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i think you will struggle to do what you want with triacs / thyristors.
3ph motor speed is generally regulated by adjusting the frequency of supply through an inverter. using triacs, you would be reducing the voltage and therefore the output torque of your lathe, which is the last thing you want on a lower speed.
your best bet would be a 3ph inverter.
if you want to try triacs, something along the lines of a high power dimmer circuit x3 with a triple gang pot may work
i think that building/designing your own inverter is not that simple
and it is not worth a try because of the costs...
inverter Lenze type 8200 series or ev82 vector series costs
about 200 euro and have a lot of feutures
input = 230vac
output 3 phase
almost all is presettable...
speed,acc, dec,pole-pairs,dc brake...
built in filter, monitoring of current/power/voltage/frekw...
therefore i rater buy one for 200 euros than building one.
To control a three phase asynchronous induction motor you, in a direct mode,( not vector control) must keep constant the Voltage/Frequency ratio as long as possible up to a particular point of frequency (under 15 Hz usually) where it is possible yet to apply overvoltage boosting characteristic which offset decreasing of inductance and so a torque reduction(due to inductance reduction).The secret is to use a device which has three pulse width modulator (8 or better ten bit) and a computational core (more uCs can do this work or you can describe them in VHDL);so if you store in memory a sine look-up table which magnitude of produces the maximun of link voltage you can always modulate this magnitude in function of a frequency set taken from AD or a communication channel such UART or other.