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Orthonormality in coordinate systems
I did a little research into the formalities of this, and then of course had to dispose of some paradoxes that brought to mind. Here is how it looks to me now. ************ I will of course paraphrase here to keep the matter of understanding moving forward. A coordinate system may be considered orthonormal if: • Its axes are all orthogonal (for a two or three dimensional system, visualized as physical, that means that they are mutually at right angles) • Its axes all have the same "scale" We can perhaps best understand this if we consider an actual physical coordinate system of three dimensions  for example, as we might use to describe a point in space with respect to some reference point on my optical laboratory bench. Suppose we define the axes thus (the intent being that they in fact all be at right angles to one another in the physical, geometric source. Perhaps X is to the East, Y to the North, and Z up. Then we say that, for all of them, one unit of the numerical coordinates equals one meter of physical space. This is an orthonormal coordinate system. (Coordinate system A). Now suppose we create another coordinate system where X and Y are in fact in a truly horizontal plane (in the directions previously mentioned) by the X axis is perpendicular to a certain tilted surface in my apparatus. This is no longer an orthonormal coordinate system. (Coordinate system B) Or suppose that we keep the original mutuallyorthogonal axes but for X and Y consider one unit to correspond to one foot, but for Z consider one unit to correspond to 2 feet. This is no longer an orthonormal coordinate system. (Coordinate system C) An impractical distinction between the first coordinate system and the other two is this (a matter than is of great importance in the field in which I recently encountered the concept). Euclidean distance The Euclidean distance between two set of values in a coordinate system is an "abstract distance" between the points representing them. If one point is our origin (X,Y,Z=0,0,0), and the other is at A, B, C (this is in abstract units of the axes, not physical feet), then the Euclidean distance between those sets of values is given by: S=sqrt(A^2+B^2+C^2) Now, in an orthonormal coordinate system (in physical space), the Euclidean distance between two sets of numbers is the same as the actual physical, spatial distance between the points represented by those sets of numbers (or is related to it by a constant factor, depending on the actual "scale" we use for all three axes). If the coordinate system is not orthonormal, that is not true. Related coordinate systems Suppose we take our first coordinate system (which we have declared to be orthonormal) and just twist it, intact, to another orientation. This means that two or perhaps all three axes now point in different directions that before (perhaps we have aligned it with some frame in our optical apparatus). This new coordinate system will be orthonormal. (Coordinate system D) As an intuitive test of that, consider two points in our physical apparatus that are exactly 1.000 inch apart (and let's assume that the scale of all our axes is 1.000 unit equals 1.000 inch. Now in our first coordinate system, system A (orthonormal), if we take the ordinate value sets that describe these two points and determine the Euclidean distance between them, we will get 1.000 unit. If, in coordinate system D (also orthonormal), we take the ordinate value sets that describe these same two points and determine the Euclidean distance between them, we will get 1.000 unit (because the actual distance is unchanged, and the "measuring frame" is unchanged, albeit shifted in orientation). A nongeometric coordinate system Of course, we often have coordinate systems that do not relate to the actual physical position of a point, but merely are a way to organize the concept of several attributes of an item. Suppose that, for a population of audio amplifiers, we are concerned with their maximum power output (P), their cost (C), and their mass (M). We can now consider a "threedimensional" coordinate system in which the axes are P, C, and M. We will consider one unit along the P axis to be one watt, one unit along the C axis to be one USD, and one unit along the M axis to be one kilogram. Now, is this an orthonormal coordinate system? The concept is not really defined (it took a couple of minutes for me to get that). For example, what about the notion that the three axes have to be at right angles to one another. Is the axis of Power at right angles to the axis of Cost? That has no meaning. Certainly, if we draw (for use in a book) an oblique representation of this threedimensional coordinate system, we will proceed as if all three axes are mutually at right angles. But that is just a graphic custom. Is the Euclidean distance between the sets of P,C,M values (in abstract form, shorn of their units of watts, USD, and kg) for two amplifiers consistent with the "actual physical distance between their points"? Well, there is no meaning to "the actual distance between their points". So we must recognize that the classification of a coordinate system as orthonormal or not is predicated on that coordinate system existing in a coordinate system in which "actual distance" has meaning. Thus, for coordinate systems that describe the location of a point in physical space (where there is a clear meaning of "actual distance"  we could more it with a tape measure, if it's not too big) we can properly describe possible coordinate systems as orthonormal or nonorthonormal. But when we have a coordinate system that just "comes out of left field" (like our P, C, M system for describing three properties of audio amplifiers) we cannot properly say whether it is orthonormal or not. Another outlook is that "Of course it is orthonormal. We think of the three axes as being mutually at right angles, and who is to say that they are not". I will relate the application of this notion to the work of Bezryadin a little later on. Best regards, Doug 
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