The idea behind the coordinate system is to use the minimal number of axes to define points on a plane, in a space, and so on. It starts with a line. To define a point on a line, you specify a value on one of two axes (one axis for one direction and another axis for the opposite direction). With this, I got rid of the negatives. Only one bit of information is necessary to pick which axis you use.
To define a point on a plane, you specify two values on each of two axes out of a possible three. This contrasts with the Cartesian coordinate system, where you would pick two axes out of a possible of four axes. Less information is necessary to specify which axes you use with my coordinate system.
If you use my coordinate system to store points in a circular or spherical space, you'll notice that there is some wasted space. The same happens with the Cartesian coordinate system. But, there is less wasted space with my coordinate system. So, it seems like if you wanted a computer to model a space, it would be most efficient to use my coordinate system. Assuming that all axes are of the same length and that the space that is used is circular or spherical, I calculated that 90.7% of the area with my planar coordinate system is usable, whereas 78.5% of the planar Cartesian system is usable. Similarly, I calculated that 69.8% of the space with my spatial coordinate system is usable, and only 52.4% is usable with the corresponding Cartesian system.
One thing that I just noticed yesterday is that the number of points in my planar coordinate system is less than in the planar Cartesian system. That was true, but not a problem. The reason it has fewer points is that it is more efficient in using them.
The other thing that I thought yesterday was that my coordinate system was usually less efficient at specifying the regions in which points lie when modeled by a computer. This was wrong. The two-dimensional variety of my coordinate system has three regions, or uses two axes out of three. The two-dimensional variety of the Cartesian coordinate system has four regions, or uses two axes out of four. Three regions mean one base-three digit, and four regions mean one base-four digit, or two bits. Two bits can also represent one base-three digit. In this case, it seems apparent that there isn't much (or any?) savings of this kind by using my coordinate system.
The three-dimensional varieties are different, though. The three-dimensional variety of my coordinate system has four regions, which mean one base-four digit, or two bits. The three-dimensional variety of the Cartesian system has eight regions, which mean one base-eight digit, or three bits. Here, we start to see what may be some improvement with using my coordinate system. I'm not certain that a reduction in the number of regions results in greater efficiency. It seems that a coordinate system with fewer regions would have fewer points. It should be pretty easy to figure out if my coordinate system would be more efficient in this way. I haven't yet done it though.
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