A machining system comprising a machining tool comprising: • at least one real tool for machining a real part; • a plurality of actuators arranged to cause said at least one real tool to move relative to a real blank for machining in order to form the real part; • a plurality of sensors for generating positioning data representative of a current position of the real tool relative to a real machine reference frame; • a memory storing shape correction data for the real tool; and • a physical controller arranged to execute a machining program and to control the actuators in compliance with instructions contained in the machining program and as a function of at least some of the shape correction data for the real tool, so as to move said at least one real tool in the real machine reference frame relative to the real blank in such a manner as to machine the real blank in order to form the real part; the system being characterized in that operator information means are connected to the physical controller in order to receive data representative: • of the current position of the real tool in the real reference frame; and • of the current position of the real blank in the real reference frame; the information means further including memory means for storing a succession of current positions of the real tool in the real reference frame and image display means and, characterized in that the image display means are arranged to generate and display at least one reconstituted image that is reconstituted on the basis of said data representative of the positions of the real tool and of the real blank in the real reference frame, the reconstituted image being representative: • of the shape at a given instant of the real blank being machined; and • of the position at the given instant of the real tool relative to the real blank; the machining system further including means for generating an alarm in the event of a reconstituted theoretical path (Tj) of the real tool penetrating into a predetermined limit zone extending around a virtual surface representative of the real part that it is desired to obtain at the end of machining, the reconstituted theoretical path (Tj) being calculated using said succession of current positions of the real tool in the real reference frame. • at least one real tool for machining a real part; • a plurality of actuators arranged to cause said at least one real tool to move relative to a real blank for machining in order to form the real part; • a plurality of sensors for generating positioning data representative of a current position of the real tool relative to a real machine reference frame; • a memory storing shape correction data for the real tool; and • a physical controller arranged to execute a machining program and to control the actuators in compliance with instructions contained in the machining program and as a function of at least some of the shape correction data for the real tool, so as to move said at least one real tool in the real machine reference frame relative to the real blank in such a manner as to machine the real blank in order to form the real part; • of the current position of the real tool in the real reference frame; and • of the current position of the real blank in the real reference frame; • of the shape at a given instant of the real blank being machined; and • of the position at the given instant of the real tool relative to the real blank;