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Re: Interpolation of complex numbers
Posted:
Sep 27, 2001 5:17 PM


In article <3bb38e75.6479659@news.iserv.net>, Joe Geluso <jgeluso@iserv.net> wrote: >On Thu, 27 Sep 2001 11:37:50 0500, "Timothy E. Vaughan" ><tvaughan@bwh.harvard.edu> wrote:
>>Can anyone give me some information on the interpolation of complex numbers?
>Is it meaningful to interpolate complex numbers? I was informed in >this group that complex numbers can't be ordered.
Yes, it is meaningful to interpolate complex numbers. It's true that the complex numbers are not an *ordered field* (not the same as claiming the complex numbers cannot be ordered as a set, since they obviously can). However, interpolation has nothing to do with order.
>Interpolation is a technique to arrive at a value that is 'close' to >the 'right' value by calculating a value 'between' two known values.
It doesn't have to be between, and betweenness is not related to order anyway. For example, say you are given that f(1) = 2 + 3i and f(3) = 4 + i. What is f(2)? If you choose to use a simple linear interpolation, you get f(2) = 3 + 2i, which happens to be on the line segment joining (2,3) and (4,1) in the complex plane and therefore lies "between" those points, even though the complex numbers are not ordered.
Worse yet, you may decide to do a nonlinear interpolation in which it turns out that f(2) is merely something close to 3 + 2i but not actually on the line joining f(1) and f(3). Then you can't say f(2) is "between" f(1) and f(3) except in an approximate sense.
>But if values can't be ordered, then there is no 'between'.
Wrong, as the example above demonstrates.
>A line can be drawn on the complex plane which appears to connect a >progression of values, so it would seem that 'betweenness' is >meaningful in some sense  yet the values supposedly cannot be >ordered.
"Cannot be ordered" is incorrect. The correct statement is that C is not an ordered field.
>Perhaps the shortest distance between two points on the complex plane >(if 'distance' can be measured there  what would its units be?) is >not necessarily a straight line?
What units do you use for measuring distance on the real line?
>The path between two points may not be a line  the path traced by a >rotating phasor, for example. Interpolating between points 180 >degrees apart could be ambiguous (at +90 or 90 degrees from a known >point) if interpolated by phase angle or at the center of the circle >(nowhere near the traced path) if interpolated by components.
You interpolate the real and imaginary parts separately. Trying to interpolate the phase angle doesn't make much sense.
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