In the machining process of conventional lathes, the turning of circular arcs is typically performed manually using experience-based techniques or with standardized turning tools (as shown in Figure 1). The former method often leads to difficulties in ensuring the accuracy of the workpiece, while the latter requires high-precision tools that must match the shape of the part exactly. This increases tooling costs, especially for low-volume and multi-standard production. With the rapid development of numerical control (NC) technology, CNC lathes have become a more efficient solution for machining arc-shaped contours on shaft parts. In this case, a standard external turning tool is used (as shown in Figure 2), which can produce accurate arc profiles. However, there are limitations—these tools can only machine arcs in the first two quadrants. If a cross-quadrant arc is required, interference may occur due to the tool's relief angle, leading to rejection of the part (as illustrated in Figure 3). To address these challenges, we analyzed the issue and proposed using a specialized arc turning tool (as shown in Figure 1) on CNC lathes to machine arc profiles. Although the resulting surfaces look aesthetically pleasing, they still face several issues, such as dimensional inaccuracies and tool interference. This article aims to explore programmatic solutions to resolve the interference problem caused by arc turning tools. Turning Trajectory Analysis Let’s take the part shown in Figure 4 as an example to analyze the turning trajectory of the arc. Normally, manual programming would generate code similar to the following for arc machining: N100 M06 T0101; N110 G00 X22 Z-16; N120 G01 X20 Z-16 F200 S500; N130 G02 X20 Z-34 I12 K-9; N140 G01 X22 Z-34; The surface appears smooth and matches the desired arc visually (as seen in Figure 5). However, when measured with a caliper, it becomes clear that the actual dimensions do not meet the design specifications. Specifically, the distance between the start and end points along the Z-axis is larger than expected. Upon investigation, it was found that the arc turning tool interfered with the workpiece surface, causing inaccuracies in the final part. To solve this, we replaced the arc turning tool with a sharp-edged tool (as shown in Figure 5), which improved the accuracy and met the design requirements. While sharp tools can be effective under certain conditions, such as when tool strength and accuracy allow, they also come with limitations. For instance, their effectiveness depends on the angle of the cutting edge, and not all arcs can be machined successfully with them. As shown in Figure 6, consider a scenario where point A is the starting point, point B is the endpoint, and point C is the highest point of the arc. The sharp tool follows the path ACB. At any given point D on this arc, when the tool touches the arc at D, a line OD is drawn, and a tangent L2 is created at that point. A vertical line L3 is then drawn from the Z-axis, forming an angle a with L2. The slope of the arc varies along its path, and as the tool moves from A to C, the angle a increases. We define the angle between the back of the sharp tool and the Z-axis as b. Once the tool is mounted, the value of P remains fixed, and the relationship between angles a and b determines whether the tool will interfere with the arc surface. As shown in Figure 6, when a > b, the tool does not interfere. However, if a < b, the tool may collide with the arc surface, causing damage or inaccuracy. Understanding this relationship is essential for optimizing the machining process and avoiding tool interference during arc turning.

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