# Symmetric Geometry

In classical sciences, such as physics, researchers have made advancements through both inductive and deductive approaches. At the turn of the century, led by Albert Einstein and other physicists, modern physics has thus accomplished revolutionary progress through both approaches.

On the other hand, in the driverless-car field, inductive approaches have dominated. *Grand Challenges* held by DARPA in 2004, 2005, and 2007 inspired us with the current standard testing method of cruising cars in deserts, highways, and streets. However, driverless science is a vast, unexplored territory, and using only the inductive approaches may reveal only limited aspects of the area. From now on, Symmetric Geometry plays an important role, as the mathematical foundation for the driverless car industry.

*Symmetric Geometry* has helped MotionLab describe and define all vehicle motion-creation problems and has found solutions to the problems. Because of the explicit goal, the Geometry is distinct from Euclidean Geometry. The following is only a small set of issues, where the Geometry was of direct help:

- Symmetric Geometry gives the best estimates of the vehicle’s frame (position and direction) at every motion sampling cycle.
- Symmetric Geometry found all of the
*Atomic Motion*algorithms. - In our daily car driving, we have to stop the car and switch moving directions (forward or backward) numerous times. A three-point turn is a typical example, where we need to determine two stop frames (position and direction). A driverless car has to do the same. For the car, it is much better to compute these stop frames geometrically using Symmetric Geometry so that the car may obtain the globally least-cost motion. That is one of the advantages a driverless car possesses. A human driver may, following an inductive approach, start moving forward using the maximum curvature and stops at a frame just before hitting an obstacle and continues. In this way, he or she cannot expect a globally optimal motion-planning.

Symmetric Geometry consists of several theories, including:

- A non-abelian, two-dimensional transformation group theory
- A theory of static placement and digital motion creation of two-dimensional rigid bodies (vehicle kinematics)
- Theory of Curves
- Differential Geometry
- The Theory of Voronoi Diagrams and Boundaries

An unpublished 156-page textbook, *An Introduction to Symmetric Geometry*, fully describes this Geometry. It includes numerous definitions, 20 theorems, 12 lemmas, 3 corollaries, and numerous examples and exercises. It also explains all algorithms in the Geometry and Atomic Motions in detail.

A set of object-oriented classes implements all of the data structures and methods (functions) of Symmetric Geometry, Atomic Motions, and Motion Abstractions.