1. In a system for controlling a path of a multi-axis machine by a mathematical model representing a curve or surface containing the path, a method of controlling the machine comprising the steps of: planning the path of the machine in parametric space using the mathematical model by calculating path data points in proper density to minimize a need of interpolation between points, by (a) selecting a time interval for computation of each point in the path, (b) selecting a desired machine path velocity, (c) for each point, determining a ratio of path velocity in parametric space to path velocity in machine coordinate space, (d) determining parametric space velocity from the ratio and the desired path velocity, and (e) calculating a point spacing from the parametric space velocity and the time interval to define a new point; translating each point to the coordinate space of the machine; and operating the machine along the path defined by the data points.