Lathe Tool Nose Radius Compensation
Tool nose radius compensation provides a way to perform precise cutting with a tool that doesn't end in a perfect point. Since no tool has a perfect point, the sharpness of the point be expressed in the radius of the tool nose (the smaller the radius, the sharper the point).
Here is an illustration of a realistic cutting tool tip. The tool nose radius is exaggerated to better illustrate the idea that the theoretical tip (commanded) position (illustrated with a red +) is not in the same place as the actual cutting edge of the tip (highlighted with purple).
Suppose you wish to cut this profile (with a tool tip of type 3):
If the cutter had a perfect point, then the commanded path would be precisely along the desired geometry path.
For a real tool, however, the nose of the cutting tool is some distance away from the point on the desired geometry. If we commanded positions along the path of the desired geometry, we would not achieve the desired geometry. The actual geometry achieved is illustrated by the blue path. Compare that with the commanded positions, illustrated with red + marks.
A real tool will not produce the desired geometry without some compensation. A compensated path (illustrated in red) would put the rounded edge of the tool nose at the correct position to produce the desired geometry (illustrated in black).