geometry. PGA features: uniform
representation of points, lines, and planes; robust, parallel-safe join and
meet operations; compact, polymorphic syntax for euclidean formulas and
constructions; a single intuitive sandwich form for isometries; native support
for automatic differentiation; and tight integration of kinematics and rigid
body mechanics. Inclusion of vector, quaternion, dual quaternion, and exterior
algebras as sub-algebras simplifies the learning curve and transition path for
experienced practitioners. On the practical side, it can be efficiently
implemented, while its rich syntax enhances programming productivity. The basic
ideas are introduced in the 2D context; the 3D treatment focus on selected
topics. Advantages to traditional approaches are collected in a table at the
end. The article aims to be a self-contained introduction for practitioners of
euclidean geometry and includes numerous examples, figures, and tables.
