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Symmetric Geometry

Why do we need a new geometry, while we have Euclidean geometry, which has a history of 3,000 years? Actually the main purpose of Euclidean geometry is to analyze properties of drawings inked on a sheet of paper. Geometrical elements are mostly static. The viewpoint of an observer of the drawings is mostly stationary.

On the other hand, a mobile robot senses and analyzes dynamic objects. Furthermore, the viewpoint of the robot changes dynamically. In these dynamic situations, the robot must solve problems in real time. A new, made-to-order geometry, should be delivered for these new requirements.

The major differences between Symmetric Geometry (SG) and Euclidean Geometry (EG) are as follows:

1. SG has no lines; only directed lines. A directed line is represented by the X axis of a 2D transformation, ((x, y), θ).
2. SG has no circles; only counterclockwise (ccw) and clockwise (cw) circles. If a circle is ccw, its radius is positive, otherwise negative. Then the equation, curvature = 1/radius, makes sense.
3. Any element in SG could be a variable of time.
4. The two-dimensional transformation theory is a core of SG.
5. Differential equations in SG have an independent variable of arc-length s, not of time t.
6. SG does not need the concepts “line,” “triangle,” or “parallel,” which are crucial in EG.

SG is totally imbedded into the MotionMind (MM) engine. Each geometrical concept in SG is translated into a Java class in MM. Each element in SG is translated into an instance of an MM class. Each function in SG is translated into a method in an MM class. In short, SG and MM are mirror images of one another.

The most important core of Symmetric Geometry is the theory of Motion Planning, which is discussed in the previous page. More frankly, one purpose of formulating Symmetric Geometry is creatinging the better-than-human motion-planning algorithms, which can also be applied to advanced automobile driving.

The sophisticated simulation of "Circle Ring" with n=12 is given here. Only Symmetric Geometry can make the complex algorithms materialized.

This simulation works with Java. If this browser does not have it, please visit java.com to download.

 

 
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