Monitoring method for the operation of a machine having a plurality of axes by means of which elements (2 to 4) of the machine are moved relative to one another in a position-controlled fashion, - wherein a monitoring facility receives a number of groups (p*, p) of position values of the axes, wherein the groups (p*, p) of position values each specify the position of the elements (2 to 4) of the machine relative to one another, - wherein on the basis of modelling of the elements (2 to 4) of the machine, modelling of the kinematics of the elements (2 to 4) of the machine for the groups (p*, p) of position values of the number of groups (p*, p) of position values in each case, the monitoring facility determines which element (2 to 4) of the machine respectively takes up which surface and/or which volume in a working space of the machine, - wherein the monitoring facility checks on the basis of the determined surfaces and/or volumes whether there is a risk of elements (2 to 4) of the machine colliding with one another, and - wherein the monitoring facility reacts in accordance with the result of the check, characterised in that - the monitoring facility models at least part of the surface of at least one of the elements (2 to 4) of the machine by means of a number of two-dimensional splines which are defined by respective nodes and respective checkpoints, if applicable plus weights, - on the basis of the checkpoints of the respective spline for the respective section, the monitoring facility determines a respective envelope which encloses the at least one element (2 to 4) in the respective section, - straight connecting lines of the checkpoints of the respective spline are boundary lines of surfaces of the respective envelopes and - the monitoring facility uses the respective envelope as a surface which is taken up by the at least one element (2 to 4) of the machine in the respective section. - wherein a monitoring facility receives a number of groups (p*, p) of position values of the axes, wherein the groups (p*, p) of position values each specify the position of the elements (2 to 4) of the machine relative to one another, - wherein on the basis of modelling of the elements (2 to 4) of the machine, modelling of the kinematics of the elements (2 to 4) of the machine for the groups (p*, p) of position values of the number of groups (p*, p) of position values in each case, the monitoring facility determines which element (2 to 4) of the machine respectively takes up which surface and/or which volume in a working space of the machine, - wherein the monitoring facility checks on the basis of the determined surfaces and/or volumes whether there is a risk of elements (2 to 4) of the machine colliding with one another, and - wherein the monitoring facility reacts in accordance with the result of the check, characterised in that - the monitoring facility models at least part of the surface of at least one of the elements (2 to 4) of the machine by means of a number of two-dimensional splines which are defined by respective nodes and respective checkpoints, if applicable plus weights, - on the basis of the checkpoints of the respective spline for the respective section, the monitoring facility determines a respective envelope which encloses the at least one element (2 to 4) in the respective section, - straight connecting lines of the checkpoints of the respective spline are boundary lines of surfaces of the respective envelopes and - the monitoring facility uses the respective envelope as a surface which is taken up by the at least one element (2 to 4) of the machine in the respective section.