The shape appears compliated by the underlying principles and design goals are very simple, and can all be calulated with a pencil and paper.
1) Set your mass flow, at the atmospheric conditions you need the velocity into the compressor cant go above the speed of sound. This sets the minimum diameter of the entry.
2) Set your desired outlet pressure, this is determined by the radius of the disc and the size/efficiency of the volute available. You can work this out from Eulers turbomachinery equation, derived in the year 1752.
You can start with just multiplying the tip speed by the tangential velocity of the air leaving the rotor tip.
3) Whats the maximum acceptable temperature rise ?
The temperature rise through the compressor goes up with the square of the tip speed. This (among other things) allows you to set a ceiling for the shaft speed.
4) Knowing all this, and your basic adabatic equation (whats the temperature of air after you squish it by a certain amount), you can then do the bit thats so simple people often forget about it. Area ruling, the velocity through the compressor wants to be constant, since its density and temperature change, you need to design the flow channels so that as these variables change, the area changes to keep the velocity constant. This keeps turblence low and improves efficiency.
In the Second World War all this was done by hand, and to speed it it, nomograms were made to quickly cross reference the characteristics to check the key dimensions.
There is a lot more to it, but the basic ingredients are all very straightforward.
Here is Bruno Eckerts WW2 German compressor design nomogram, onto which I`ve added how you draw the lines (Had is adiabatic head, which is what the German compressor designers used instead of Pressure Ratio, of "reasons"*)
Photo is a German aeroplane engine compressor, in mid-manufacture stage before the inducer vanes are cold-bent, this is the raw forging. The only really significant advance is the forward lean to the blades, which helps lower tip leakage and also helps prevent blade resonance by buttressing the blades as "beam" elements.
* See my translation of the German engineering professors (Prof. Dr.-Ing Kollmann), book on WW2 compressor design published by the American Society of Mechanical Engineers for more end-boss level geek details. I found his original typewritten manuscript in a house in Stuttgart, which I then translated and published with ASME with much aid from world renowned compressor expert Dr Gülen who really assisted a lot by putting in the mathematical bridging between the WW2 theory and contemporary practise.
centrifugal impellers are truly one of those crazy things that just looks alien too me
every curve on the part is solving something. the blade twist, the backsweep, the fillet blending into the hub, the passage narrowing toward the tip.
and somehow the result is just *chef's kiss*. form is just physics that got resolved.