Hubert Holin wrote:
It is interesting if, for instance, you have to track an object along an orbit for a very long time. Using the canonical ("cartesian") representation usually entails transcendental functions (say sine and cosine), for which it is very hard to guaranty the accuracy (of a given, one-size-fits-all, implementation) for big values of the argument. If you integrate numerically, you all the more want to stay in the polar domain (unless we cross the singularity, but we are not modeling missiles, are we ;-) ? ).
Excuse me, but this does not convince me. First I cannot see easily how complex numbers enter into the picture (but this might be due to my lack of knowledge in that domain), and then you need to use sine and cosine only ever if you are using polar _and_ cartesian coordinates at the _same_ time. Even if so, you could formulate your algorithm in the cartesian domain and need sine and cosine only at the last stage to transform the whole trace back to polar representation (if ever needed). But I admit these arguments are all very vaguely and don't make sense as long as not refering to a concrete algorithm. The situation is similar to the concept of number types in general: You may formulate an algorithm in terms of decimals, but do not yet specify whether you will use short int or long. You will specify the type when refering to a certain algorithm. You usually will want to be able to trade off accuracy for size and speed and choose the apropriate type. I would expect from a complex type then that is should be possible to specify e.g.: polar comlex double myvar; when wanting a magnitude phase representation, or complex double myvar; when needing real imag representation. Of course it should be easy to convert between these two representations similar as beeing able to convert between int and long int. Again I am still thinking that even this will not even justify the need for a dedicated polar representation of complex numbers. Take e.g. the FFT algorithm. Altough you have complex representation at the core, you won't gain anything by polar representation because you need addition and multiplication (as is common in any linear algorithm). Rather than optimize the representation you need to optimze the so called "butterfly operation" which is a combination of multiplication and addition, to speed things up. (This reminds me more to what I've seen in ublas when doing late evaluation of composite terms.) Even complex linear equation systems are usually solved by representing them as a (larger) real system. (Polar representation again does not help much.) I am somewhat knowledgeable in the domain of signal processing, and as far as I can tell one will not really benefit much from a polar representation in this field at all. Having said this I do not think it is of no use to think about having such a type. I'd rather suggest pointing out first some concrete examples or algorithms that really benefit from it. Roland BTW.: Wouldn't tracking of an object in an orbit rather require rotations (i.e. quaternions) instead of complex numbers?