Duality and Symmetry

One of the striking features of Symmetric Geometry is that it contains duality and symmetry regarding geometrical elements and their relationships.

Euclidean Geometry begins with the definition of a “line.” However, Symmetric Geometry defines only oriented lines, such as north-bound or south-bound Interstate Highway 5. A frame, ((x, y), θ), where (x, y) is a position and θ a direction, represents an oriented line. Symmetric Geometry deals with circles with orientations, clockwise and counterclockwise, but not a circle, as in Euclidean Geometry.

Symmetric Geometry assumes there is an active “agent” (people or sensors) that observes objects. We consider that there exist counterclockwise and clockwise polygons in the world. An agent observes a square pillar; we regard its intersection with a horizontal plane as a CCW square. On the other hand, an agent observes a square room from inside; we regard this as a CW square.

A floor and ceiling in a room are both horizontal planes. An agent observes the floor from the top, while the agent observes the ceiling from the bottom. Thus, the ceiling and floor are planes with opposite orientations, which are conveniently represented by their positions and normal directions in Symmetric Geometry.

Sometimes, it is useful to attribute an orientation to a point, such as a counterclockwise point P+ and a clockwise point P– for a point P. If we make the obstacles A, B, C, D, E in the slalom world in Path Classes and Their Symbolic Representation extremely small, the obstacles converge to points, but the path-class concept with these “oriented points” is still meaningful.

Symmetric Geometry also conveniently describes and computes the abstract and symbolic  concepts of “left” and “right.” The meaning of the left/right concept depends on the relationship between an observer’s viewpoint and the frame of a geometrical element.

An algorithm that tracks the bisector of
left and right featured points
creates this Swan motion.

Symmetric Geometry deals with such duality and symmetry; the world is abundant in duality and symmetry. The naming “Symmetric” refers to these characteristics.