How do they cut potatoes in a potato processing plant?

Good question, but no easy answer. One thing is sure, most industrial cutting systems are quite amazing. But different products require different cutting machines.
 

Regular French Fries, wedges and steak Fries

Regular french fries and wedges are cut using a water gun. This is the work horse for a regular french fries plant.

The image above shows such a cutting system.

After potatoes have been sorted on size, a big pump pumps water with the potatoes in it at high speed through a static knife block. Each size of potatoes goes to a set-up that is specifically tuned for that size. E.g the set-up shown above has three different lanes.

A special aligning device right in front of the knife block makes sure the potatoes hit the center of the block are cut length-wise.

Speeds are something like 80 km/hour.

The potato strips are separated from the water and the water is returned to the pump, to be used again with new uncut potatoes.
 

Crinkle cut Fries, Thick slices

Since you can only cut straight products with a water gun (put your 3D insight to work and you can probably figure out why), crinkle cut fries are cut with a mechanical cutter.

Such a cutter type is typically used for thick slices as well.
 

Curly Fries

Curly Fries are cut by a rotating cutter specifically designed to cut curly fries.
 

Some people have difficulty believing all french fries are just cut from potatoes and I can especially forgive you if you think this for curly fries. But really they are! The principle is more or less the same as for the apple slicer below.