# Symmetric Geometry

In classical sciences, such as physics, researchers have made advancements through both inductive and deductive approaches. For instance, at the turn of the century, Albert Einstein surprised the world with epoch-making hypotheses led by a deductive approach.

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 particular 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. - Symmetric Geometry deals (1) curves, continuous geometrical entity, and (2) two-dimensional transformations, discrete geometrical entity. These independent theories work together to produce novel theorems.
- Symmetric Geometry can quantitatively evaluate the complexity of curves and motions.
- In our daily car driving, we have to stop and switch directions numerous times. We need to calculate these s
*top frames*. A driverless car has to do the same. For the car, it is much more natural to compute these static frames using Symmetric Geometry rather than to compute them as a part of dynamic vehicle motion-control problems.

Symmetric Geometry consists of several fundamental 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An unpublished 161-page PFD textbook, *An Introduction to Symmetric Geometry*, fully discusses this Geometry. It includes numerous definitions, 20 theorems, 12 lemmas, 3 corollaries, and numerous examples and exercises. It also explains all algorithms in Symmetric 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.