The word orthogonal means pependicular. The cartesian coordinate system is orthogonal as are the cylindrical, spherical, and polar coordinate systems. Polysign numbers form a nonorthogonal geometry.

In conversations about nonorthogonal representations I've run into some repeated misunderstandings which I believe spring form the standardized usage of orthogonal representations. We have been trained on the usage of perpendicular measurement but nonorthogonal treatment requires one to use parallels and in actuality the parallel method works in the orthogonal representation as well. This simple point goes beneath cognition since it has become habit.

One of the three graphs above is not appropriate for clean mathematical work. It is the middle graph, which holds attachment to perpendiculars rather than following the natural flow of unit vectors and their meaning.Perhaps the most concise reasoning is in terms of unit vectors. When we instantiate a coordinate such as

( 1.1, 2.2, 3.4 )

in a space ( i, j, k ) we see the correspondence of 1.1 to i, 2.2 to j, and 3.4 to k. This is appropriate to both orthogonal and nonorthogonal representations. While for a cartesian system it will be true that the position denoted will be found 1.1 units perpendicular to j and k it is also true that the position will be found 1.1 units along i. This latter consideration is the one we should focus on since it is the general form that representation ( i, j, k ) originally intended.

Additionally in terms of quadrants the insistence on usage of perpendicular measures leads to inconsistent results whereas the usage of parallel measures yields perfect correspondence.